Andrián Pertout

Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition

Volume 1

Submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy

Produced on acid-free paper

Faculty of Music The University of Melbourne

March, 2007

Abstract

Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition encompasses the work undertaken by Lou Harrison (widely regarded as one of America’s most influential and original composers) with regards to just intonation, and tuning and scale systems from around the globe – also taking into account the influential work of Alain Daniélou (Introduction to the Study of Musical Scales), Harry Partch (Genesis of a Music), and Ben Johnston (Scalar Order as a Compositional Resource). The essence of the project being to reveal the compositional applications of a selection of Persian, Indonesian, and Japanese musical scales utilized in three very distinct systems: theory versus performance practice and the ‘Scale of Fifths’, or cyclic division of the octave; the equally-tempered division of the octave; and the ‘Scale of Proportions’, or harmonic division of the octave championed by Harrison, among others – outlining their theoretical and aesthetic rationale, as well as their historical foundations. The project begins with the creation of three new microtonal works tailored to address some of the compositional issues of each system, and ending with an articulated exposition; obtained via the investigation of written sources, disclosure of compositional technique, mathematical analysis of relevant tuning systems, spectrum analysis of recordings, and face-to-face discussions with relevant key figures.

THE UNIVERSITY OF MELBOURNE Faculty of Music

TO WHOM IT MAY CONCERN

This is to certify that (i) the thesis comprises only my original work towards the PhD except where indicated in the Preface*, (ii) due acknowledgement has been made in the text to all other material used, (iii) the thesis is less than 80,000 words in length, exclusive of tables, maps, bibliographies and appendices or the thesis is [number of words] as approved by the RHD Committee.

Signature:

Name in Full: Andrián Pertout

Date: 2 March, 2007

Dedicated to my father, the late Aleksander Herman Pertout (b. Slovenia, 1926; d. Australia, 2000)

Acknowledgements

A special thanks to the supervisors:

Professor Brenton Broadstock (Coordinator of Composition, Faculty of Music, University of Melbourne) and Associate Professor Neil McLachlan (School of Behavioural Science, Faculty of Medicine, Dentistry and Health Sciences, University of Melbourne). Brenton Broadstock should be especially thanked for being an inspirational force not only during the last four years of the PhD candidature, but throughout the last ten years of my composition studies at the University of Melbourne. His encouragement, support, and direction have exceeded well beyond his duties as supervisor and composition teacher, and consequently remain forever grateful.

Professor Andrew Schultz (Dean of the Faculty of Creative Arts, University of Wollongong, NSW, Australia) also deserving a mention with regards to L’assaut sur la raison for Symphony Orchestra (2003) and Bénédiction d’un conquérant for Symphony Orchestra (2004), which were especially composed for ACOF 2003 and 2004 (Australian Composers’ Orchestral Forum – Composition workshops with Brenton Broadstock, Andrew Schultz, and the Tasmanian Symphony Orchestra).

A special thanks also to Dr. Julian Yu who was the official mentor for the 2003 and 2004 ACOF project.

A special thanks to the following people for their direct assistance to the composition folio:

Stephen Adams (Presenter, ABC Classic FM) for producing an excellent program featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) on his ABC Classic FM radio program.

Susan Batten (Presenter, 3MBS FM) for producing two excellent programs featuring L’assaut sur la raison for Symphony Orchestra (2003), Navigating the Labyrinth for String Orchestra (2002), and Aristotle’s Rhetoric, Suite for Orchestra (2001-02, Rev. 2005) – together with an interview with the composer – on her 3MBS FM Radio ‘Music in Melbourne’ program, in celebration of the Betty Amsden Award – 2005 3MBS FM National Composer Awards.

APRA (Australasian Performing Right Association) for recognizing L’assaut sur la raison for Symphony Orchestra (2003) with the APRA Encouragement Award – 2004 3MBS FM National Composer Awards.

÷××× Acknowledgements

Andrew Blackburn (Artistic Director, 2007 Melbourne Town Hall Organ Project), Jean Penny and the Melbourne City Council for commissioning Symétrie intégrante for Flute, Organ and Electronics (2005-06) for the upcoming 2007 Melbourne Town Hall Organ Project, Melbourne, Australia.

Enmanuel Blanco (Executive Director, Festival Internacional de Música Electroacústica) for selecting Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to be performed at the XI Festival Internacional de Música Electroacústica ‘Primavera en la Habana’ 2006, 6-12 March, 2006, Habana, Cuba.

Associate Professor Jack Body (Artistic Director, 2007 Asia Pacific Festival, 26th Asian Composers League Festival & Conference, and Associate Professor of Composition, New Zealand School of Music, Victoria University, Wellington, New Zealand) and the festival organizers for selecting Àzàdeh for Santär and Tape (2004, Rev. 2005) to be performed at the 2007 Asia Pacific Festival (26th Asian Composers League Festival & Conference), 8-16 February, 2007, Wellington, New Zealand. Also, for selecting the conference paper Theory Versus Performance Practice: Àzàdeh for Santär and Tape to be presented at the 2007 Asia Pacific Festival ‘Tradition/Transformation’ Conference.

Warren Burt (Wollongong, NSW, Australia) for his generous support and contribution to the direction of the PhD research, and especially with regards to Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005).

Ao Changqun (Organizing Committee Chairman, 2005 Second Sun River Student New Composition Competition, and President, Sichuan Conservatory of Music, Chengdu, People’s Republic of China) for recognizing La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) with the Third Prize in the 2005 Second Sun River Student New Composition Competition (Chengdu, People’s Republic of China).

Phyllis Chen for requesting a Toy Piano and Tape arrangement of Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005), and for her incredible talent, evident in her virtuosic interpretations of the work in Bloomington, Indiana and Chicago, Illinois, USA, as well as at the 2007 International Gaudeamus Interpreters Competition & Chamber Music Week in Amsterdam, The Netherlands.

David Claman (Assistant Professor, Music Department, College of the Holy Cross, Worcester, Massachusetts, USA) and Matt Malsky (Associate Professor of Music, Department of Visual and Performing Arts, Clark University, Worcester, Massachusetts, USA) for selecting Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to be part of the Extensible Toy Piano Project, 5-6 November, 2005, Department of Visual and Performing Arts, Clark University, Worcester, Massachusetts, USA; and the Acknowledgements ×Ø

Extensible Toy Piano Festival, 4 March, 2007, Performing Arts Center, Department of Music, State University of New York, Albany, New York, USA.

Barry Cockroft (tenor saxophone) and Adam Pinto (pianoforte) for commissioning and performing Digressioni modali for Tenor Saxophone and Pianoforte (2003) at the Melbourne International Festival of Single Reeds, 26-29 March, 2005, Victorian College of the Arts, Southbank, Melbourne, Australia. Also, for recording the work for the ‘rompduo’ Crazy Logic CD release. Barry Cockroft (tenor saxophone) and Marc Ryser (pianoforte) for performing the work at The Banff Centre, Banff, Alberta, Canada, and finally Barry Cockroft (Reed Music) for publishing the work with Reed Music.

Professor Barry Conyngham (former Emeritus Professor of the University of Wollongong and Southern Cross University, Lismore, NSW, Australia) for his compositional direction during his residency at the University of Melbourne in 2005.

David Collins (Technical Officer, Faculty of Music, University of Melbourne) for technical assistance throughout the PhD candidature, as well as invaluable advice with regards to sound diffusion concepts.

David B. Doty (Author of The Just Intonation Primer, Founder of the Just Intonation Network, and Editor of the Network’s Journal, 1/1, San Francisco, California, USA) for making time for me during my 2004 visit to San Francisco, California, USA, and for his compositional guidance with regards to just intonation concepts.

Ensamble Contemporáneo (Aliocha Solovera [artistic director], Cristián Gonzáles [flute], Dante Burotto [bass clarinet], Alexandros Jusakos [pianoforte], Davor Miric [violin], and Celso López [violoncello]) for performing La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) at the XV Festival de Música Contemporánea Chilena (15th Chilean Festival of Contemporary Music), 21-27 November, 2003, Santiago, Chile.

Ivano Ercole (Presenter, Rete Italia) for producing an excellent program featuring L’assaut sur la raison for Symphony Orchestra (2003), Navigating the Labyrinth for String Orchestra (2002), Gèrëémeler for Amplified Èrhú, Sampled Harmonium, Cajón and Bombo (2001), Bénédiction d’un conquérant for Symphony Orchestra (2004), An Honourable Silence for Solo Santär (2001), Renascence for Violin, Violoncello, Piano and Percussion (2001, Rev. 2006), and Seeds of Passion for Amplified Violoncello (1999) – together with an interview with the composer – on his Rete Italia radio program.

Ø Acknowledgements

The Ónix Ensamble (Alejandro Escuer [flute], Fernando Domínguez [clarinet], Abel Romero [violin], Edgardo Espinosa [violoncello], and Krisztina Deli [pianoforte]) for selecting La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to be performed at the XXIX Foro Internacional de Música Nueva (29th International Forum of New Music), “Manuel Enríquez”, 2007, May-June, 2007, México City, México.

Isabel Ettenauer (St. Poelten, Austria) and Goska Isphording (Eindhoven, The Netherlands) for inspiring the arrangement of Exposiciones for Toy Piano and Spinet (2005), and for performing the work at the 2006 BMIC Cutting Edge Series, London, UK, and at Axes/Jazzpower, Eindhoven, The Netherlands.

Rodolfo Fischer (Conductor, Basel, Switzerland) for selecting Bénédiction d’un conquérant for Symphony Orchestra (2004) to be performed by the Orquestra Petrobras Sinfônica at the Theatro Municipal do Rio de Janeiro, in Rio de Janeiro, Brazil as part of the Orquestra Petrobras Sinfônica ‘Série Ouro Negro’ 2006 concert series, and also for his excellent direction during the rehearsals and final performance.

Robert Franz (Associate Conductor, Louisville Orchestra, Louisville, Kentucky, USA) and the Louisville Orchestra for recognizing L’assaut sur la raison for Symphony Orchestra (2003) as the winner of the First Prize in the 2004 ISU Contemporary Music Festival/Louisville Orchestra Composition Competition, and also for presenting a memorable performance of the work at the Indiana State University 38th Annual Contemporary Music Festival, ‘Plugged In: Music With an Electric Edge’, November 10-12, 2004, Terre Haute, Indiana, USA. Indiana State University for sponsoring the award, and providing an opportunity to conduct a lecture at the festival.

Professor Don Freund (Professor of Music Composition, Indiana University School of Music, Bloomington, Indiana, USA) and Sandra Freund for their hospitality and enormous generosity during my weekend stay in 2004 with the Freunds in Bloomington, Indiana, USA. A further warm thanks to Don Freund for his contribution to the development of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04).

Kyle Gann (Associate Professor of Music, Faculty, Bard College, Annandale-on-Hudson, New York, USA) for his support during my 2004 visit to Bard College (Annandale-on-Hudson, New York, USA), and for his compositional guidance with regards to just intonation concepts.

Dr Noah Getz (Instructor of Saxophone, American University, and Jazz Saxophone Instructor, Levine School of Music, Washington, DC, USA) and the judges of the 2005 American University Saxophone Acknowledgements Ø×

Symposium Composition Contest for recognizing Digressioni modali for Tenor Saxophone and Pianoforte (2003) as the winner of the Third Prize in the international composition competition. Noah Getz (tenor saxophone) and John Kilkenny (marimba) for inspiring the arrangement of Digressioni modali for Tenor Saxophone and Marimba (2003), and for performing the work in Alexandria, Virginia and Washington, DC, USA. Noah Getz (tenor saxophone) and Laurence Gingold (pianoforte) for performing the work in Lancaster, Pennsylvania, USA, and finally, Noah Getz (tenor saxophone) and Jeffrey Chappell (pianoforte) for recording the work for CD release.

Brooke Green (Presenter, ABC Classic FM) for producing an excellent program featuring L’assaut sur la raison for Symphony Orchestra (2003) on her ABC Classic FM ‘Composers Emerging’ program, together with an interview with the composer, as part of ACOF 2003 (Australian Composers’ Orchestral Forum – Composition workshops with Brenton Broadstock, Andrew Schultz, and the Tasmanian Symphony Orchestra).

Dr. Stuart Greenbaum (Lecturer in Composition, Faculty of Music, University of Melbourne) for his incredible support throughout the PhD candidature.

Alejandro Guarello (Artistic Director, XV Festival de Música Contemporánea Chilena, Instituto de Música, Facultad de Artes, Pontificia Universidad Católica de Chile, Santiago, Chile) for selecting La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to be performed at the XV Festival de Música Contemporánea Chilena (15th Chilean Festival of Contemporary Music), 21-27 November, 2003, Santiago, Chile.

Christian Haines (Lecturer and Unit Coordinator, Electronic Music Unit, Elder Conservatorium of Music, University of Adelaide) for selecting Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to be part of the Medi(t)ations: Computers, Music and Intermedia, Australasian Computer Music Association Conference 2006, 11-13 July, 2006, Adelaide, Australia.

Michael Harrison (New York, NY, USA) for his demonstration of the ‘harmonic piano’ – a modified seven- foot Schimmel grand piano – during my visit to New York, NY, USA in 2004.

The international jury of the ISCM (consisting of Stanko Horvat [Croatia], Zygmunt Krauze [Poland], Giampaolo Coral [Italy], Frank Corcoran [Ireland/Germany], Arne Nordheim [Norway], and Berislav Šipuš [Croatia]) for selecting La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to Ø×× Acknowledgements be performed at the International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia.

Jerome Kitzke (New York, NY, USA) for his compositional direction during his McGeorge Fellowship residency at the University of Melbourne in 2005.

Jennifer Logan (Co-Artistic Director, Los Angeles Sonic Odyssey, Electronic and Computer Music Concert Series 2006, Los Angeles, California, USA) for selecting Paåc hazàr chakêà kaâ andar for Prepared Multi- tracked Disklavier (2000), Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005), Àzàdeh for Tape (2004, Rev. 2005), La Homa Kanto for Tape (2005), and Sonic Junk Yard for Tape (1999) to be part of the Los Angeles Sonic Odyssey Electronic and Computer Music Concert Series 2005, 2006, and 2007 Los Angeles and Pasadena, California, USA.

Jana Haluza Lucic (Producer, HRT, Hrvatska Radio, Zagreb, Croatia) for producing an excellent program featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) – together with an interview with the composer – on her HRT, Hrvatska Radio (Croatian Radio) ‘World of Music’ program in Zagreb, Croatia.

Dr. Susan McDonald (Department of Fine Arts, Philadelphia, Pennsylvania, USA) for selecting Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to be performed at the ‘La Salle University: Electroacoustic Works Inspired by Popular Music’ concert in November, 2005, Philadelphia, Pennsylvania, USA.

Marshall McGuire (Artistic Director, Sonic Art Ensemble, Sydney, NSW, Australia) and the Sonic Art Ensemble (Christine Draeger [flute], Margery Smith [bass clarinet], Rowan Martin [violin], Adrian Wallis [violoncello], and Bernadette Balkus [pianoforte]) for programming the Australian premier of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) within the 2006 ‘Southern Stars’ concert in Sydney, Australia. Marshall McGuire for also inspiring and presenting the world premier of Zambalogy for Harp (2004) in Sydney.

Pavel Mihelœiœ (Artistic Director, Ensemble MD7, and Dean of the Ljubljana Academy of Music, Ljubljana, Slovenia) and Ensemble MD7 (Steven Loy [conductor], Anamarija Tomac [flute], Jože Kotar [clarinet], Katja Krajnik [viola], Igor Mitrovic [violoncello], Uroš Polanc [trombone], Luca Ferrini [pianoforte], and Franci Krevh [percussion]) for commissioning and performing Aequilibrium for Flute, Clarinet, Viola, Cello, Acknowledgements Ø×××

Trombone, Piano and Percussion (2006) at the Ljubljana Festival 2006, 19 June – 31 August, 2006, Ljubljana, Slovenia.

Adam Muller (Associate Professor of Saxophone, Florida International University, Miami, Florida, USA) and Matthew Van Hoose (Accompanist in Residence, Department of Performing Arts, College of Arts and Sciences, American University, Washington, DC, USA) for performing Digressioni modali for Tenor Saxophone and Pianoforte at the First American University Saxophone Symposium, 26 March, 2005, Washington, DC, USA.

Anne Norman (shakuhachi) and Peter Hagen (harpsichord) for assisting in the development of Tres Imágenes Norteñas for Shakuhachi and Harpsichord (2006), and for performing the work at the Melbourne Composers’ League ‘From a Silence Well’ concert as part of the 2006 Australia-Japan Year of Exchange Celebrations.

Juan Miranda (Presenter, SBS Radio, ‘Spanish Radio’ Program) for producing an excellent program featuring Navigating the Labyrinth for String Orchestra (2002), Seeds of Passion for Amplified Violoncello (1999), and Bénédiction d’un conquérant for Symphony Orchestra (2004) – together with an interview with the composer – on his SBS Radio, ‘Spanish Radio’ program.

Peter Neville (Head of Percussion, School of Music, Victorian College of the Arts) for his incredible enthusiasm for contemporary music and Australian composition, as well as for his insight into polyrhythmic science.

John D. Nugent (Music Editor, Oregon Literary Review: An Online Collection of Literature, Hypertext, Art, Music, and Hypermedia, Portland, Oregon, USA) for publishing Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) in the Winter/Spring 2006, Vol. 1, No. 1 edition of the Oregon Literary Review.

The Omni Ensemble (David Wechsler [flute], Paul Garment [bass clarinet], Olivier Fluchaire [violin], Deborah Sepe [violoncello], and Jim Lahti [pianoforte]) for presenting the American premier, as well as a follow-up performance of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) in Brooklyn and New York, NY, USA during their 2006 concert series.

José Oplustil Acevedo (Presenter, Radio Beethoven [Radioemisoras], Siglo XX, Santiago, Chile) for producing an excellent program featuring Bénédiction d’un conquérant for Symphony Orchestra (2004), Ø×÷ Acknowledgements

L’assaut sur la raison for Symphony Orchestra (2003), Görüsmeler for Amplified Èrhú, Sampled Harmonium, Cajón and Bombo (2001), and Pañc hazar chakra kai andar for Prepared Disklavier (2000) – together with an interview with the composer – on his Radio Beethoven (Radioemisoras) ‘Programa Siglo XX’ program in Santiago, Chile.

Alex Pertout (Head of Improvisation, School of Music, Victorian College of the Arts) for his invaluable advice with regards to Afro-Latin percussion, rhythm and improvisation.

Katija Farac-Pertout, my wife, for her amazing belief and understanding not only during the last four years of the PhD degree, but throughout the last ten years of my composition studies at the University of Melbourne.

Maritza Pertout (Library Technician, State Library of Victoria) for her assistance with Spanish grammar, as well as countless other aspects of music publishing dilemmas.

Qmars Piraglu (formerly Siamak Noory) for his great inspiration and dedication to the realization of Àzàdeh for Santär and Tape (2004, Rev. 2005), as well as for the performance of the work at the 2007 Asia Pacific Festival (26th Asian Composers League Festival & Conference), 8-16 February, 2007, Wellington, New Zealand.

Glen Riddle (Coordinator, Foundation Program, Music Performance, School of Music, Victorian College of the Arts) for the French lessons.

Hans Roels (Concert Program Manager and Producer, Logos Foundation, Ghent, Belgium) for presenting the European premier of Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) at the Logos Foundation 2006 ‘Tape Tum & Heleen Van Haegenborgh’ concert in Ghent, Belgium.

Johanna Selleck for her incredible support throughout the PhD candidature.

Berislav Šipuš (Artistic Director, International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia) for his hospitality during the International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd Music Biennale Zagreb, Croatia.

Acknowledgements Ø÷

The Sonemus Ensemble [Bosnia-Herzegovina] (Ališer Sijaric [Artistic Director], Boris Previšic [flute], Vedran Tuce [bass clarinet], Julia Gubaidulina [pianoforte], Petar Haluza [violin], and Conradin Brodbek [violoncello]) for the performance of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) at the International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia.

Dr. Todd E. Sullivan (Chairperson, Department of Music, Indiana State University, Terre Haute, Indiana, USA) for his incredible hospitality during the Indiana State University 38th Annual Contemporary Music Festival, ‘Plugged In: Music With an Electric Edge’, November 10-12, 2004. A further warm thanks for driving me all the way from Terre Haute to Bloomington, Indiana.

Natasha Talmacs (Presenter, SBS Radio, ‘Croatian Radio’ Program, Sydney, Australia) and Silvio Rivier (Presenter, Narrator and Series Producer, Global Village, SBS TV, Sydney, Australia) for producing an excellent program featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003- 2004), L’assaut sur la raison for Symphony Orchestra (2003), and Seeds of Passion for Amplified Violoncello (1999) – together with an interview with the composer – on her SBS Radio, ‘Croatian Radio’ program.

Antonio Tenace for his incredible support throughout the PhD candidature, and more importantly, for fixing my scientific calculator.

Kenneth Young (Conductor, Wellington, New Zealand) and The Tasmanian Symphony Orchestra for the performance of L’assaut sur la raison for Symphony Orchestra (2003) and Bénédiction d’un conquérant for Symphony Orchestra (2004), which were especially composed for ACOF 2003 and 2004 (Australian Composers’ Orchestral Forum – Composition workshops with Brenton Broadstock, Andrew Schultz, and the Tasmanian Symphony Orchestra).

A special thanks to the following people for their general assistance, advice and support:

Betty Amsden (OAM), Celia Anderson, Dr. Jeri-Mae Astolfi (Assistant Professor, Department of Music, Henderson State University, Arkadelphia, Arkansas, USA), Rachel Atkinson (Trio Erytheia), Peter Aviss (Conductor and Musical Director, Oare String Orchestra, Faversham, UK), Laura Baker-Goldsmith, Pip Barry, Natasha Bennett, Jennifer Bird (New Audience Ensemble), David Black (Rarescale, London, UK), Ellen Bottorff (Orenunn Trio, Kansas City, Missouri, USA), Julianne Boren (Orenunn Trio, Kansas City, Missouri, USA), Mark Boren (Orenunn Trio, Kansas City, Missouri, USA), James Bradley (Doubling Up Trio), Le Brass Ø÷× Acknowledgements

Band du Nord-Pas de Calais (Calais, France), Stuart Brownley (Doubling Up Trio), Gary Robert Buchanan (Conductor and Musical Director, The Foundation Orchestra, Reno, Nevada, USA), David C. Bugli (Conductor and Musical Director, Carson City Symphony, Carson City, Nevada, USA), Stuart Byrne (Doubling Up Trio), Isin Cakmakcioglu (Trio Erytheia), José Miguel Candela (Coordinator, Comunidad Electroacústica de Chile [CECh], Santiago, Chile), Erik Carlson (New York Miniaturist Ensemble, New York, NY, USA), Robert Casteels (Dean of the Faculty of Performing Arts, LASALLE-SIA College of the Arts, Singapore), Robert Chamberlain (Trio Erytheia), La Chapelle Musicale de Tournai (Tournai, Belgium), Radiance Chen (New Audience Ensemble), Penelope Clarke (Thunder Bay, Ontario, Canada), Dr. Christopher Coleman (Radio Television Hong Kong Radio 4, Hong Kong), Andrew Conley, Rolando Cori (Associate Professor of Music, Facultad de Artes, Departamento de Música, Universidad de Chile, and President, Asociacion Nacional de Compositores de Chile, Santiago, Chile), Nicholas Cowall (Conductor, Victorian Youth Symphony Orchestra), Patricia Da Dalt (Quinteto CEAMC, Buenos Aires, Argentina), Lerida Delbridge (The Tin Alley String Quartet), Madonna Douglas (Thunder Bay, Ontario, Canada), Eve Duncan (President, The Melbourne Composers’ League), Shannon Ebeling, Mark Engebretson (Conference Chair, 2005 Society of Composers [SCI] National Conference, School of Music, University of North Carolina at Greensboro, Greensboro, North Carolina, USA), Ed Ferris (New Audience Ensemble), Barbara Finch (Thunder Bay, Ontario, Canada), The Foundation Orchestra (Reno, Nevada, USA), Johannes Fritzsch (Nürnberg, Germany), Steve Gibson (Open Space Art Society, Victoria, British Columbia, Canada), Yves Gigon (Canadian Electroacoustic Community [CEC], Montréal, Québec, Canada), Ian Godfrey (Lecturer in Music and Education, Faculty of Music, University of Melbourne), Ben Goudy, Alejandro Guarello (Artistic Director, XIII Festival de Música Contemporánea Chilena, Instituto de Música, Facultad de Artes, Pontificia Universidad Católica de Chile, Santiago, Chile), Elías Gurevich (Quinteto CEAMC, Buenos Aires, Argentina), Steven Heyman (The Syracuse Ensemble, Syracuse, New York, USA), Nancy Hosking, Luke Howard, Ashley Hribar (Speak Percussion), Frédéric Inigo (Artistic Director, 3èmes Rencontres Musiques Nouvelles, Lunel, France), Jason Kenner, Danae Killian, Victoria Jacono (3 Lines String Trio, Sydney), Jérôme Joy (Coordinator, Locus Sonus – Audio in Art, École Nationale Supérieure d’Art de Nice-Villa Arson, Nice, France), Stijn Kuppens (Artistic Director, Violoncello 2005, Brussels, Belgium), Laura Lentz (Crossroads Trio, New York, NY, USA), Jennifer Logan (Co-Artistic Director, Los Angeles Sonic Odyssey, Electronic and Computer Music Concert Series 2005, Los Angeles, California, USA), Phillipe Lorthios (Conductor, Le Brass Band du Nord-Pas de Calais, Calais, France), Eric Lyon (Assistant Professor, Dartmouth College, Hanover, New Hampshire, USA), George Macero (The Syracuse Ensemble, Syracuse New York, USA), Briony Mackenzie (New Audience Ensemble), Marco Antonio Mazzini (Duo Dicto and Diversity, Ghent, Belgium), John McMurtery (Doctoral Fellow, The Juilliard School of Music, New York, NY, USA), Nyssa McPhail, The Melbourne University Orchestra, Natsuko Mineghishi, Patrick Murphy (3 Lines String Trio, Sydney), Simona Musiani (Crossroads Trio, Rome, Italy), Tom Nelson (Southhampton, UK), Cliff Ojala (Thunder Bay, Ontario, Acknowledgements Ø÷××

Canada), Jorge Pérez (Quinteto CEAMC, Buenos Aires, Argentina), Sonni Petrovski (Musical Director, The Alea Contemporary Music Ensemble, Skopje, Republic of Macedonia), Marina Phillips (3 Lines String Trio, Sydney), Timothy Phillips (Speak Percussion), Judy Pile, Vladimir Pritsker (The Syracuse Ensemble, Syracuse New York, USA), Aleksander Pusz, Ryszard Pusz, Sabina Rakcheyeva (Diversity, Ghent, Belgium), Carla Rees (Rarescale, London, UK), Darlene Chepil Reid (President, New Music North, Thunder Bay, Ontario, Canada), Dr. James Romig (Co-Musical Director, The Society for Chromatic Art, New York, NY, USA), Joelene Rzepisko, Guillermo Sánchez (Quinteto CEAMC, Buenos Aires, Argentina), Naomi Sato (The Netherlands), Ginevra Schiassi (Ensemble Octandre, Bologna, Italy), José Schiller (Rádio MEC ‘Concerto das Américas’, Rio de Janeiro, Brazil), Sam Schmetterer (New Audience Ensemble), Phillip Schroeder (Associate Professor, Department of Music, Henderson State University, Arkadelphia, Arkansas, USA), Haydée Schvartz (Quinteto CEAMC, Buenos Aires, Argentina), Johanna Selleck, Gemma Sherry, Tarko Sibbel, Robert Sipos-Ori, Frank Sita (Plenty Valley FM), Emma Skillington (The Tin Alley String Quartet), Laura Sullivan, Gabriella Swallow (Rarescale, London, UK), Matt Tait, Gaspare Tirincanti (Ensemble Octandre, Bologna, Italy), Jo To, Eugene Ughetti (Speak Percussion), Josephine Vains, Amy Valent, Carlos Vera (Santiago, Chile), Lauren Van Der Werff, Orchestra Victoria, The Victorian Youth Symphony Orchestra, Professor Cirilo Vila Castro (Facultad de Artes, Departamento de Música, Universidad de Chile), Ward de Vleeschhouwer (Duo Dicto, Ghent, Belgium), Carina Voly (Crossroads Trio, Buenos Aires, Argentina), Cory Wagstaff, Koen Walraevens (Diversity, Ghent, Belgium), Russell Ward, Anneliese Weibel (Artistic Director, 2004 Society of Composers [SCI] Region II Conference, University of New York, School of Performing Arts, Geneseo, New York, USA), Larissa Weller (New Audience Ensemble), Justin Williams (The Tin Alley String Quartet), Elissa Wilson, Michelle Wood (The Tin Alley String Quartet), Larry Zimmerman (Minneapolis, Minnesota, USA).

Table of Contents

Volume 1

Introduction ...... 1

Microtonality ...... 1 Pitch Audibility and Discrimination ...... 3 Three Microtonal Compositions ...... 4 Folio of Compositions ...... 5 Methodology ...... 6 Interval Nomenclature and Notation System ...... 9

1. Theory Versus Performance Practice: Àzàdeh for Santñr and Tape ...... 11

A Brief History of Persian Classical Music ...... 11 The Seventeen-Note Gamut ...... 12 Persian Musical Scholarship in the Twentieth Century ...... 15 The Twenty-Four Equally-Tempered Quarter-Tone Scale ...... 16 The Pythagorean Division of the Octave ...... 18 Alain Daniélou’s Scale of Fifths ...... 22 The Twenty-Two Note Division of the Octave ...... 26 The Theory of Flexible Intervals ...... 27 Àzàdeh for Santñr and Tape ...... 30 The Artist ...... 30 The Instrument ...... 31 The Persian Modal System ...... 32 Tuning Analysis Protocols ...... 36 Tuning of the Santñr ...... 38 Spectrum Analysis Results...... 42 Analysis of Variance ...... 49 Tuning System Comparison ...... 52 Performance Practice and Tuning ...... 53 The Piano Tuner’s Octave and Inharmonicity ...... 55 The Tuning of Unisons ...... 56 ØØ Table of Contents

Climate and Tuning ...... 58 Gušes of Dastgàh-e Segàh ...... 60 Sampling of the Santñr and Vocals ...... 63

2. The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut Toy Piano ...... 67

Equal Temperaments ...... 67 Studies of Microtonal Equal Temperaments ...... 68 Nicolas Mercator’s Fifty-Three-Tone Equally-Tempered Division of the Octave ...... 71 Pietro Aron’s Quarter-Comma Meantone Tempered Division of the Octave ...... 74 Joseph Sauveur’s Forty-Three-Tone Equally-Tempered Division of the Octave ...... 80 Origins of Equal Temperament ...... 83 The Twelve-Tone Equally-Tempered Division of the Octave ...... 87 Exposiciones for Sampled Microtonal Schoenhut Toy Piano ...... 92 A Brief History of the Toy Piano ...... 92 The Schoenhut Toy Piano Sample ...... 94 Sound Diffusion ...... 95 Polyrhythmic Theory ...... 96 Alain Daniélou’s Scale of Proportions ...... 99 Notation for the Twenty-Four Equal Temperaments ...... 102 Sléndro and Pélog Scales ...... 104 One-Tone Equal Temperament ...... 107 Two-Tone Equal Temperament ...... 109 Three-Tone Equal Temperament ...... 111 Four-Tone Equal Temperament ...... 113 Five-Tone Equal Temperament ...... 115 Six-Tone Equal Temperament ...... 117 Seven-Tone Equal Temperament ...... 119 Eight-Tone Equal Temperament ...... 121 Nine-Tone Equal Temperament...... 123 Ten-Tone Equal Temperament ...... 127 Eleven-Tone Equal Temperament ...... 129 Twelve-Tone Equal Temperament ...... 132 Thirteen-Tone Equal Temperament ...... 135 Table of Contents ØØ×

Fourteen-Tone Equal Temperament...... 139 Fifteen-Tone Equal Temperament ...... 144 Sixteen-Tone Equal Temperament ...... 150 Seventeen-Tone Equal Temperament ...... 153 Eighteen-Tone Equal Temperament ...... 157 Nineteen-Tone Equal Temperament ...... 162 Twenty-Tone Equal Temperament ...... 168 Twenty-One-Tone Equal Temperament ...... 171 Twenty-Two-Tone Equal Temperament ...... 174 Twenty-Three-Tone Equal Temperament ...... 179 Twenty-Four-Tone Equal Temperament ...... 184 Blackwood’s Dictum ...... 188

3. The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer Quartet ...... 189

Just Intonation ...... 189 The Harmonic and Subharmonic Series ...... 190 The Monochord ...... 198 Combinational Tones ...... 200 Periodicity Pitch ...... 201 Prime Numbers, Primary Intervals, and Prime Limits ...... 202 The Just Diatonic Scale ...... 202 The Just Chromatic Scale ...... 205 Ben Johnston’s Fifty-Three-Tone Just Intonation Scale ...... 209 Harry Partch’s Forty-Three-Tone Just Intonation Scale ...... 212 Adriaan Daniël Fokker’s Thirty-One-Tone Equally-Tempered Division of the Octave ...... 216 La Homa Kanto for Harmonically Tuned Synthesizer Quartet ...... 218 The Harpsichord Sample ...... 221 Ben Johnston’s System of Notation ...... 222 Compositional Strategy ...... 223 Composing With Melodicles ...... 224 Three-Limit Just Intonation ...... 231 Five-Limit Just Intonation ...... 236 Seven-Limit Just Intonation ...... 243 ØØ×× Table of Contents

Eleven-Limit Just Intonation ...... 252 Thirteen-Limit Just Intonation ...... 259 Seventeen-Limit Just Intonation ...... 266 Nineteen-Limit Just Intonation ...... 272 Twenty-Three-Limit Just Intonation ...... 278 Twenty-Nine-Limit Just Intonation...... 284 Thirty-One-Limit Just Intonation ...... 289 Johnston’s Dictum ...... 296

Conclusion ...... 297

‘Manual’ of Microtonal Composition ...... 297 A Vast Universe of Subtle Intervallic Relationships ...... 297

Bibliography ...... 301

Appendices ...... 311

Appendix A: Comparative Table of Musical Intervals ...... 311 Appendix B: Microtonal Notation Font ...... 345

Volume 2

Recordings – Folio of Compositions 2003-2007: Volume 2 ...... vii

1. Àzàdeh for Santär and Tape, no. 389 (2004, Rev. 2005) ...... 1

2. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, no. 392 (2005) ...... 47

3. La Homa Kanto for Harmonically Tuned Synthesizer Quartet, no. 393 (2005) ...... 91

4. Symétrie intégrante for Flute, Organ and Electronics, no. 394 (2005-2006) ...... 153

5. Tres Imágenes Norteñas for Shakuhachi and Harpsichord, no. 396 (2006) ...... 203 Table of Contents ØØ×××

Volume 3

Recordings – Folio of Compositions 2003-2007: Volume 3 ...... vii

1. L’assaut sur la raison for Symphony Orchestra, no. 386 (2003) ...... 1

2. Digressioni modali for Tenor Saxophone and Pianoforte, no. 387 (2003) ...... 71

3. La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte, no. 388 ...... 97 (2003, Rev. 2004) 4. Bénédiction d’un conquérant for Symphony Orchestra, no. 390 (2004) ...... 175

5. Zambalogy for Harp, no. 391 (2004) ...... 245

6. Aequilibrium for Flute, Clarinet, Viola, Cello, Trombone, Piano and Percussion, no. 395 ..... 257 (2006)

Introduction

Microtonality

In a Perspectives of New Music article, Douglas Keislar states that the term microtonality “conjures up images of impossibly minute intervals, daunting instruments with hundreds of notes per octave, and wildly impractical performance instructions,” but that “such difficulties in fact characterize only a small percentage of the music that uses tunings other than standard twelve-note equal temperament.” Keislar then suggests that American composer Ivor Darreg’s proposal of the Greek term ‘xenharmonic’ or ‘unfamiliar modes’ is perhaps better suited to music utilizing “radically different tunings.”1 Alternative language for the term ‘microtonal’ is presented by Lydia Ayers in Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional Applications, with the following list of expressions: “tuning; microintervals; macrointervals or macrotones, such as 5-tone, 7-tone, and 10-tone equal temperaments; omnitonal; omnisonics; neoharmonic; xenharmonic; ‘exploring the sonic spectrum’; and non-twelve.” Although in spite of Ayers’s general attraction to the broadness of ‘omnitonal’, ‘microtonal’ is nevertheless espoused for its universality.2 The actual term ‘microtonal’ is generally reserved for music utilizing “scalar and harmonic resources” outside of Western traditional twelve-tone equal temperament, with “music which can be performed in twelve-tone equal temperament without significant loss of its identity” not considered “truly microtonal” by some theorists. Most non-western musical traditions (intonationally disengaged from contemporary Western musical practice) almost certainly accommodate this description. In the online Encyclopedia of Microtonal Music Theory, Joe Monzo provides the following discussion about the etymology of ‘microtonal’:

“Strictly speaking, as can be inferred by its etymology, ‘microtonal’ refers to small intervals. Some theorists hold this to designate only intervals smaller than a semitone (using other terms, such as ‘macrotonal’, to describe other kinds of non-12-edo intervals), while many others use it to refer to any intervals that deviate from the familiar 12-edo scale, even those which are larger than the semitone – the extreme case being exemplified by Johnny Reinhard, who states that all tunings are to be considered microtonal.”3

In the West, the concept of microtonality was notably given prominence to during the Renaissance by Italian composer and theorist Nicola Vicentino (1511-1576), in response to “theoretical concepts and

1 Douglas Keislar, “Introduction,” Perspectives of New Music 29.1 (Winter, 1991): 173. 2 Lydia Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional Applications,” (DMA diss., U. of Illinois, Urbana-Champaign, 1994, PA 9512292) 1-2. 3 Joe Monzo, “Encyclopedia of Microtonal Music Theory,” Microtonal, Just Intonation Electronic Music Software, 2005, Tonalsoft, 17 Nov. 2006, . 2 Introduction materials of ancient Greek music,”4 and later, by music theorists R. H. M. Bosanquet (1841-1912), as well as Hermann L. F. Helmholtz (1821-1894), and his “translator and annotator” Alexander John Ellis (1814- 1890).5 With regards to the adoption of microtonality by composers in more recent times, according to The New Harvard Dictionary of Music:

“The modern resurgence of interest in microtonal scales coincided with the search for expanded tonal resources in much 19th-century music. Jacques Fromental Halévy was the first modern composer to subdivide the semitone, in his cantata Prométhée enchâiné (1847). The first microtonal piece to use Western instrumental forms is a string quartet by John Foulds (1897); and the earliest known published quarter-tone composition, Richard Stein’s Zwei Konzertstücke, op. 26 (1906), is for cello and piano.”6

Gardner Read offers the following historical perspective:

“The history of microtonal speculation during the first half of the twentieth century displays six names above all others: Julián Carrillo, Adriaan Fokker, Alois Hába, Harry Partch, Ivan Wyschnegradsky, and Joseph Yasser. All six contributed extensive studies on microtones – historical, technical, and philosophical – and all but Yasser composed a significant body of music based on their individual explorations into microtonal fragmentation of the traditional twelve-tone chromatic scale. Later theorist-composers – notably Easley Blackwood, Ben Johnston, Rudolf Rasch, and Ezra Sims – have extended those explorations into various tuning systems and temperaments, and each has devised a personal notation for various unorthodox divisions of the octave.”

Read identifies five essential strategies for the procurement of microtonal intervals, which include: quarter- and three-quarter-tones, or the division of the octave into twenty-four equal intervals; eighth- and sixteenth-tones, or forty-eight and ninety-six equal intervals; third-, sixth-, and twelfth-tones, or eighteen, thirty-six, and seventy-two equal intervals; and fifth-tones, or thirty-one equal intervals; as well as “extended and compressed microtonal scales” with forty-three, fifty-three, sixty, seventy-two, or more equal or unequal intervals in the octave.7 J. Murray Barbour on the other hand pronounces Pythagorean (“excellent for melody, unsatisfactory for harmony”), just intonation (“better for harmony than for melody”), meantone (“a practical substitute for just intonation, with usable triads all equally distorted”), and equal temperament (“good for melody, excellent for chromatic harmony”) as the “four leading tuning

4 Accounts of the arcicembalo (a two-manual harpsichord capable of producing thirty-six distinct pitches per octave) and arciorgano (organ adaptation) were presented by Nicola Vicentino in his treatises L’antica musica ridotta a la moderna prattica of 1555 and Descrizione dell’ arciorgano (1561). For a further discussion, see Don Michael Randel, ed., The New Harvard Dictionary of Music (Cambridge, Mass.: Belknap Press of Harvard U Press, 1986) 47. 5 John H. Chalmers, Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales (Hanover, NH: Frog Peak Music, 1993) 1-2. 6 Randel, ed., The New Harvard Dictionary of Music 492. 7 Gardner Read, 20th-Century Microtonal Notation (Westport, CT: Greenwood Press, 1990) 2-127. Introduction 3 systems,” or the “Big Four.” Barbour also makes mention of the “more than twenty varieties of just intonation,” and “six to eight varieties of the meantone temperament,” as well as the “geometric, mechanical, and linear divisions of the line” for the mathematical approximation of equal temperament.8 According to Barbour, tuning systems may be classified into two distinct classes: the first being ‘regular’, where all fifths but one are equal in size; and the second, ‘irregular’, where more than one fifth is unequal in size. The former includes Pythagorean, meantone, and equal temperament, while the latter (as classified by Barbour) excludes just intonation.9

Pitch Audibility and Discrimination

Although it may be stated that the human ear has a general capacity to hear frequencies between the ranges of 16Hz and 16,000Hz (equal to 16 to 16,000 cycles per seconds, and approximately C0 and B9), it must be noted that numerous factors influence the actual outcomes. The 16Hz lower limit is dependent on two principal factors, being wave intensity and shape; with the inclusion and exclusion of pure tones displacing the figures for the lower limit to anywhere between 12Hz and 100Hz (approximately Gþ0 and G2). The 16,000Hz upper limit is generally reserved for a healthy population under the age of forty, with adolescent capacity as high as 25,000Hz (approximately G10); a supposed ‘normal hearing’ population in some cases not surpassing a 5,000Hz (approximately DÚ8) upper limit; and another probable large percentage incapable of hearing beyond 10,000Hz (approximately DÚ9).10 The frequency range of the 88-key pianoforte is between 27.5Hz and 4,186Hz, or A0 to C8, and therefore encompasses pitch material with a range of over seven octaves. The seven-octave range additionally represents the range embodied within the collection of instruments that constitute the traditional symphony orchestra.11 The pitch discrimination threshold for an average adult is around 3Hz at 435Hz, which is approximately one seventeenth of an equal tone, or 11.899 cents, although a “very sensitive ear can hear as small a difference as 0.5Hz or less” (approximately a hundredth of a tone, or 1.989 cents). Tests conducted in 1908 by Norbert Stücker (Zeitschrift für Sinnesphysiologie 42: 392-408) of sixteen professional musicians in the Viennese Royal Opera conclude a pitch discrimination threshold between one five-hundred-and-fortieth (0.1Hz) and one forty-ninth of a tone (1.1Hz), or 0.370 and 4.082 cents,

8 J. Murray Barbour, “Irregular Systems of Temperament,” Journal of the American Musicological Society 1.3 (Autumn, 1948): 20. 9 J. Murray Barbour, Tuning and Temperament: A Historical Survey (New York: Dover Publications, 2004) x-xi 10 Carl E. Seashore, Psychology of Music (New York: Dover Publications, 1967) 54-55. 11 Harry F. Olson, Music, Physics and Engineering, 2nd ed. (New York: Dover Publications, 1967) 123. 4 Introduction with an average of 0.556Hz (approximately a hundredth of a tone), or 2.060 cents.12 In Tuning, Timbre, Spectrum, Scale William A. Sethares adds the following to the discussion:

“The Just Noticeable Difference (JND) for frequency is the smallest change in frequency that a listener can detect. Careful testing such as that of E. Zwicker and H. Fastl (Psychoacoustics, Springer-Verlag, Berlin [1990]) has shown that the JND can be as small as two or three cents, although actual abilities vary with frequency, duration and intensity of the tones, training of the listener, and the way in which JND is measured.”13

Three Microtonal Compositions

Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition encompasses the work undertaken by Lou Harrison (widely regarded as one of America’s most influential and original composers) with regards to just intonation, and tuning and scale systems from around the globe – also taking into account the influential work of Alain Daniélou (Introduction to the Study of Musical Scales), Harry Partch (Genesis of a Music), and Ben Johnston (Scalar Order as a Compositional Resource). The essence of the project being to reveal the compositional applications of a selection of Persian, Indonesian, and Japanese musical scales utilized in three very distinct systems: theory versus performance practice and the ‘Scale of Fifths’, or cyclic division of the octave; the equally-tempered division of the octave; and the ‘Scale of Proportions’, or harmonic division of the octave championed by Harrison, among others – outlining their theoretical and aesthetic rationale, as well as their historical foundations. The project begins with the creation of three new microtonal works tailored to address some of the compositional issues of each system, and ending with an articulated exposition; obtained via the investigation of written sources, disclosure of compositional technique, mathematical analysis of relevant tuning systems, spectrum analysis of recordings, and face-to-face discussions with relevant key figures. The three microtonal works discussed in the thesis include Àzàdeh for santñr and tape, no 389 (2004, Rev. 2005) – composed for Iranian santñrist Qmars Piraglu (formerly Siamak Noory) – which features the Persian santär (72-string box zither), and serves as a practical study of Persian tuning systems, with its presentation of both ‘theoretical’ and ‘performance practice’ tunings; an ‘acousmatic’ work entitled Exposiciones for sampled microtonal Schoenhut toy piano, no. 392 (2005), which attempts to explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut model 6625, 25-key toy piano, a complex polyrhythmic scheme, and sequential tuning modulations

12 “Pitch discrimination is measured by sounding two pure tones in quick succession and gradually reducing the difference in frequency until the observer is unable to tell which of the two tones is higher. The steps usually employed in such a series are 30, 23, 17, 12, 8, 5, 3, 2, 1, and 0.5Hz, at the level of international (standard) pitch.” For a further discussion, see Seashore, Psychology of Music 56-57. 13 William A. Sethares, Tuning, Timbre, Spectrum, Scale, 2nd ed. (London: Springer-Verlag, 2005) 44. Introduction 5 featuring the first twenty-four equally-tempered divisions of the octave; and La Homa Kanto (or ‘The Human Song’ in Esperanto) for harmonically tuned synthesizer quartet, which derives its pitch material from Lou Harrison’s five-tone scales (presented in Lou Harrison’s Music Primer: Various Items About Music to 1970) and features ten distinct tuning modulations: 3-limit through to 31-limit just intonation systems based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth, twenty-third, twenty-ninth, and thirty-first partials of the harmonic series. The aim of the dissertation is to present an articulated exposition of three ‘original’ and unique microtonal composition models individually exploring the expanded tonal resources of Pythagorean intonation, equal temperament, and just intonation. It is also proposed that the thesis outlines their theoretical and aesthetic rationale, as well as their historical foundations, with mathematical analysis of relevant tuning systems, and spectrum analysis of recordings providing further substance to the project. Theory versus performance is also taken into account, and the collaboration with an actual performer is intended to deliver the corporeal perspective. It is anticipated that the thesis will not represent current acoustic and psychoacoustic research at any great depth, and therefore should not be seen to serve as a comprehensive study of physics and music. It will nevertheless provide a foundation for the exploration of tuning systems, and additionally, present a composer’s perspective – as opposed to a musicological or ethnomusicological study – of microtonal music composition.

Folio of Compositions

Other works incorporated into volume two and three of ‘Folio of Compositions 2003-07’ include: Symétrie intégrante for Flute, Organ and Electronics, no. 394 (2005-06); Aequilibrium for flute, clarinet, viola, cello, trombone, piano and percussion, no. 395 (2006); Tres imágenes norteñas for shakuhachi and harpsichord, no. 396 (2006); L’assaut sur la raison for symphony orchestra, no. 386 (2003); Digressioni modali for tenor saxophone and pianoforte, no. 387 (2003); La flor en la colina for flute, clarinet, violin, violoncello and pianoforte, no. 388 (2003-04); Bénédiction d’un conquérant for symphony orchestra, no. 390 (2004); and Zambalogy for harp, no. 391 (2004). These works do not represent the microtonal models of the first three compositions, yet certainly adhere to an exploration of alternative scalar and harmonic materials, and their application in contemporary compositional practice. Pitch material for these works has been generated via a selection of methods such as multi-octave grouping (pitch material based on multi- octave scales constructed of dissimilar tetrachords), modality (modes generated by the major, in, hirajoshi and kumoijoshi scales), aleatoric formation (pitch material generated via indeterminate means), pitch class set theory (pitch material derived from the 208 basic pitch-class sets of set theory), synthetic symmetry (hexatonic and octatonic major and minor scales), cluster generation (pitch material derived from five-note chords and inversions), physical and psychological concepts of consonance and 6 Introduction dissonance (the harmonic language of the twelve primary intervals), polymodal and polytonal juxtaposition (multiple scales and tonalities), as well as cross-cultural abstraction (non-Western music theoretical concepts).

Methodology

Chapter one (theory versus performance practice) begins with a brief history of Persian music, and is followed by the presentation of Éafå al-Dån Urmawå’s seventeen-note gamut and division of the whole- tone, and an explanation of the significance of the tetrachord in the construction of melodic and harmonic structures. A discussion of Persian musical scholarship in the twentieth century then introduces the three separate theories on intervals and scales of Persian music proposed in the twentieth century: the twenty-four equally-tempered quarter-tone scale proposed by Ali Naqi Vaziri in the 1920s, the alternative twenty-two-note scale proposed by Mehdi Barkešli in the 1940s based on Pythagorean principles, as well as the theory of the five primary intervals of performance practice presented by Hormoz Farhat in the 1990 publication of his doctoral thesis The Dastgàh Concept in Persian Music.14 The division of the octave into twenty-four equally-tempered quarter-tones is given a historical perspective, as well as a mathematical exposition, while the concept of Pythagorean intonation is firstly illustrated via the construction of a twenty-seven-note Pythagorean scale with the necessary intervals to facilitate the general modulations of Western tonal music; and secondly, via Daniélou’s ascending ‘scale of fifths’, or cyclic division of the octave, which presents a series of fifty-nine consecutive fifths, or sixty lü. The BCE Chinese origins of Pythagoreanism and its philosophical significance according to theorist King Fâng are also subsequently discussed.15 The development of the seventeen-note gamut by Mehdi Barkešli into a twenty-two-note Pythagorean scale is then presented, which is followed by Farhat’s theory of flexible intervals, or of the five primary intervals of performance practice – advocated by Farhat in opposition to both twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and Barkešli.16 The work, Àzàdeh for santñr and tape, is then introduced, together with a brief biography of the artist, Qmars Piraglu; a description of instrument, the Persian santär (a 72-string [or 18 quadruple-stringed] box zither); and a discussion of the essence of the Persian modal system. Following the establishment of the tuning analysis protocols, a detailed exposition of the tuning process of the santñr for dastgàh-e segàh (on F) is presented. Spectrum analysis results collected on three separate occasions (with a

14 Hormoz Farhat, The Dastgàh Concept in Persian Music (New York: Cambridge U. Press, 1990) 7. 15 Alain Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness (Rochester, VT: Inner Traditions, 1995) 35-37. 16 Farhat, The Dastgàh Concept in Persian Music 15-16. Introduction 7 periodicity of 3-6 months) for each of the twenty-seven sets of strings are then analyzed with regards to the intervallic size of octaves, perfect fifths, perfect fourths, tempered perfect fourths, and neutral thirds. An analysis of variance is then conducted with the data collected, which in turn produces mean measurements with the capacity to characterize tuning characteristics. A tuning system comparison then concludes a relationship between Farhat’s and Piraglu’s division of the whole-tone, with Farhat’s theory of flexible intervals accorded as the most plausible hypothesis. In view of the fact that stretched, as well as compressed octaves are a common occurrence in Piraglu’s tuning of the santñr, the theory of the ‘piano tuner’s octave’ is discussed, along with the natural phenomenon of inharmonicity – a factor especially affecting plucked and struck strings (along with other musical sounds with a short decay).17 A comparison is also made between the tuning of a triple-string unison of a piano and a quadruple-string unison of a santñr. Climate and its effects on tuning are then considered, and especially in order to substantiate Piraglu’s claims of the climatic conditions of Melbourne, Australia being “unsatisfactory” for the tuning of the santär in comparison to Tehran, Iran. The twenty-four gušes for dastgàh-e segàh according to a prominent radif associated with Mñsà Marñfi are then presented, followed by the pitch organization of the adopted six most prominent elements of the radif of dastgàh-e segàh. Finally, the structural scheme of the work and its basis on ‘golden mean’ proportions are explained, as well as the sampling process of the santär and vocals, and digital processing that culminates in the tape element of Àzàdeh for santñr and tape. Chapter two (the equally-tempered archetype) begins with a discussion about Partch’s notion of two distinct classes of equal temperaments: those that produce equal third-tones, quarter-tones, fifth- tones, sixth-tones, eighth-tones, twelfth-tones, and sixteenth-tones; as opposed to those that divide the octave into nineteen, thirty-one, forty-three, and fifty-three equally-tempered intervals.18 This is followed by a brief history of some important studies of the equally-tempered paradigm, namely by Julián Carrillo Trujillo, Ferruccio Busoni, Ramon Fuller, and Easley Blackwood, with the latter two serving as benchmarks for the establishment of the criteria to properly assess the musical virtues of a particular equal temperament. The deviation of basic equally-tempered intervals from just intonation, Fuller’s eight best equal temperaments, and Blackwood’s concept of ‘recognizable diatonic tunings’ are then discussed. Nicolas Mercator’s fifty-three-tone equally-tempered division of the octave, which is Fuller’s recommendation for a temperament with the capacity to approximate just intervals, is consequently presented, along with an opposing view by Dirk de Klerk. In order to illustrate the principal evolutionary markers leading up to the adoption of equal temperament in the West – from Pythagorean intonation, meantone and well temperament, to equal

17 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 166-67. 18 Harry Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments, 2nd ed. (New York: Da Capo, 1974) 425. 8 Introduction temperament – Pietro Aron’s quarter-comma meantone temperament is introduced, as well as Joseph Sauveur’s forty-three-tone equal temperament, which approximates fifth-comma meantone temperament. The origins of equal temperament are then traced back to 1584 China, and Prince Chu Tsai-yü’s monochord. What follows is a discussion of the geometrical and numerical approximations of Marin Mersenne and Simon Stevin, which culminate in Johann Faulhaber’s monochord, and the first printed numerical solution to equal temperament based on the theory of logarithmic computation.19 The mathematical formula for twelve-tone equal temperament, the equally-tempered monochord, and beating characteristics of the twelve-tone equally-tempered major and minor triads are then sequentially presented, which are followed by the equal thirds, sixths, fifths, and fourths in piano tuning. The work, Exposiciones for Sampled Microtonal Schoenhut Toy Piano, is then introduced, together with a brief history of the toy piano, the Schoenhut toy piano sample, as well as concepts of sound diffusion and polyrhythmic theory utilized in the composition. In order to illustrate the design of the proposed notation for the twenty-four equal temperaments, Daniélou’s ‘scale of proportions’, or harmonic division of the octave, which presents a series of sixty-six unique intervals is introduced. Paul Rapoport’s Pythagorean notation then provides an alternative to the system of notation based on Daniélou’s subdivision of the whole-tone. Sléndro and pélog scales are then discussed from a historical perspective, with the gamelan gedhé sléndro and pélog tunings from Sri Wedhari theatre auditorium in Solo, Central Java serving as the ‘performance practice’ model. The harmonic characteristics of the sléndro and pélog scales are then presented in accordance to five-limit intonation principles. What follows is a systematic exposition of the compositional application of each equal temperament between one and twenty-four. Chapter three (the harmonic consideration) begins with a basic outline of just intonation and ‘extended just intonation’, or the incorporation of partials beyond the sixth harmonic.20 A historical and scientific perspective of the harmonic series is then presented, together with examples of the beating characteristics of the first eight partials of the harmonic series, as well as of the mistuned and properly tuned unison, and mistuned and properly tuned octave. Dissonance, with special reference to the theory of beats, is defined according to James Tenney, Helmholtz, Bosanquet, and Johnston. The complement or mirror image of the harmonics series, or the ‘subharmonic series’, is also discussed, together with Partch’s theory of ‘otonalities’ (pitches derived from the ascending series) and ‘utonalities’ (pitches derived from the descending series).21 A comparative table of intonation then provides interval, ratio, and cents data for the twelve basic intervals of just intonation, Pythagorean intonation, meantone temperament, and equal temperament.

19 Barbour, Tuning and Temperament: A Historical Survey 78. 20 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 106-07. 21 David D. Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 3rd ed. (San Francisco: Other Music, 2002) 28-30. Introduction 9

In order to illustrate the basic principles of proportions and string lengths, the traditional structure and function of the monochord is explained, with the generation of simple octaves and fifths utilized to demonstrate the theoretical basis for the Pythagorean monochord. A table depicting all the intervals of the harmonic series from the first partial through to the one-hundred-and-twenty-eighth partial is then presented. Combinational tones, or differential and summation tones, are also subsequently explained, together with their implications on the intervals of the octave, just perfect fifth, just perfect fourth, just major third, just minor sixth, just minor third, and just major sixth. This is followed by a discussion of periodicity pitch, and its theoretical significance in relation to JND, or Just Noticeable Difference. The relationship of prime numbers, primary intervals, and prime limits to just intonation principles is subsequently explained. The concept of just intonation is then illustrated via the construction of a seven-note just diatonic scale, and the presentation of the beating characteristics of the just major triad. This is followed by the construction of a twenty-five-note just enharmonic scale, and its development into Johnston’s fifty-three- tone just intonation scale. Harry Partch’s forty-three-tone just intonation scale, and his rationale for the consequential harmonic expansion to eleven-limit is then explained. The twenty unique triads, fifteen unique tetrads, and six unique pentads made possible via the inclusion of the eleven-limit intervals are additionally presented. The final octave division discussed in the chapter is Adriaan Daniël Fokker’s thirty- one-tone equally-tempered division of the octave, and in view of its capability to approximate the tonal resources of seven-limit just intonation. The work, La Homa Kanto for Harmonically Tuned Synthesizer Quartet, is then introduced, together with a presentation of Harrison’s five pentatonic scales, which serve as the pitch material, the ‘1967 William Dowd French Double Harpsichord’ sample, and Johnston’s system of notation, which serves as the system of notation utilized in the score. Compositional strategy is then discussed, together with Harrison’s concept of composing with melodicles, or neumes, which is adopted and developed into a system incorporating three categories of motivic manipulation: melodic transformation of motive, rhythmic transformation of motive, and harmonic transformation of motive. What follows is a systematic exposition of the compositional application of each just intonation limit between three and thirty-one.

Interval Nomenclature and Notation System

Intervals based on Pythagorean intonation have been simply named according to their cyclical position, and therefore follow an either ascending 3/2 incremental progression from natural, sharp, double sharp, to triple sharp; or a descending 4/3 incremental progression from natural, flat, double flat, to triple flat. The procedure is exemplified via the twenty-seven-note Pythagorean scale, which incorporates fifteen intervals generated by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!, 10 Introduction

B!, F#, C#, and G#; and another eleven intervals, by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$. The method adopted in equal temperament on the other hand is a nomenclature based on the comma approximations to Daniélou’s ‘scale of proportions’, or sixty-six-note just intonation scale, with every interval not characterized by the equal semitones and quarter-tones of 12-et and 24-et further indentified via its origin (for example: 5-et supermajor second, 7-et grave or small tone, and 9-et great limma, or large half-tone). Exceptions to this rule include 31-et, 43-et, and 53-et, which because are not discussed in the thesis with relation to other intervals, do not require a differential prefix with the same conditions. Adriaan Daniël Fokker’s thirty-one-tone equally-tempered division of the octave introduces a further element to intervallic nomenclature. The system, which was developed by David C. Keenan, involves the prefixes: double diminished, subdiminished, diminished, sub, perfect, super, augmented, superaugmented, and double augmented for unisons, fourths, fifths, and octaves; while subdiminished, diminished, subminor, minor, neutral, major, supermajor, augmented, and superaugmented for seconds, thirds, sixths, sevenths, and ninths. Perfect and major, or “the ones implied when there is no prefix,” represent the central position of a range based on comma or diesis increments from ß4 to +4 (for example: diminished third, subminor third, minor third, neutral third, major third, supermajor third, and augmented third).22 For intervals beyond five-limit intonation, James B. Peterson’s recommendations for the naming of bases has been adopted, which results in the following additional prefixes for seven-, eleven-, thirteen-, seventeen-, nineteen-, twenty-three-, twenty-nine-, and thirty-one- limit: septimal, undecimal, tridecimal, septendecimal, nonadecimal, trivigesimal, nonavigesimal, and untrigesimal (for example: septimal superfifth, undecimal subfifth, tridecimal subfifth, septendecimal superfifth, nonadecimal superfifth, trivigesimal superfifth, nonavigesimal subfifth, and untrigesimal superfifth).23 The classification of 724 unique intervals incorporated into the comparative table of musical intervals (see Appendix A) includes all the intervals cited in the current study. The notation symbols utilized in the thesis include the five standard accidental signs of Western music; four common quarter-tone and three-quarter-tone symbols; twenty-three unique symbols based on Daniélou’s division of the whole-tone; Ali Naqi Vaziri’s notation system, or four accidentals of Persian music; Johnston’s system of notation, which contains twenty-three unique symbols for the notation of just intonation up to the thirty-first harmonic; as well as Fokker’s nine symbols for the notation of thirty-one equal temperament. All these symbols have been incorporated into a 238-character microtonal notation PostScript Type 1 font (see Appendix B), which was created via the modification of a selection of symbols in the Coda Music Finale’s Maestro font utilizing CorelDraw 13.0 and FontMonger 1.0.8.

22 David C. Keenan, “A Note on the Naming of Musical Intervals,” David Keenan’s Home Page, 3 Nov. 2001, 22 Nov. 2006, . 23 James B. Peterson, “Names of Bases,” The Math Forum: Ask Dr. Math, 15 Apr. 2002, Drexel U., Philadelphia, PA, 22 Nov. 2006, . 1. Theory Versus Performance Practice: Àzàdeh for Santñr and Tape

A Brief History of Persian Classical Music

Modern Persian scholarship on the theory of intervals and scales may be mainly attributed to the theoretical writings of medieval music scholars Éafå al-Dån ‘Abd al-Mu’min al-Urmawå’ (d. 1294) and Quðb al-Dån ‘Maämñd ibn Mas’ñd al-Shåràzå’ (1236-1311). “The latter half of the thirteenth century constitutes one of the most important periods in the history of Arab and Persian musical theory,” notes Owen Wright. “It witnessed the emergence of a corpus of theoretical writings that not only demonstrate a considerable degree of originality, but also provided the framework within which all the major theorists of the following two centuries were to operate.” Éafå al-Dån in particular is acknowledged for founding the ‘Systematist school’ with his two influential treatises: Kitàb al-adwàr of 1252 (‘Book of Cycles’) and Risàla al-sharafiyya fi al-nisàb al-ta’lifiyya of 1267 (‘Sharafian Treatise on Intervallic Relations’), while Quðb al-Dån for his further contribution to the theory within a section about music contained in his encyclopedia Durrat al-tàj (‘Pearl of the Crown’), published circa 1300.24 In the spirit of their predecessors – Al-Kindå (d. 873), and celebrated author of Kitàb al-mñsåqå al-kabår (‘Great Book on Music’), Abu Nasr Fàràbå (872-950); as well as Ibn Sånà (980-1037) – their findings were essentially based on the musical theories of the classical Greeks; from Pythagoras of Samos (fl. 530 B.C.) to Aristoxenus of Tarentum (fl. 400 B.C.).25 Cultural links between Persia and Ancient Greece existed between 500 B.C. and 300 A.D., and were further infused by Alexander the Great’s conquest of the Achaemenid Empire in 330 B.C. that generated the hundred years of Greek rule in Persia. As a consequence, “the works of Euclid, Aristoxenus, Ptolemy, and others translated into Arabic at Baghdad during the ninth century,26 served as models for the great Islamic theorists,” notes Ella Zones.27

24 Owen Wright, The Modal System of Arab and Persian Music A.D. 1250-1300, London Oriental Series, vol. 28 (Oxford: Oxford U. Press, 1978) 1-20. 25 Hormoz Farhat, “Iran: Classical Traditions,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 531. 26 “During the Abbasid period (750-1258) many branches of Islamic scholarship developed rapidly, among them medicine, astronomy, alchemy, geography, mathematics, and also music theory. This development was stimulated by contact with ancient Greek writings which became available to Islamic scholars through translations done in the Bait al- Äikma (House of Wisdom), a library, astronomical observatory, and translation institute established in Baghdad by caliph al Ma’mñn.” For a further discussion, see by Josef M. Pacholczyk, “Secular Classical Music in the Arabic Near East,” Musics of Many Cultures (Berkeley, CA: U. of California Press, 1980) 255. 27 Ella Zonis, “Contemporary Art Music in Persia,” The Music Quarterly 51.4 (Oct., 1965): 636-37. 12 Theory Versus Performance Practice

The Seventeen-Note Gamut

In Kitbag al-adware, Éafå al-Dån proposes that a Pythagorean whole-tone (equal to the frequency ratio of 9/8, or 203.910 cents) should only be subdivided into either one Pythagorean limma (256/243, or 90.225 cents), or two Pythagorean limmas (equal to a Pythagorean diminished third, 65536/59049, or 180.450 cents). This in effect generates a theoretical basis for a whole-tone constructed from the sum of two limmas and a Pythagorean comma (531441/524288, or 23.460 cents), and a tetrachord made up of two whole-tones and a limma that is implemented in an octave as two conjunct tetrachords, plus an additional whole-tone. The result is a seventeen-note scale,28 and modality based on two conjunct tetrachords, which may be theoretically referred to as a bitetrachordal system.29

Table 1. Éafå al-Dån Urmawå’s division of the whole-tone

PATTERN L L + L L + L + C INTERVAL Pythagorean Pythagorean just limma diminished third major tone RATIO 256/243 65536/59049 9/8 CENTS 90.225 180.450 203.910

The tetrachord (a four-note series enclosed within the range of a perfect fourth) is a concept borrowed from ancient Greek music, where as part of the ‘Greater Perfect System’ – a two-octave system made up of four conjunct and disjunct tetrachords (Hypaton, Meson, Diezeugmeson and Hyperbolaeon), as well as an additional whole-tone (Proslambanomenos) to complete the lower part of the range – was “the basic building block of Greek music,” and therefore at the core of Greek theory on intervals and scales.30 The essence of Ancient Greek music and its proponents is summarized by R. P. Winnington-Ingram thus:

Ancient Greek music was purely or predominantly melodic; and in such music subtleties of intonation count for much. If our sources of information about the intervals used in Greek music are not always easy to interpret, they are at any rate fairly voluminous. On the one hand we have Aristoxenus, by whom musical intervals were regarded spatially and combined and subdivided by the processes of addition and subtraction; for him the octave consisted of six tones, and the tone was exactly divisible into fractions such

28 Owen Wright, “Arab Music: Art Music,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 806. 29 Dariush Talai, “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System,” ed. Virginia Danielson, Scott Marcus and Dwight Reynolds, The Garland Encyclopedia of World Music: The Middle East, vol. 6 (New York: Routledge, 2002) 871. 30 André C. Barbera, “Greece,” ed. Don Michael Randel, The New Harvard Dictionary of Music (Cambridge, Mass.: Belknap Press of Harvard U Press, 1986) 347-49. Theory Versus Performance Practice 13

as the half and quarter, so that the fourth was equal to two tones and a half, the fifth to three tones and a half, and so on. On the other hand we have preserved for us in Ptolemy’s Harmonics the computations of a number of mathematicians, who realized correctly that intervals could only be expressed as ratios (e.g. of string-lengths), that the octave was less than the sum of six whole tones and that this tone could not be divided into equal parts. These authorities are Archytas, the Pythagorean of the early fourth century, Eratosthenes (third century), Didymus (first century), and Ptolemy himself (second century A.D.). To these we must add the scale of Plato’s Timaeus (35B) and, closely related to it, the computations of the pseudo- Philolaus (ap. Boethium, Mus. III, 8) and of Boethius himself (IV, 6).31

With regards to the function of tetrachords in the construction of melodic and harmonic structures, John H. Chalmers presents the following discussion:

“Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world’s music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran, and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world’s music.”

Chalmers then further expands on the issue with the subsequent definition:

“The tetrachord is the interval of a perfect fourth, the diatessaron of the Greeks, divided into three subintervals by the interposition of two additional notes. The four notes, or strings, of the tetrachord were named hypate, parhypate, lichanos, and mese in ascending order from 1/1 to 4/3 in the first tetrachord of the central octave of the ‘Greater Perfect System’, the region of the scale of most concern to theorists. Ascending through the second tetrachord, they were called paramese, trite, paranete, and nete.”32

Stringed instruments are recognized as a major factor in the design of tuning and scale systems. “The fretting and tuning of stringed instruments was directly connected to the development of modes. We can deduce this from the fact that stringed instruments have been used to study intervals and tetrachords from antiquity, and from the fact that in the past, musicians were poet-singers first of all, accompanying their poetry and song with stringed instruments,” explains Dariush Talai. It is interesting to note that “the tetrachord also corresponds to a physical area on the neck of instruments such as the ‘ñd, tàr, and setàr, where the fingers can reach the notes without changing position.”33

31 R. P. Winnington-Ingram, “Aristoxenus and the Intervals of Greek Music,” The Classical Quarterly 26.3/4 (Jul.-Oct., 1932): 195. 32 Chalmers, Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales 4. 33 Talai, “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System,” The Garland Encyclopedia of World Music: The Middle East 868-69. 14 Theory Versus Performance Practice

Table 2. The seventeen-note gamut.34

TRADITIONAL PHONETIC PYTHAGOREAN CONTEMPORARY RATIO CENTS PERSIAN SYSTEM TRANSCRIPTION NOTATION NOTATION (FRACTION) A C C 1/1 0.000 b D" D" 256/243 90.225 j E$ Dî 65536/59049 180.450 d D D 9/8 203.910 h E" E" 32/27 294.135 v F" Eî 8192/6561 384.360 z E E 81/64 407.820 ä F F 4/3 498.045 ë G" G" 1024/729 588.270 y A$ Gî 262144/177147 678.495 yà G G 3/2 701.955 yeb A" A" 128/81 792.180 yej B$ Aî 32768/19683 882.405 yed A A 27/16 905.865 yeh B" B" 16/9 996.090 yu C" Bî 4096/2187 1086.315 yez B B 243/128 1109.775

yeä C C 2/1 1200.000

Illustrated in following music example is the scale of Éafå al-Dån, which is referred to as the seventeen-note gamut and based on Al-Fàràbå’s first tetrachord division on the Khorasan long-necked lute, or ðunbñr khuràsànå. The seventeen-note gamut (utilizing G as the tonic) is further characterized by the following intervallic pattern: L, L, C – L, L, C – L – L, L, C – L, L, C – L + L, L, C. The transposition to C of the intervallic pattern reorganized thus: L, L, C – L, L, C – L + L, L, C – L, L, C – L, L, C – L.35

Ex. 1. Subdivision of the tetrachord and the seventeen-note gamut (Éafå al-Dån Urmawå, 13th century)

34 Taqi Bineš, The Short History of Persian Music (Tehran: Àrvin Publication, 1995) 106. 35 Wright, “Arab Music: Art Music,” The New Grove Dictionary of Music and Musicians 806. Theory Versus Performance Practice 15

Although the theoretical notion of a seventeen-note division of the octave became widely adopted by the Islamic world for centuries to come, there is much speculation amongst musicologists today about the actual implementation of this synthetic scale by musicians of the time in the context of performance practice, as the employment of an intuitive system of flexible intervals seems more probable. It should be noted that no mode has ever contained all seventeen notes (heptatonic being the dominant character), and that the Pythagorean comma was not utilized as an actual successive pitch in a scale formation but rather as an addition to another intervallic value to form a composite interval.36 From a historical perspective, with regards to the adoption of a theoretical scale by the Arabs and Persians in the fourteenth century and the “so-called messel or octave of seventeen third tones,”37 A. J. Hipkins writes: “The arithmetical reasonings of philosophers who sought to explain the musical scale could never have been, excepting in the larger intervals, the practical art of the musicians.”38

Persian Musical Scholarship in the Twentieth Century

In the sixteenth century, with Shi’a rule imposed by the Sadavid dynasty (1501-1722) generating a certain hostility towards music, Persian musical scholarship enters its period of decline, and it is not until the beginning of the twentieth century during the Pahlavi dynasty (1925-1979) that we begin to see the resurgence of theoretical research into the Persian theory of intervals and scales.39

“In the beginning of the twentieth century, Iran entered a period during which the Western world was idealized. Western music had already been taught in Iran for over three decades; thus, Iranian musicians were thoroughly educated in it,” explains Talai. “In fact, some people began to question the value of Iranian music as serious and dignified, doubting that its instruction should be institutionalized. Others, motivated in part by a spirit of nationalism, tried to revive Persian music. They wanted to create a theory relating it to the perspective of what they regarded as the more progressive, Western classical music.”40

36 Farhat, The Dastgàh Concept in Persian Music 12-13. 37 “La Borde, Villoteau, and Kiesewetter, studying Arab music from obscure treatises of medieval Arab philosophers such as Khalil, Al-Kindå, Ibn Khaldñn, and Al-Fàràbå, maintained that Arab music is based on the so-called messel or octave of seventeen third tones, and such was purely Oriental and fundamentally different from the Greek diatonic system, which is based on tones and semitones. They proceed, therefore, to see the origin of Arab music in Persia, where we find a scale of semitones, demi-semitones, and even semidemi-semitones, their theory being that Persia, after being conquered by the Arabs, had imposed its music upon the conquerors.” For a further discussion, see Joseph Reider, “Jewish and Arabic Music,” The Jewish Quarterly Review 7.4 (Apr., 1917): 640. 38 Francesco Salvador-Daniel, and Henry George Farmer, The Music and Musical Instruments of the Arab: With Introduction on How to Appreciate Arab Music (Portland, ME: Longwood Press, 1976) 182. 39 Farhat, The Dastgàh Concept in Persian Music 5. 40 Dariush Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah, trans. Manoochehr Sadeghi (Coasta Mesa, CA: Mazda, 2000) 7-8. 16 Theory Versus Performance Practice

This ideological shift in effect being the impetus for three separate theories on intervals and scales of Persian music proposed in the twentieth century: the twenty-four equally-tempered quarter-tone scale proposed by Ali Naqi Vaziri in the 1920s, the alternative twenty-two-note scale proposed by Mehdi Barkešli in the 1940s based on Pythagorean principles, as well as the theory of the five primary intervals of performance practice presented by Hormoz Farhat in the 1990 publication of his doctoral thesis The Dastgàh Concept in Persian Music.41 In spite of the almost universal rejection of the proposal for a twenty- four-tone equally-tempered scale, one cannot underestimate Vaziri’s influence to Persian musical scholarship in the twentieth century, as some of the reforms implemented to the art form by the theorist did gain wide acceptance. Vaziri was born in Tehran, in 1887, and following his traditional studies in classical Persian music, went on to study piano, harmony, and voice at the École Supérieure de Musique in Paris, and counterpoint and composition at the Hochschule für Musik in Berlin. Upon his return to Iran in 1923, he founded his conservatory, the Madresse-ye Ali-ye Musiqi, and amongst his many achievements is today credited for the first published transcriptions of Persian music in Western staff notation.42 As theorist, his most notable writings on the twenty-four equally-tempered quarter-tone scale and the Persian modal system are contained within his 1934 publication Musiqi-ye Nazari.43

The Twenty-Four Equally-Tempered Quarter-Tone Scale

The concept of the division of the octave into twenty-four equally-tempered quarter-tones is not a new phenomena to the region, with Lebanese theorist Måkhà’ål Måshàqàh (1800-1889)44 proposing an Arabian quarter-tone system in the previous century.45 Earlier examples include Zalzal (d. after 842) – a prominent ñd teacher of the ninth century who reformed Pythagorean intonation of the time with his introduction of eleven-limit ratios to the fretting of the lute (an intonation system limited to the eleventh harmonic with intervals closely resembling those of equal temperament). The Zalzal intervals included the undecimal grave or small neutral second (12/11, or 150.637 cents), undecimal subfifth (16/11, or 648.682 cents), and undecimal grave or small neutral sixth (18/11, or 852.592 cents); and represents a system subscribed

41 Farhat, The Dastgàh Concept in Persian Music 10. 42 Ella Zonis, “Classical Iranian Music,” Musics of Many Cultures (Berkeley, CA: U. of California Press, 1980) 186-87. 43 Farhat, The Dastgàh Concept in Persian Music 9. 44 Habib Hassan Touma, The Music of the Arabs, trans. Laurie Schwartz (Portland, OR: Amadeus Press, 1957) 19. 45 Måshàqàh’s treatise proposing the twenty-four equally-tempered division of the octave for Turko-Arabian music translated into English by American protestant missionary and scholar Eli Smith (1801-57), and then consequently published in the Journal of the American Oriental Society (1849). For a further discussion, see Alexander J. Ellis, and Alfred J. Hipkins, “Tonometrical Observations on Some Existing Non-Harmonic Musical Scales,” Proceedings of the Royal Society of London 37 (1884): 371. Theory Versus Performance Practice 17 to for seven centuries prior to the reforms of fourteenth century theorists Quðb al-Dån and Abd al Qàdir al-Maràghå (d 1435).46 The twenty-four-tone equally-tempered division of the octave is based on the Western system of twelve-tone equal temperament – the division of the octave into twelve equal intervals, technically referred to as tempered half-tones, while the division of the octave into twenty-four equal intervals, technically referred to as tempered quarter-tones; the frequency ratio of each quarter-tone therefore mathematically representing the twenty-fourth root of two, or in different terms, the distance between any two tones representing twenty-four times the logarithm on the base of two of the frequency ratio.47 The ratio of the equally-tempered quarter-tone may be expressed in mathematical terms as 1: 24 2 =1:1.029302237, or the fraction approximations: 527/512 (17×31/2ù) and 35/34 (5×7/2×17).48 The unit of a cent being the method to further subdivide the semitone, and equal to the twelve- hundredth root of two; with 12 semitones per octave, 1200 cents per octave, and hence, 100 cents per semitone. The ratio of the cent may be expressed in mathematical terms as 1: 1200 2 =1:1.000577789, or approximately 1731/1730. The calculation of cents (a system of measurement devised by Alexander J. Ellis [1814-90]) is obtained via the formula: 1200ïlog2Ïlogf=cents.49 The methodology involved in the construction of a twenty-four equally-tempered quarter-tone scale requires the establishment of twenty-four degrees via the equation f= 24 2 ; the equation producing the figure 1.029302237 (the frequency ratio of one equally-tempered quarter-tone), which when multiplied by the powers of 2, 3, 4, etc., generates the ratios for the remainder of the scale degrees.50 It should be noted that the multiplication by the power of 18 ( [24 ]2 18 ) generates the frequency ratio of

46 “In view of the fact that the division into ‘quarter-tones’ gives two of the ten ratios of 11 with a falsity of only 0.6 cent (12/11, or 150.637 cents; and 11/6, or 1049.363 cents), two with a falsity of only 1.3 cents (11/8, or 551.318 cents; and 16/11, or 648.682 cents), and two with a falsity of 2.6 cents (11/9, or 347.408 cents; and 18/11, or 852.592 cents), the probabilities are that Måshàqàh’s claims for Arabic folk melodies are valid. But the fate of the other ratios of 11, and of the 5 and 7 identities (the 5th and 7th partials of the harmonic series) in ‘quarter-tones’ is another story.” For a further discussion see, Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 426. 47 Olson, Music, Physics and Engineering 46-47. 48 Alain Daniélou, Tableau Comparatif des Intervalles Musicaux (Pondichéry, India: Institut Français d’Indologie, 1958) 14. 49 Hermann L. F. Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music, 2nd ed. (New York: Dover Publications, 1954) 431. 50 “‘Quarter-tones’ are obtained by a simple duplication of the twelve-tone tempered scale a ‘quarter-tone’ higher. The original, since it is the usual scale, has good 3 identities (the 3rd partial of the harmonic series) and two fairly good 9 identities (the 9th partial of the harmonic series), but poor 5 identities (13.7 and 15.6 false). The doubling, a ‘quarter- tone’ higher, gives six of the ten 11 identities (the 11th partial of the harmonic series) almost perfectly, but fails to represent the other four at all; it comes closer to the 7 identities than the original, but not close enough, since they show a maximum falsity of 18.8 cents; it improves the 5s not at all.” For a further discussion see, Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 428. 18 Theory Versus Performance Practice

A=440Hz (1.68179283), or equal major sixth ( 4 ]2[ 3 , approximately 37/22, or 900.000 cents),51 while 440ï1.68179283 presents the relative frequency of middle C (261.6255654Hz), or 1/1. Once data is available for all frequency ratios, it simply becomes a matter of multiplying every ratio by 261.6255654Hz (1/1) in order to obtain the frequency values in hertz (Hz) of every scale degree. The mathematical formula for cents is then utilized to generate the cent values of every scale degree, although unnecessary in equal temperament because of the obvious outcomes.

The Pythagorean Division of the Octave

‘Pythagorean’ intonation, and the generation of a scale based on a series of ‘pure’ fifths (3/2s), is in reference to Greek philosopher Pythagoras (570-504 B.C.), who according to Llewelyn Southworth Lloyd, and Hugh Boyle is “usually credited with the discovery that a vibrating string, stopped at two thirds or one-half of its length, sounds the fifth (3/2) or octave (2/1) of the note it produces when vibrating freely (1/1).”52 The system of intonation “prevailed in China and Arabia almost to the exclusion of anything else,” notes Partch, and “was for many centuries the only system in ecclesiastical Roman and Byzantine music.”53 According to Fritz A. Kuttner, the Pythagorean conceptual basis for the construction of a scale of twelve semitones via the generation of twelve consecutive fifths was recognized in China “several centuries before Pythagoras, probably in the ninth century B.C., or even earlier.” Kuttner explains:

“The twelve semitones (lüs) received pitch names which are mentioned for the first time in the Kuo Yü, a work usually dated towards the end of the Chou Dynasty (c. 1030-722 B.C). However, the Kuo Yü is believed to contain a good deal of much ‘earlier material from ancient written sources,’ so the origin of the pitch names must be assigned to about 900 B.C. or earlier.”54

In order to construct a Pythagorean scale with the necessary intervals to facilitate the general modulations of Western tonal music one must produce twenty-seven distinct pitches.55 Fifteen intervals are generated by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!, B!, F#, C#, and G#; while the other eleven intervals, by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$. The mathematical procedure thus begins with the generation of the first ascending fifth, G; which has a ratio of 3/2 (the just and Pythagorean perfect fifth, measuring 701.955 cents) in relation to the

51 Daniélou, Tableau Comparatif des Intervalles Musicaux 79. 52 Llewelyn Southworth Lloyd, and Hugh Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation (London: McDonald and Jane’s, 1978) 2. 53 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 399. 54 Fritz A. Kuttner, “A Musicological Interpretation of the Twelve Lüs in China’s Traditional Tone System,” Ethnomusicology 9.1 (Jan., 1965): 22. 55 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 433. Table 3. The twenty-four-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ CË / Dì equal quarter-tone 24 2 1.029302 269.292 50.000 +50 ÐÓ CÚ / DÝ equal semitone, or minor second 12 2 1.059463 277.183 100.000 +00 ÐÔ CÍ / Dí equal three-quarter-tone 8 2 1.090508 285.305 150.000 +50 ÐÕ D equal tone 6 2 1.122462 293.665 200.000 +00 ÐÖ DË / Eì five equal quarter-tones (24 2)5 1.155353 302.270 250.000 +50 Ð× DÚ / EÝ equal augmented second, or minor third 4 2 1.189207 311.127 300.000 +00 ÐØ DÍ / Eí seven equal quarter-tones (24 2)7 1.224054 320.244 350.000 +50 ÐÙ E equal major third 3 2 1.259921 329.628 400.000 +00 ÑÐ EË / Fí nine equal quarter-tones (8 2)3 1.296840 339.286 450.000 +50 ÑÑ F equal perfect fourth (12 2)5 1.334840 349.228 500.000 +00 ÑÒ FË / Gì eleven equal quarter-tones (24 2)11 1.373954 359.461 550.000 +50 ÑÓ FÚ / GÝ equal tritone, augmented fourth, or diminished fifth 2 2 1.414214 369.994 600.000 +00 ÑÔ FÍ / Gí thirteen equal quarter-tones (24 2)13 1.455653 380.836 650.000 +50 ÑÕ G equal perfect fifth (12 2)7 1.498307 391.995 700.000 +00 8 5 ÑÖ GË / Aì fifteen equal quarter-tones ( 2) 1.542211 403.482 T 750.000 +50 19 Practice Performance Versus heory Ñ× GÚ / AÝ equal augmented fifth, or minor sixth (3 2)2 1.587401 415.305 800.000 +00 ÑØ GÍ / Aí seventeen equal quarter-tones (24 2)17 1.633915 427.474 850.000 +50 ÑÙ A equal major sixth (4 2)3 1.681793 440.000 900.000 +00 ÒÐ AË / Bì nineteen equal quarter-tones (24 2)19 1.731073 452.893 950.000 +50 ÒÑ AÚ / BÝ equal augmented sixth, or minor seventh (6 2)5 1.781797 466.164 1000.000 +00 ÒÒ AÍ / Bí twenty-one equal quarter-tones (8 2)7 1.834008 479.823 1050.000 +50 ÒÓ B equal major seventh (12 2)11 1.887749 493.883 1100.000 +00 ÒÔ BË / Cí twenty-three equal quarter-tones (24 2)23 1.943064 508.355 1150.000 +50

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 20 Theory Versus Performance Practice fundamental C (1/1). 2/1 (measuring 1200.000 cents) is consequently subtracted from intervals exceeding the 2/1 ratio in order to bring the interval within the octave. These procedures are repeated consecutively, producing the following series of fifteen ascending fifths:

i. G (just and Pythagorean perfect fifth) generated by ratio 3/2 ii. D (just major tone) by the equation (3/2×3/2)/2=9/8 iii. A (Pythagorean major sixth) by 9/8×3/2=27/16 iv. E (Pythagorean major third, or ditone) by (27/16×3/2)/2=81/64 v. B (Pythagorean major seventh) by 81/64×3/2=243/128 vi. F! (Pythagorean tritone, or augmented fourth) by (243/128×3/2)/2=729/512 vii. C! (Pythagorean apotome) by (729/512×3/2)/2=2187/2048 viii. G! (augmented fifth) by 2187/2048×3/2=6561/4096 ix. D! (augmented second) by (6561/4096×3/2)/2=19683/16384 x. A! (augmented sixth) by 19683/16384×3/2=59049/32768 xi. E! (augmented third) by (59049/32768×3/2)/2=177147/131072 xii. B! (Pythagorean comma) by (177147/131072×3/2)/2=531441/524288 xiii. F# (double augmented fourth) by 531441/524288×3/2=1594323/1048576 xiv. C# (double augmented octave) by (1594323/1048576×3/2)/2=4782969/4194304 xv. G# (double augmented fifth) by 4782969/4194304×3/2=14348907/8388608

The process is then reversed, with the generation of the first descending fifth (F) by the mathematical equation of 4/3 (the just and Pythagorean perfect fourth, measuring 498.045 cents). These procedures are repeated consecutively, producing the following concluding series of eleven descending fifths:

i. F (just and Pythagorean perfect fourth) generated by the ratio 4/3, or 2/3×2=4/3 ii. B" (just minor seventh) by the equation (4/3×2/3)×2=16/9 iii. E" (Pythagorean minor third, or trihemitone) by 16/9×2/3=32/27 iv. A" (Pythagorean minor sixth) by (32/27×2/3)×2=128/81 v. D" (Pythagorean limma) by 128/81×2/3=256/243 vi. G" (diminished fifth) by (256/243×2/3)×2=1024/729 vii. C" (diminished octave) by (1024/729×2/3)×2=4096/2187 viii. F" (diminished fourth) by 4096/2187×2/3=8192/6561 ix. B$ (diminished sixth) by (8192/6561×2/3)×2=32768/19683 x. E$ (diminished third) by 32768/19683×2/3=65536/59049 xi. A$ (diminished sixth) by (65536/59049×2/3)×2=262144/177147 Table 4. The Pythagorean division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ BÚ Pythagorean comma (A) XII 531441/524288 1.013643 265.195 23.460 +23 ÐÓ DÝ Pythagorean limma, or diatonic semitone (D) V 256/243 1.053498 275.622 90.225 ß10 ÐÔ CÚ Pythagorean apotome, or chromatic semitone (A) VII 2187/2048 1.067871 279.382 113.685 +14 ÐÕ EÞ Pythagorean diminished third (D) X 65536/59049 1.109858 290.367 180.450 ß20 ÐÖ D just major tone (A) II (9th harmonic) 9/8 1.125000 294.329 203.910 +04 Ð× CÛ Pythagorean double augmented octave (A) XIV 4782969/4194304 1.140349 298.344 227.370 +27 ÐØ EÝ Pythagorean minor third, or trihemitone (D) III 32/27 1.185185 310.075 294.135 ß06 ÐÙ DÚ Pythagorean augmented second (A) IX 19683/16384 1.201355 314.305 317.595 +18 ÑÐ FÝ Pythagorean diminished fourth (D) VIII 8192/6561 1.248590 326.663 384.360 ß16 ÑÑ E Pythagorean major third, or ditone (A) IV (81st harmonic) 81/64 1.265625 331.120 407.820 +08 ÑÒ F just and Pythagorean perfect fourth (D) I 4/3 1.333333 348.834 498.045 ß02 ÑÓ EÚ Pythagorean augmented third (A) XI 177147/131072 1.351524 353.593 521.505 +22 ÑÔ GÝ Pythagorean diminished fifth (D) VI 1024/729 1.404664 367.496 588.270 ß12 ÑÕ FÚ Pythagorean tritone, or augmented fourth (A) VI 729/512 1.423828 372.510 611.730 +12 ÑÖ AÞ Pythagorean diminished sixth (D) XI 262144/177147 1.479811 387.156 678.495 ß22 Ñ× G just and Pythagorean perfect fifth (A) I (3rd harmonic) 3/2 1.500000 392.438 701.955 +02

ÑØ FÛ Pythagorean double augmented fourth (A) XIII 1594323/1048576 1.520465 397.792 725.415 +25 T er essPromnePatc 21 Practice Performance Versus heory ÑÙ AÝ Pythagorean minor sixth (D) IV 128/81 1.580247 413.433 792.180 ß08 ÒÐ GÚ Pythagorean augmented fifth (A) VIII 6561/4096 1.601807 419.074 815.640 +16 ÒÑ BÞ Pythagorean diminished seventh (D) IX 32768/19683 1.664787 435.551 882.405 ß18 ÒÒ A Pythagorean major sixth (A) III (27th harmonic) 27/16 1.687500 441.493 905.865 +06 ÒÓ GÛ Pythagorean double augmented fifth (A) XV 14348907/8388608 1.710523 447.517 929.325 +29 ÒÔ BÝ Pythagorean minor seventh (D) II 16/9 1.777778 465.112 996.090 ß04 ÒÕ AÚ Pythagorean augmented sixth (A) X 59049/32768 1.802032 471.458 1019.550 +20 ÒÖ CÝ Pythagorean diminished octave (D) VII 4096/2187 1.872885 489.995 1086.315 ß14 Ò× B Pythagorean major seventh (A) V 243/128 1.898438 496.680 1109.775 +10

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 22 Theory Versus Performance Practice

Alain Daniélou’s Scale of Fifths

An alternative method for deriving a Pythagorean scale is Alain Daniélou’s ‘scale of fifths’ (the cyclic division of the octave), as presented by Daniélou in Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness (1995) – the current edition of his 1943 monumental work Introduction to the Study of Musical Scales. The theoretical basis for the ‘cyclic division of the octave’ is the ascending scale of fifths.56 The theory states that when a length of bamboo (called ‘81 parts’ for symbolic reasons) is reduced by a third (2/3), it will produce the perfect fifth (3/2, or 54 parts). If the process is continued by increasing the length of bamboo by a third (4/3), and then alternated between the two ratios, it will result in the following five-note series of pipe lengths and frequency ratios: 81 (1/1), 54 (3/2), 72 (9/8), 48 (27/16), and 64 (81/64). The juxtaposition of two sets of six lü, or pitch pipes a Pythagorean major half-tone apart generates a Pythagorean twelve-tone scale, or twelve lü (alternatively achieved via the simple expansion of the series of consecutive fifths initially forming a heptatonic scale). Han Dynasty historian Ssu-ma Chi’en (145-86 B.C.) attributes the “mathematical formula for the pentatonic scale to Ling Lun, minister or court musician under Emperor Huang-Ti, of the twenty-seventh century B.C.”57 Chinese theorist King Fâng (c. 45 B.C.) followed the series up to the sixtieth sound (equal to fifty-nine consecutive fifths), although mainly for symbolic reasons. The fifty-third fifth is the cyclic octave (3õó/2øô, or 3.615 cents), but the series being of an infinitive nature, it is musically impractical to follow it beyond this given point, so it is rounded off to the ratio of 2/1. Fang based his system of the sixty lü on the “analogy of the eight gua or mystical trigrams of the I ching, which united two by two, form sixty-four distinct combinations.” Sixty-four hexagrams allow for the “representation of all aspects of existence.” The twelve original lü, multiplied by five (the number of elements), also equal to sixty lü. Daniélou makes the following statement with regards to the infinite nature of the series:

“However, after the 52nd fifth, the octave is filled up and the 53rd fifth (note 54) comes out of the octave and inserts itself between the octave C (2/1) and the twelfth fifth C¢ (531441/524288), thus forming, above C (1/1), a small interval of 0.84 savarts (3.349 cents [1 savart = 3.986313725 cents, or 1 cent =

56 “Alain Daniélou (1907-94), the founder of the International Institute for Comparative Musicology in Berlin, elucidated for tens of thousands of readers the meanings of the arts and religious traditions of both East and West. He was an accomplished player on the vånà and taught in the music department at the University of Benares. His numerous books, the product of a career spanning six decades, include The Myths and Gods of India; Gods of Love and Ecstasy; While the Gods Play; Virtue, Success, Pleasure, and Liberation; The Phallus; Mastering the Secrets of Matter and the Universe; and The Complete Kàma Sñtra.” For a further discussion, see Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 35. 57 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 362. Theory Versus Performance Practice 23

0.250858329 savarts]).58 Thus begins a new cycle, which in its turn, with a period of 53 fifths, will divide the octave into small intervals of 0.84 savarts. The next cycle will appear during the seventh series of 53, when the 359th fifth (note 360) comes out of the octave and forms, with C (1/1), an interval of 0.47 savarts (1.874 cents). The next cycle is of 666 notes, with a basic interval of 0.035 savarts (0.140 cents). Then comes a cycle of 25,524 notes with a basic interval of 0.0021174 savarts (0.008 cents). This cycle is very near to that of the precession of the equinoxes, or the Pythagorean great year, which is of 25,920 solar years. The small difference between the twelfth fifth and octave, similar to that of the lunar and the solar year, leaves the door open for further cycles.”59

Table 5. Some correspondences of the first twelve lü

NOTE NAME OF SEVEN FUNDAMENTAL ANNUAL GENDER MOON LÜ BEGINNINGS FOR SACRIFICES CYCLE C huang zhong (I) heaven to the yellow emperor, winter masculine 11th (December) to heaven solstice D¸ da lü (VIII) feminine 12th (January)

D tai cu (male form man to the white emperor, spring masculine 1st (February) of jia zhong (III) to ancestors equinox EÝ jia zhong or feminine 2nd (March) yuan zhong (X) E¢ gu xian (V) spring to the green emperor, masculine 3rd (April) to the first agriculturists F¢ zhong lü (XIII) feminine 4th (May)

F¥ rui bin (VII) summer summer masculine 5th (June) solstice G lin zhong or earth to the red emperor, feminine 6th (July) han zhong (II) to earth AÝ yi tze (IX) masculine 7th (August)

A¢ nan lü (IV) autumn to the black emperor, to autumn feminine 8th (September) the empress, to the moon equinox BÝ wu yi (XI) masculine 9th (October)

B¢ ying zhong (VI) winter feminine 10th (November)

58 “An alternative method (to cents) of interval measurement, but one used far less frequently, is the savart. Named after the French scientist Félix Savart (1791-1841), this method assigns a total of 25 savarts to each semitone (or 25.08583297), one savart being approximately equal to four cents.” For a further discussion, see Read, 20th-Century Microtonal Notation 7. 59 Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 20-50. 4Ter essPromnePractice Performance Versus Theory 24 Table 6. Alain Daniélou’s scale of fifths (the cyclic division of the octave)

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison 1ñ/1ñ 1.000000 261.626 0.000 +00 cyclic octave (A) LIII 3õó/2øô 1.002090 262.172 3.615 +04 ÐÒ C¢ Pythagorean comma (A) XII 3ñò/2ñù 1.013634 265.195 23.460 +23 ÐÓ C£ great diesis (A) XXIV 3òô/2óø 1.027473 268.813 46.920 +47 ÐÔ CÚ cyclic grave or small chromatic semitone, or minor half-tone (A) XXXVI 3óö/2õ÷ 1.041491 272.481 70.380 ß30 ÐÕ D¹ cyclic Pythagorean limma (A) XLVIII 3ôø/2÷ö 1.055700 276.198 93.840 ß06 ÐÖ D¸ Pythagorean apotome, diatonic semitone, or major half-tone (A) VII 3÷/2ññ 1.067871 279.382 113.685 +14 Ð× DÝ cyclic great limma, acute or large half-tone (A) XIX 3ñù/2óð 1.082440 283.194 137.145 +37 ÐØ D³ cyclic grave or small tone (A) XXXI 3óñ/2ôù 1.097208 287.058 160.605 ß39 ÐÙ D² cyclic minor tone (A) XLIII 3ôó/2öø 1.112178 290.974 184.065 ß16 ÑÐ D just major tone (A) II (9th harmonic) 3ò/2ó 1.125000 294.329 203.910 +04 55th cyclic fifth (A) LV 3õõ/2ø÷ 1.127352 294.944 207.525 +08 ÑÑ D¢ acute or large tone (A) XIV 3ñô/2òò 1.140349 298.344 227.370 +27 ÑÒ D£ cyclic supermajor second (A) XXVI 3òö/2ôñ 1.155907 302.415 250.830 +51 ÑÓ DÚ cyclic augmented second (A) XXXVIII 3óø/2öð 1.171677 306.541 274.290 ß26 ÑÔ E¸ cyclic Pythagorean minor third, or trihemitone (A) L 3õð/2÷ù 1.187663 310.723 297.750 ß02 ÑÕ EÝ cyclic minor third (A) IX 3ù/2ñô 1.201355 314.305 317.595 +18 ÑÖ E³ cyclic neutral third (A) XXI 3òñ/2óó 1.217745 318.593 341.055 ß59 Ñ× E² cyclic grave or small major third (A) XXXIII 3óó/2õò 1.234359 322.940 364.515 ß35 ÑØ E cyclic major third (A) XLV 3ôõ/2÷ñ 1.251200 327.346 387.975 ß12 ÑÙ E¢ Pythagorean major third, or ditone (A) IV (81st harmonic) 3ô/2ö 1.265625 331.120 407.820 +08 57th cyclic fifth (A) LVII 3õ÷/2ùð 1.268271 331.812 411.435 +11 ÒÐ E£ cyclic acute or large major third (A) XVI 3ñö/2òõ 1.282892 335.637 431.280 +31 ÒÑ F³ cyclic subfourth (A) XXVIII 3òø/2ôô 1.300395 340.217 454.740 ß45 ÒÒ F² cyclic grave or small fourth (A) XL 3ôð/2öó 1.318137 344.858 478.200 ß22 ÒÓ F cyclic perfect fourth (A) LII 3õò/2øò 1.336120 349.563 501.660 +02 ÒÔ F¢ cyclic acute or large fourth (A) XI 3ññ/2ñ÷ 1.351524 353.593 521.505 +22 ÒÕ F£ cyclic superfourth (A) XXIII 3òó/2óö 1.369964 358.417 544.965 +45 ÒÖ FÚ cyclic grave or small augmented fourth (A) XXXV 3óõ/2õõ 1.388654 363.307 568.425 ß32 DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) Ò× F¤ cyclic tritone, or augmented fourth (A) XLVII 3ô÷/2÷ô 1.407600 368.264 591.885 ß08 ÒØ F¥ Pythagorean tritone, or augmented fourth (A) VI 3ö/2ù 1.423828 372.510 611.730 +12 59th cyclic fifth (A) LIX 3õù/2ùó 1.426804 373.289 615.345 +15 ÒÙ GÝ cyclic acute or large diminished fifth (A) XVIII 3ñø/2òø 1.443254 377.592 635.190 +35 ÓÐ G³ cyclic subfifth (A) XXX 3óð/2ô÷ 1.462944 382.744 658.650 ß41 ÓÑ G² cyclic grave or small fifth (A) XLII 3ôò/2öö 1.482904 387.966 682.110 ß18 ÓÒ G just and Pythagorean perfect fifth (A) I (3rd harmonic) 3ñ/2ñ 1.500000 392.438 701.955 +02 54th cyclic fifth (A) LIV 3õô/2øõ 1.503135 393.259 705.570 +06 ÓÓ G¢ cyclic acute or large fifth (A) XIII 3ñó/2òð 1.520465 397.792 725.415 +25 ÓÔ G£ cyclic superfifth (A) XXV 3òõ/2óù 1.541209 403.220 748.875 +49 ÓÕ GÚ cyclic augmented fifth (A) XXXVII 3ó÷/2õø 1.562236 408.721 772.335 ß28 ÓÖ A¸ cyclic minor sixth (A) XLIX 3ôù/2÷÷ 1.583550 414.297 795.795 ß04 Ó× AÝ Pythagorean minor sixth (A) VIII 3ø/2ñò 1.601807 419.074 815.640 +16 ÓØ A³ cyclic neutral sixth (A) XX 3òð/2óñ 1.623661 424.791 839.100 ß61 ÓÙ A² cyclic grave or small major sixth (A) XXXII 3óò/2õð 1.645813 430.587 862.560 ß37 ÔÐ A cyclic major sixth (A) XLIV 3ôô/2öù 1.668267 436.461 886.020 ß14 ÔÑ A¢ Pythagorean major sixth (A) III (27th harmonic) 3ó/2ô 1.687500 441.493 905.865 +06 56th cyclic fifth (A) LVI 3õö/2øø 1.691027 442.416 909.480 +09 ÔÒ A£ cyclic acute or large major sixth (A) XV 3ñõ/2òó 1.710523 447.517 929.325 +29 ÔÓ AÚ cyclic augmented sixth (A) XXVII 3ò÷/2ôò 1.733860 453.622 952.785 ß47 ÔÔ B¹ cyclic grave or small minor seventh (A) XXXIX 3óù/2öñ 1.757516 459.811 976.245 ß24

ÔÕ B¸ (A) LI 1.781494 466.084 999.705 T cyclic minor seventh 3õñ/2øð +00 25 Practice Performance Versus heory ÔÖ BÝ cyclic acute or large minor seventh (A) X 3ñð/2ñõ 1.802032 471.458 1019.550 +20 Ô× B³ cyclic neutral seventh (A) XXII 3òò/2óô 1.826618 477.890 1043.010 ß57 ÔØ B² cyclic grave or small major seventh (A) XXXIV 3óô/2õó 1.851539 484.410 1066.470 ß34 ÔÙ B cyclic diatonic major seventh (A) XLVI 3ôö/2÷ò 1.876800 491.019 1089.930 ß10 ÕÐ B¢ Pythagorean major seventh (A) V 3õ/2÷ 1.898480 496.680 1109.775 +10 58th cyclic fifth (A) LVIII 3õø/2ùñ 1.902406 497.718 1113.390 +13 ÕÑ B£ cyclic acute or large major seventh (A) XVII 3ñ÷/2òö 1.924338 503.456 1133.235 +33 ÕÒ C³ cyclic suboctave (A) XXIX 3òù/2ôõ 1.950593 510.325 1156.695 ß43 ÕÓ C² cyclic grave or small octave (A) XLI 3ôñ/2öô 1.977205 517.287 1180.155 ß20

ÐÑ C octave 2ñ/1ñ 2.000000 523.251 1200.000 +00 26 Theory Versus Performance Practice

The Twenty-Two Note Division of the Octave

The twenty-two note division of the octave was conceptualized by Persian scholar and physicist Barkešli in the 1940s, following his personal analysis of contemporary tradition, and is based on the Pythagorean cycle of fifths, as well as the writings of medieval theorists Éafå al-Dån Urmawå and Abu Nasr Fàràbå. After scientific analysis of intervals in recordings of vocal music by five respected traditional musicians, Barkešli concluded that the intervallic structure of the whole-tone and semitone in Persian music were reasonably fixed, and closely resembled the first tetrachord of the Pythagorean diatonic model with the figures: 206+204+89=499 cents. In exact Pythagorean terms this is expressed as 203.910 (two limmas and a comma) + 203.910 + 90.225 (limma) = 498.045, or 9/8×9/8×256/243=4/3. In other tetrachords he also identified a “120-cent interval as the characteristic interval of Persian music”, which is close to a Pythagorean apotome (2187/2048, or 113.685 cents), as well another less popular interval measuring 181 cents similar to a Pythagorean diminished third (65536/59049, or 180.450 cents). Éafå al-Dån’s seventeen-note gamut, with its whole-tone pattern of L, L + L, and L + L + C is therefore extended by Barkešli into one exhibiting the following intervallic pattern: L, L + C, L + L, and L + L + C; or in more elaborate terms as: 256/243, 256/243×531441/524288=2187/2048 (Pythagorean apotome), 256/243×256/243=65536/59049 (Pythagorean diminished third), and L + L + C = 9/8, or just major tone.

Table 7. Mehdi Barkešli’s division of the whole-tone

PATTERN L L + C L + L L + L + C INTERVAL Pythagorean Pythagorean Pythagorean just limma apotome diminished third major tone RATIO 256/243 2187/2048 65536/59049 9/8 CENTS 90.225 113.685 180.450 203.910

It may therefore be summarized that the first eight intervals of the twenty-two-note division of the octave are derived from the construction of a Pythagorean C mixolydian scale (the pattern of a ratio of 3/2, or ascending fifths generating the pitches G, D, A, E, and B, while the ratio 4/3, or descending fifths generating the pitches F and B"). The basic design essentially allows for the formation of both major (C, D, E, and F) and minor (G, A, B", and C) tetrachords. Another fourteen intervals are then derived from the construction of a whole-tone based on the four-microtone model of L, L + L, and L + L + C. The twenty-two-note division of the octave is characterized by the following intervallic pattern: L, C, DDT, C – L, C, DDT, C – L + L, C, DDT, C – L, C, DDT, C – L, C, DDT, C – L. ‘DDT’, or Pythagorean double diminished Theory Versus Performance Practice 27 third (134217728/129140163, or 66.765 cents) represents the interval between the Pythagorean diminished third (65536/59049) and apotome (2187/2048).

Ex. 2. Subdivision of the tetrachord and the 22-note division of the octave (Mehdi Barkešli, 20th century)

The Theory of Flexible Intervals

The third proposition for a theory on intervals and scales of Persian music – the theory of flexible intervals, or of the five primary intervals of performance practice – advocated by Farhat in opposition to both twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and Barkešli. Farhat offers the following critical assessment:

“Both theories suffer equally from a tendency to accommodate certain Western concepts. Each theory, by suggesting very exact intervals, remains oblivious to the fluidity and flexibility of Persian intervals. Vaziri did not take account of this instability, as his apparent objective was to make Persian music adhere to a process of equal temperament so that it can be harmonized. Barkešli, on the other hand, was not interested in the Westernisation of Persian music but was committed to prove that today’s music is still rooted in the medieval system. He has taken the exactness of the medieval theory very seriously – as have many others – and has proposed a system vested with even greater precision.”

Utilizing a stroboconn and a melograph as measurement devices, and fretted instruments (tàrs and setàrs) as the sources, Farhat identifies five primary intervals, with semitones and whole-tones representative of relatively fixed intervals, while the rest, of a flexible nature. His classification of Persian intervals include: (1) the semitone or minor second, measuring approximately 90 cents; (2) the small neutral tone, measuring between 125 and 145 cents, or the average of 135 cents; (3) the large neutral tone, measuring between 150 and 170 cents, or the average of 160 cents; (4) the whole-tone or major second, measuring approximately 204 cents; and (5) the plus-tone, measuring approximately 270 cents.60

60 Farhat, The Dastgàh Concept in Persian Music 10-16. 8Ter essPromnePractice Performance Versus Theory 28 Table 8. Mehdi Barkešli’s twenty-two-note division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ CÎ superoctave [6] (D) V 256/243 1.053498 275.622 90.225 ß10 ÐÓ CÚ Pythagorean apotome, or chromatic semitone [7] (A) VII 2187/2048 1.067871 279.382 113.685 +14 ÐÔ Dî neutral second [8] (D) X 65536/59049 1.109858 290.367 180.450 ß20 ÐÕ D just major tone (A) II (9th harmonic) 9/8 1.125000 294.329 203.910 +04 ÐÖ DÎ supermajor second [9] (D) III 32/27 1.185185 310.075 294.135 ß06 Ð× DÚ Pythagorean augmented second [10] (A) IX 19683/16384 1.201355 314.305 317.595 +18 ÐØ Eî neutral third [11] (D) VIII 8192/6561 1.248590 326.663 384.360 ß16 ÐÙ E Pythagorean major third, or ditone (A) IV (81st harmonic) 81/64 1.265625 331.120 407.820 +08 ÑÐ F just and Pythagorean perfect fourth (D) I 4/3 1.333333 348.834 498.045 ß02 ÑÑ FÎ superfourth [12] (D) VI 1024/729 1.404664 367.496 588.270 ß12 ÑÒ FÚ Pythagorean tritone, or augmented fourth [13] (A) VI 729/512 1.423828 372.510 611.730 +12 ÑÓ Gî subfifth [14] (D) XI 262144/177147 1.479811 387.156 678.495 ß22 ÑÔ G just and Pythagorean perfect fifth (A) I (3rd harmonic) 3/2 1.500000 392.438 701.955 +02 ÑÕ GÎ superfifth [1] (D) IV 128/81 1.580247 413.433 792.180 ß08 ÑÖ GÚ Pythagorean augmented fifth [2] (A) VIII 6561/4096 1.601807 419.074 815.640 +16 Ñ× Aî neutral sixth [3] (D) IX 32768/19683 1.664787 435.551 882.405 ß18 ÑØ A Pythagorean major sixth (A) III (27th harmonic) 27/16 1.687500 441.493 905.865 +06 ÑÙ BÝ Pythagorean minor seventh (D) II 16/9 1.777778 465.112 996.090 ß04 ÒÐ AÚ Pythagorean augmented sixth [4] (A) X 59049/32768 1.802032 471.458 1019.550 +20 ÒÑ Bî neutral seventh [5] (D) VII 4096/2187 1.872885 489.995 1086.315 ß14 ÒÒ B Pythagorean major seventh (A) V 243/128 1.898438 496.680 1109.775 +10

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 Theory Versus Performance Practice 29

Table 9. Five-limit approximation of Hormoz Farhat’s division of the whole-tone

PATTERN L L + TSC L + GJCS L + L + C INTERVAL Pythagorean great limma, acute or grave or just limma large half-tone small tone major tone RATIO 256/243 27/25 800/729 9/8 CENTS 90.225 133.238 160.897 203.910

The five-limit approximation of Farhat’s division of the whole-tone exhibits the following intervallic pattern: L, L + TSC, L + GJCS, and L + L + C. ‘TSC’, or two syntonic commas (also referred to as the Mathieu

0.5 81 superdiesis, and produced by the factors 80 and 6561/6400, or 43.013 cents), and ‘GJCS’, or grave just chromatic semitone (25/24, or 70.672 cents) represent the intervals between the Pythagorean great limma (27/25, or 133.238 cents) and limma (256/243), and grave or small tone (800/729) and limma (256/243) respectively. The interval between the great limma (27/25) and limma (256/243) is recognized by Manuel Op de Coul as the minimal diesis (20000/19683, or 27.660 cents).61 Farhat’s research concludes that “in Persian music, intervals are often unstable: they tend to fluctuate, within a certain latitude, depending on the mode and according to the performer’s tastes and inclination.” Farhat also acknowledges that the organization of melodic material beyond a tetrachord or pentachord, and the actual notion of a ‘Persian scale’ has “no practical applications” in Persian music, and is therefore misleading.62 The statement that, “these theories were written by eminent scholars who, in most cases, were not practicing musicians,” additionally suggestive of the divide between theorists and practitioners; a point further marked by Farhat’s insistence that “today’s musical traditions do not support the exactitude of those theories.”63 With regards to the relationship between frequency ratios, string lengths, or cents and the performer, it is interesting to note that as Scott Marcus suggests:

“The term ‘comma’ is not meant to signify an interval of any specific size and is thus never defined in terms of frequency ratios, string lengths, or cents. Rather, like the term æruti as it occurs in North Indian music today, kñmà is used to refer to slight changes in the pitch of a given note. Thus when a specific note is said to be ‘minus a comma’, it means only that the note is slightly lowered from what is perceived to be the note’s standard position.”64

61 Manuel Op de Coul, “Huygens-Fokker: List of intervals,” Huygens-Fokker Foundation: Centre for Microtonal Music, 2006, Huygens-Fokker Foundation, 22 Nov. 2006, . 62 Farhat, “Iran: Classical Traditions,” The New Grove Dictionary of Music and Musicians 532. 63 Farhat, The Dastgàh Concept in Persian Music 15. 64 Scott Marcus, “The Interface Between Theory and Practice: Intonation in Arab Music,” Asian Music 24.2 (Spring- Summer, 1993): 41. 30 Theory Versus Performance Practice

Àzàdeh for Santñr and Tape

Àzàdeh for santñr and tape – composed for Iranian santñrist Qmars Piraglu – features the Persian santär (a 72-string [or 18 quadruple-stringed] box zither), and will hereby serve as a practical study of Persian tuning systems, with its presentation of both ‘theoretical’ and ‘performance practice’ tunings. Within the context of an analysis of the tuning methodology of performer Qmars Piraglu, an attempt will be made to make comparisons not just with the three propositions outlined previously, but also with other existing alternative tuning systems. This discussion presented in order to generate a greater understanding of the musical processes that govern the theory of Persian intervals, at the same time disclosing some of the strategies employed in the general contemporary ‘cross-cultural’ compositional process. The enquiry into the basic structure of the Persian modal system reveals some of this music’s essential characteristics, while also providing the necessary organisational framework for the composition.

The Artist

Iranian artist Qmars Piraglu (formerly Siamak Noory) was born in Tehran in 1961, and initiated his musical education at the age of five via Orff instruments (children’s instruments designed by German composer Carl Orff [1895-1982] such as the xilophon, metalophon, and vibraphon).65 Following a seven-year period, he then adopted the Persian santär (a 72-string [or 18 quadruple-stringed] box zither), initially learning the instrument under the guidance of Ms Farzaneh Noshad (a student at the Persian Traditional Music Conservatory in Tehran). Three years later, he is accepted as a student of “one of Iran’s greatest contemporary composers and master santärists,” Ostàd Faràmarz Pàyvar (1933-). For the next eight years Qmars develops his knowledge and understanding of the complete Persian classical music repertoire. As well as this, he synchronically studies Western music – the pianoforte with Taher Djalili (a pianist from the Music Conservatory), and for a year, the bassoon with Khosrow Soltani. In 1979, the Islamic revolution in Iran bestowed the study of music with an illegal status, consequently bringing about the premature closure of the Music Conservatory. “Because of the Islamic revolution in 1979, which caused an undemocratic situation, more than three million Iranians became obliged to escape from the country. Of these three million nearly sixty thousand came to Sweden during the 1980s. More than a million people fled to USA, and the rest mostly went to Germany, France, and other European countries,” explains Qmars. After the fact, for his own ‘anti-establishment’ beliefs in

65 German composer Carl Off (1895-1982) developed Orff-Schilwerk, “a system of music education intended for groups of children singing and playing together,” which emphasized “the development of creativity and the ability to improvise.” For a further discussion, see Randel, ed., The New Harvard Dictionary of Music 577-78. Theory Versus Performance Practice 31 artistic freedom, he too becomes a victim of the ongoing institutionalised persecution, and in 1981 is consequently arrested and imprisoned for a period of two years. Qmars ultimately escaped from Iran in 1986 (during the Iran-Iraq war of the eighties), arriving in Turkey as a refugee, to then temporarily settle in Belgium, where he studied the piano at the Music Academy in Antwerp with Hedvig Vanvarenberg, before establishing a permanent base in Sweden in 1989. Here he studied the piano for two years at the Birkagårdens Folkhögskolan in Stockholm with Stella Tchaikowsky, and went on to graduate from the University of Göteborg with a Bachelor of Science in Musicology. Since 2001, Qmars has been living in Melbourne, Australia. In 2003, he completed a Master of Music (MMus) degree at the University of Melbourne, and is currently undertaking a Doctor of Philosophy (PhD) degree at Monash University.66

Fig. 1. Andrián Pertout, Qmars Piraglu.67

The Instrument

The Persian or Iranian santär is an integral part of the traditional orchestra, sharing the lute repertoire of the tàr and setàr. It is also utilized in the motrebi ‘entertainment music’ genre, but folk styles are excluded. An article by Jean During, Scheherazade Q. Hassan, and Alastair Dick describes the santär’s construction thus:

66 Andrián Pertout, “Siamak Noory: The Santurist – Part 1,” interview with Siamak Noory (Qmars Piraglu) Mixdown 97 (May 2002): 25. 67 Qmars Piraglu, personal photograph of Andrián Pertout, 22 Oct. 2006. 32 Theory Versus Performance Practice

“The santär consists of a trapeziform case made of walnut wood, approximately 90cm wide at the broad end, 35cm wide at the narrow end and 6cm deep. The sides form an angle of 45 degrees to the wider end. The strings are fixed to hitch-pins along the left-hand side and wound round metal wrest-pins on the right by means of which they are tuned with a tuning-key. Each quadruple set of strings rests on a movable bridge of hardwood (kharak). These bridges are aligned almost parallel with the sides of the case. The right-hand rank corresponds to the bass strings and that on the left to the treble strings. In the centre of the santär the low-pitched strings on the right cross the high-pitched strings on the left.”

Adding to this, the instrument features three courses of strings (the bass strings, made of brass, while the treble ones, steel), with a total of 72 strings, or 18 groups of strings, capable of producing 27 different pitches, and is played “by striking the strings with two hammers (mezràb) held in three fingers of each hand.”68

Ex. 3. Range of the santñr

Persian 72-string (or 18 quadruple-stringed) box zither

The Persian Modal System

An important aspect of understanding the basic concept behind the Persian modal system is the radif (row, series), being the term used to describe the complete collection of melodies that constitute the repertoire of Persian traditional music, as well as the separate issue of melodic patterns associated with each individual mode represented within the subsystem of the twelve dastgàhs.69 According to Talai:

“To understand the radif, we must first understand that it is something different from the modal system. This repertory is not like Western art music, which is composed and intended to be played exactly as written. The radif consists of traditional melodies, many of which are derived from popular and folk sources, and whose origins have been obscured with the passage of time.”70

On the other hand, the notion of dastgàh (organisation, system) – the subsystem of the radif – further delineates the twelve groupings of modes, with their own collection of associated melodies, or guêes (corner, section, piece). The introductory section of each mode referred to as the daràmad (opening, introduction), while the conclusive, as the forñd (descent, cadence). Another factor of note includes

68 Jean During, Scheherazade Q Hassan, and Alastair Dick, “Santñr,” The New Grove Dictionary of Musical Instruments, ed. Stanley Sadie, vol. 3 (London: Macmillan Reference, 1984) 291-92. 69 Farhat, The Dastgàh Concept in Persian Music 21. 70 Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah 4. Theory Versus Performance Practice 33 the specific role of certain tones within a mode, which include the ist, or ‘stop’ note used to conclude phrases; the šàhed (witness), or prominent note; as well as the àqàz (beginning), or introductory note in improvisation. The moteqayyer (changeable), or variable note further represents a tone reintroduced to a mode in an altered form. It is quite common for example for a mode to utilize A koron (a microtonal inflection on the tone of A) in the bottom registers, while A flat, in the top; or for there to be two microtonal possibilities for a note within the same register, with the use of one or both pitches governed by attributes of a particular guše.71 Habib Hassan Touma offers the following discussion with regards to the compositional and improvisational nature of the dastgàh system in the music of the Middle East:

“The realization of a truly convincing and original maqam requires a creative faculty like that of a composer of genius. Nevertheless, this phenomenon can be considered only partly as a composed form, because no maqam, makam, mugam, or dastgàh can be identical to any other. Each time it is re-created as a new composition. The compositional aspect is demonstrated in the pre-determined tonal-spatial organization of a fixed number of tone-levels without repetitions, while the improvisation aspect unfolds itself freely in the rhythmic-temporal scheme. Thus, the interplay of composition and improvisation is one of the most distinctive features of the maqam phenomenon.”72

The presentation of the twelve dastgàhs that follows (the basic pitch material of the modes) adheres to the range of the santñr, and utilizes Vaziri’s system of accidentals, which was originally intended for the microtonal sharpening and flattening of tones in equal quarter-tone ( 24 2 ) increments, but consequently adopted as a general standard for Persian music notation.73

Fig. 2. The Ali Naqi Vaziri notation system (the accidentals of Persian music)

sori – raised by one Pythagorean limma (90.225 cents)

sharp – raised by one limma and one Pythagorean comma (113.685 cents)

koron – lowered by one Pythagorean limma (90.225 cents)

flat – lowered by one limma and one Pythagorean comma (113.685 cents)

71 Farhat, The Dastgàh Concept in Persian Music 19-26. 72 Habib Hassan Touma, “The Maqam Phenomenon: An Improvisation Technique in the Music of the Middle East,” Ethnomusicology 15.1 (Jan., 1971): 47. 73 Zonis, Classical Persian Music: An Introduction 56-57. 34 Theory Versus Performance Practice

Ex. 4. The twelve dastgàhs i. Šhñr

ii. Dašhtå

iii. Abñ atà

iv. Màhñr

v. Ràst-Panjgàh

Theory Versus Performance Practice 35 vi. Homàyñn

vii. Bayàt-e Eéfahàn

viii. Bayàt-e Tork

ix. Segàh

x. Áahàrgàh

36 Theory Versus Performance Practice xi. Afšhàrå

xii. Navà

Tuning Analysis Protocols

The obvious consequential procedure in the compositional process now being the selection of the appropriate dastgàh, which is an exercise delivering an outcome based on definite instinctive principles, although assisted somewhat by the audition of modes in workshops organized with Piraglu. Upon the decision to select dastgàh-e segàh, it simply becomes a matter of tuning the santñr to concert performance standard so as to acquire twenty-seven naturally decaying (approximately six seconds each) samples. The sound of every string therefore captured in the digital domain (in this case utilizing an Audio-Technica AT4050/CM5 condenser microphone [with switchable cardioid, omnidirectional, or figure-8 operation, and a frequency response from 20Hz-20kHz], Mackie 32×8×2 8-Bus Mixing Console, and a Fostex D-160 Digital Multitrack Recorder), to be then digitally transferred (via optical cable) to a digital audio editor PC software environment (Sony Sound Forge 8.0). The strategy adopted in order to obtain the frequencies (in hertz) of each string with the aid of spectrum analyzing software, which will provide an invaluable source of data for later extensive analysis. This data (collected on three separate occasions), once subjected to an analysis of variance (utilizing average and standard deviation criteria) will then form a ‘mean’ scale, which in essence will represent the intervallic nature of dastgàh-e segàh according to Persian performer Qmars Piraglu. The first collection of samples will additionally serve as the source material for the creation of a multi-sampled santñr for the Akai S3000XL Midi Stereo Digital Sampler. In view of the fact that Sony Sound Forge 8.0 is unable to provide accurate enough frequency readings, a series of tests is then implemented in order to locate the appropriate software package. These tests include the multiplatform readings of 1Hz, 10Hz, 100Hz, 1KHz, 10KHz, and 440Hz sine wave Theory Versus Performance Practice 37 sync tones, with the results nominating Adobe Audition 2.0 as the most error free vehicle. Utilizing the frequency analysis window function of the program it is now possible to generate FFT (Fast Fourier Transform) computations that translate frequency and amplitude analysis into a simple display of the mean frequency of the most prominent frequency peak, or the fundamental.74 The Adobe Audition manual describes the FFT process thus: “Fourier Theory states that any waveform consists of an infinite sum of sin and cos functions, allowing frequency and amplitude to be quickly analyzed.” The program uses an algorithm based on the Fourier Theory for filtering, spectral view, and frequency analysis. The FFT size menu includes 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, and 65536 sample options; while the FFT type menu provides eight type of FFT windows – Triangular, Hanning, Hamming, Blackmann, Welch (Gaussian), Blackmann-Harris, Kaiser (120dB), and Kaiser (180dB). According to the manual:

“The Triangular window gives a more precise frequency estimate, but it’s also the noisiest, meaning that other frequencies will be shown as present, even though they may be much lower in volume. At the other extreme, the Blackmann-Harris window has a broader frequency band, which isn’t as precise, but the sidelobes are very low, making it easier to pick out the major frequency components.”

An analysis of the test tone readings (all one hundred percent accurate apart from the 0.94257Hz result for 1Hz) reveal that a FFT size of 65536 and a Hamming75 type is required. At 44.1KHz, or 44,100 samples per second, a 65536 FFT sample window length allows for the equivalent of 673 milliseconds of sample time. One further test procedure is nevertheless implemented in order to confirm the above results, and this involves obtaining data (utilizing a smaller 4096 FFT sample window length, or 93 milliseconds) from all Cs (C3, C4, C5, and C6) at multiple start points (in 50 millisecond intervals beginning from 0 and ending on 500 milliseconds). The vibrating strings are shown to be extremely stable over time, with approximately ß99.923564% and +100.122296% deviations from the 65536 FFT sample window length reading of the first sample (C3), which represents fluctuations within 0.26Hz, or 3.440 cents.

74 “Frequency, which is a measurement expressed in hertz (Hz), is derived from German physicist Heinrich Rudolf Hertz (1857-94), and represents the rate of vibrations per second of a sound wave, with each cycle a sonic structure incorporating positive and negative amplitudes.” For a further discussion, see Arthur H. Benade, Fundamentals of Musical Acoustics, 2nd ed. (New York: Dover Publications, 1990) 159. 75 The Hamming window is “a very popular compromise between simplicity and effectiveness. It’s named after Richard W. Hamming, a pioneer in the application of computers to practical computation. Mathematically, it consists of a single cycle of a cosine, raised and weighted so that it drops to 0.08 at the end-points and has a peak value of one: ht=0.54ß0.46cos(24t/nß1),  <0 nt .” For a further discussion, see Ken Steiglitz, A DSP Primer: with Applications to Digital Audio and Computer Music (Menlo Park, CA: Addison-Wesley Pub., 1995) 206. 38 Theory Versus Performance Practice

Tuning of the Santñr

The tuning of the santñr for dastgàh-e segàh (on F) presents the following series of pitches: C3, F3, G3, Aù3, B"3, C4, Dù4, E"4, F4 (first position); Eù4, F4, G4, Aù4, B"4, C5, Dù5, E"5, F5 (second position); and Eù5, F5, G5, A"5, B"5, C6, Dù6, E6, F6 (third position). The ‘position’ is in reference to each of the instrument’s three courses of strings. The instrument’s design incorporates eighteen quadruple set of strings, with the division via movable bridges of the second set of nine strings generating the second and third courses (two individual pitches generally tuned octaves apart), to produce twenty-seven distinct pitches in total. The santñrist generally commences with the tuning of the first front string, which then becomes the basis for the generation of unison intervallic relationships (removing beating as much as possible) with the other three strings of each set. An interesting point of note is that although the primary structure of dastgàh-e segàh can be essentially stated to be heptatonic, in performance practice the scale is expanded into a three-octave scale with dissimilar tetrachords. Some points of interest include the low brass string tuned to C'; the A’s in the first two octaves tuned to Aù, while in the third, tuned to A"; E" and Eù alternatives provided in two octaves; and the duplication of F pitches in the second and third octaves. The design of the instrument – presenting a total of nine strings per octave, with two strings, E and F, tunable to either E', E" or Eù; and F' or F! respectively – allows for the performance of heptatonic scales (inclusive of their auxiliary pitches) from the complete Persian modal system.76 Dastgàh-e segàh is mode number nine of the twelve dastgàhs (according to Piraglu’s own codification derived from the santñr tradition of Pàyvar), and its name (“literally ‘third place’”) is in reference to “the positioning of the central pitch relative to other modes.” According to Zonis, “se is the Persian word for ‘three’; gàh, the word for ‘time’ or ‘place’,” and alludes to the fact that “this kind of designation was formerly used both in Persia and in the Arab countries to indicate the degrees of the scale.”77 The guêe nomenclature and design of the dastgàh is extensively interconnected with dastgàh-e áahàrgàh (“literally ‘fourth place’”).78 “Segàh and áahàrgàh allied to a degree unparalleled in any other two dastgàhs,” notes Farhat. “Virtually every piece performed in segàh can be performed in áahàrgàh, although áahàrgàh includes a few guêes peculiar to that dastgàh and not performed in segàh.”79 The following three examples present dastgàh-e segàh’s basic structure, as well as the tuning of the santñr, and harmonic characteristics (in Pythagorean terms).

76 Ella Zonis, Classical Persian Music: An Introduction (Cambridge, MA: Harvard U. Press, 1973) 57-58. 77 Zonis, Classical Persian Music: An Introduction 88. 78 Laudan Nooshin, “The Song of the Nightingale: Processes of Improvisation in Dastgàh Segàh (Iranian Classical Music),” British Journal of Ethnomusicology 7 (1998): 76. 79 Farhat, The Dastgàh Concept in Persian Music 56. Theory Versus Performance Practice 39

Ex. 5. Dastgàh-e segàh i. Basic mode

ii. Tuning of the santñr for dastgàh-e segàh (on F)

iii. The harmonic characteristics of dastgàh-e segàh

The tuning of the santñr for dastgàh-e segàh (on F) involves a specific tuning scheme, which begins with the acquisition of a fundamental C' in second position (C5) calibrated to A=440Hz. The notion of standard pitch only idealistic as in actuality on one occasion demonstrating a 10.223 cent deviation from A=440Hz (or A=442.606Hz), while on other occasions, 10.387 cents (or A=442.648Hz), and 1.584 cents (or A=440.403Hz). In spite of Piraglu’s insistence of possessing perfect pitch and therefore the ability to tune to a sung high C at standard pitch,80 something that becomes clearly evident via this experience is that in this particular context the solo performer is rarely that concerned about the procurement of A=440Hz in literal terms, hence it is a musical pitch certainly not in the domain of obligatory criteria. Although it must be noted that in a performance setting involving the santñr in

80 “Absolute pitch, while extremely rare in the general population (its incidence has been estimated as 1 in 10,000), is in reality common among professional musicians, and to a limited extent may serve as a marker for musicality. But as N. Slonimsky, in his autobiography, Perfect Pitch, writes: ‘The lack of it does not exclude musical talent, or even genius. Neither Wagner nor Tchaikovsky had absolute pitch, while a legion of mediocre composers possessed it to the highest degree.’ There is a greatly heightened occurrence of absolute pitch in some other populations: among the autistic the incidence may be about one in 20; and among those with savant syndrome, more than a third have musical gifts – and all musical savants, apparently, have absolute pitch.” For a further discussion, see Oliver Sacks, G. Schlaug, L. Jäncke, Y. Huang, and H. Steinmetz, “Musical Ability,” Science 268.5211 (May 5, 1995): 621. 40 Theory Versus Performance Practice conjunction with Western traditional instruments, some form of pitch calibration will be required in order to accommodate the limitations of some of the instruments. A performance setting involving a Persian ensemble on the other hand will require the whole group to calibrate their pitch to the santñr, for the simple reason of practicality – the instrument requiring the tuning of seventy-two strings. It is interesting to note that following a discussion with the performer about the rationale of the last measurement of A=440.403Hz, and its obvious discrepancy in comparison with the mean of the first two (442.627Hz, and therefore a difference of almost nine cents), it is revealed that a duet performance with a flautist had required the santñr to be calibrated down to A=440Hz. Stage one of the tuning process encapsulates the following procedures:

I. i. C5 is tuned to A=440Hz ii. C4 to C5 iii. C3 to C4 iv. C6 to C5 II. i. F4 (2nd position) is tuned to C5, generating a perfect fifth ii. F3 to F4 (2nd position) iii. F5 (3rd position) to F4 (2nd position) iv. F5 (2nd position) to F5 (3rd position) v. F6 to F5 (2nd position) vi. F4 (1st position) to F4 (2nd position) III. i. G4 is tuned to C5, generating a perfect fourth ii. G3 to G4 iii. G5 to G4 IV. i. B"4 is tuned to F4, generating a tempered perfect fourth ii. B"3 to B"4 iii. B"5 to B"4 V. i. Aù4 is tuned to first tetrachord melodic patterns (F4, G4, Aù4 and B"4) ii. Aù3 to Aù4 VI. i. Dù5 is tuned to Aù4, generating a tempered perfect fourth ii. Dù4 to Dù5 iii. Dù6 to Dù5 VII. i. E"5 is tuned to B"4, generating a tempered perfect fourth ii. E"4 to E"5 iii. E"6 to E"5 VIII. i. Eù4 is tuned to Aù4, generating a tempered perfect fourth Theory Versus Performance Practice 41

ii. Eù5 to Eù4 IX. i. A"5 is tuned to second tetrachord melodic patterns (Eù5, F5, G5 and A"5)

It may be further noted that although the technique applied to the tuning of Aù4 has a basis of first tetrachord melodic patterns (F4, G4, Aù4, and B"4), the interval is additionally calculated according to the evaluation of the perfect fourth simultaneous sonorities of F4 (2nd position) and B"4, as well as F4 (2nd position) and Aù4. As a matter of interest, it must be stated that the perfect fourth is “the smallest invariable interval” in Persian music, and recognized as the most important.81 The duplication of Fs (F4 in first and second positions, and F5 in second and third positions) serve merely as alternatives for the equivalent pitches, although in performance practice, F4 (first position) and F5 (second position) are generally avoided in dastgàh-e segàh (on F) due to their inferior timbral quality (the strings, shorter in this region of the instrument, hence dynamically weaker). In the context of the pitch material requirements of other dastgàhs, this additional pitch serves to provide a mandatory chromatic alternation. The synchronous utilization of Eù and E" in dastgàh-e segàh (on F) illustrates the application of that principle. The fact that the integrity of second and third position pitches are determined by the placement of individual bridges dividing the string into specific ratios, these bridges may also have to be manually adjusted in order to produce the desired sonorities. Third position A"5 is produced via this method, with the adjustment of the bridge dividing the relevant string essentially simultaneously producing second position Aù4 and third position A"5. This also holds true for other pitches acquired during the dipartite tuning process not conforming to the aesthetics of dastgàh-e segàh (on F), which may have to be alternatively fine-tuned via the adjustments of the bridge. The final pitch of interest in need of some explanation is C3 in first position, which according to Piraglu, is a relatively modern phenomenon – adopted post 1980 by a new generation of santñr players such as Parviz Meshkàtiàn (1955-) and Pashang Kàmkàr (1951-). The practice favours the tuning of the low string to C or D (depending on the dastgàh), as opposed to the earlier schools of Abol-Äasan Éabà (1902-1957) and Faràmarz Pàyvar (1933-) subscribing to the acquisition of E or Eù. The C3 in first position of course provides the facility for the instrument to produce both perfect fourth (between C3 and F3) and perfect fifth (between C3 and G3) dyads in the low tessitura of the instrument.82 Stage two of the tuning process – the evaluation of simultaneous sonorities (perfect fifths, perfect fourths, tempered perfect fourths, and neutral thirds) – then encapsulates the following procedures:

I. i. F4 (2nd position) is evaluated with C5, generating a perfect fifth

81 Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah 10. 82 Qmars Piraglu, “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties,” diss. Göteborg U., Swed., 1998, 33. 42 Theory Versus Performance Practice

ii. C4 with F4 (2nd position), generating a perfect fourth iii. C4 with F4 (1st position), generating a perfect fourth iv. C5 with F5 (3rd position), generating a perfect fourth v. C5 with F5 (2nd position), generating a perfect fourth vi. F5 (3rd position) with C6, generating a perfect fifth vii. F3 with C4, generating a perfect fifth viii. C3 with F3, generating a perfect fourth ix. C6 with F6, generating a perfect fourth II. i. C4 is evaluated with G4, generating a perfect fifth ii. G4 with C5, generating a perfect fourth iii. C5 with G5, generating a perfect fifth iv. G5 with C6, generating a perfect fourth v. G3 with C4, generating a perfect fourth vi. C3 with G3, generating a perfect fifth III. i. F4 (2nd position) is evaluated with B"4, generating a tempered perfect fourth ii. F4 (2nd position) with Aù4, generating a neutral third iii. Aù4 with Dù5, generating a tempered perfect fourth iv. F3 with B"3, generating a tempered perfect fourth v. F3 with Aù3, generating a neutral third vi. Aù3 with Dù4, generating a tempered perfect fourth vii. F5 (3rd position) with B"5, generating a tempered perfect fourth IV. i. B"4 is evaluated with E"5, generating a tempered perfect fourth ii. B"3 with E"4, generating a tempered perfect fourth iii. B"5 with E"6, generating a tempered perfect fourth V. i. Eù4 is evaluated with Aù4, generating a tempered perfect fourth

Spectrum Analysis Results

The following three tables present the spectrum analysis results collected on three separate occasions (with a periodicity of 3-6 months) for each of the twenty-seven sets of strings, and therefore denotes all data for string set, pitch, order, frequency (Hz), ratio (decimal), and cents. The calculation of frequency represents prime (not normalized) data, and therefore ratio and cents are the only two comparative frames of reference. Ratios have been calculated from the relationship of frequencies to the base pitch of the second position C5 (the reference pitch, or 526.350Hz, 526.400Hz, and 523.730Hz respectively for each of the three tunings), while cents are a derivative of ratio data. Theory Versus Performance Practice 43

Table 10. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 1

1st position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÐÑ C3 3 (I-iii) 130.830 1.988487 1190.005 ÐÒ F3 6 (II-ii) 174.990 1.329838 493.500 ÐÓ G3 12 (III-ii) 196.480 1.493151 694.032 ÐÔ Aî3 18 (V-ii) 213.800 1.624774 840.287 ÐÕ BÝ3 15 (IV-ii) 232.060 1.763541 982.171 ÐÖ C4 2 (I-ii) 261.390 1.986435 1188.218 Ð× Dî4 20 (VI-ii) 286.780 1.089693 148.706 ÐØ EÝ4 23 (VII-ii) 310.080 1.178227 283.942 ÐÙ F4 10 (II-vi) 351.200 1.334473 499.524

2nd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÑÐ Eî4 25 (VIII-i) 322.540 1.225572 352.147 ÑÑ F4 5 (II-i) 350.090 1.330256 494.044 ÑÒ G4 11 (III-i) 394.970 1.500788 702.865 ÑÓ Aî4 17 (V-i) 429.610 1.632412 848.406 ÑÔ BÝ4 14 (IV-i) 465.950 1.770495 988.983 ÑÕ C5 1 (I-i) 526.350 1.000000 0.000 ÑÖ Dî5 19 (VI-i) 578.200 1.098509 162.655 Ñ× EÝ5 22 (VII-i) 620.720 1.179291 285.504 ÑØ F5 8 (II-iv) 703.100 1.335803 501.249

3rd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÑÙ Eî5 26 (VIII-ii) 645.220 1.225838 352.522 ÒÐ F5 7 (II-iii) 697.240 1.324670 486.759 ÒÑ G5 13 (III-iii) 788.260 1.497597 699.179 ÒÒ AÝ5 27 (IX-i) 827.400 1.571958 783.075 ÒÓ BÝ5 16 (IV-iii) 930.310 1.767474 986.027 ÒÔ C6 4 (I-iv) 1049.300 1.993540 1194.399 ÒÕ Dî6 21 (VI-iii) 1144.400 1.087109 144.596 ÒÖ EÝ6 24 (VII-iii) 1246.000 1.183623 291.852 Ò× F6 9 (II-v) 1406.200 1.335803 501.249 44 Theory Versus Performance Practice

Table 11. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 2

1st position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÐÑ C3 3 (I-iii) 130.790 1.987690 1189.311 ÐÒ F3 6 (II-ii) 174.260 1.324164 486.098 ÐÓ G3 12 (III-ii) 197.400 1.500000 701.955 ÐÔ Aî3 18 (V-ii) 213.740 1.624164 839.637 ÐÕ BÝ3 15 (IV-ii) 231.990 1.762842 981.484 ÐÖ C4 2 (I-ii) 260.930 1.982751 1185.004 Ð× Dî4 20 (VI-ii) 286.690 1.089248 147.999 ÐØ EÝ4 23 (VII-ii) 310.050 1.178002 283.610 ÐÙ F4 10 (II-vi) 351.170 1.334233 499.212

2nd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÑÐ Eî4 25 (VIII-i) 322.490 1.225266 351.714 ÑÑ F4 5 (II-i) 348.840 1.325380 487.687 ÑÒ G4 11 (III-i) 394.670 1.499506 701.385 ÑÓ Aî4 17 (V-i) 428.470 1.627926 843.642 ÑÔ BÝ4 14 (IV-i) 465.940 1.770289 988.782 ÑÕ C5 1 (I-i) 526.400 1.000000 0.000 ÑÖ Dî5 19 (VI-i) 578.150 1.098309 162.341 Ñ× EÝ5 22 (VII-i) 620.620 1.178989 285.061 ÑØ F5 8 (II-iv) 703.090 1.335657 501.060

3rd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÑÙ Eî5 26 (VIII-ii) 645.160 1.225608 352.197 ÒÐ F5 7 (II-iii) 697.280 1.324620 486.694 ÒÑ G5 13 (III-iii) 788.220 1.497378 698.927 ÒÒ AÝ5 27 (IX-i) 829.380 1.575570 787.049 ÒÓ BÝ5 16 (IV-iii) 930.230 1.767154 985.714 ÒÔ C6 4 (I-iv) 1048.700 1.992211 1193.245 ÒÕ Dî6 21 (VI-iii) 1144.200 1.086816 144.129 ÒÖ EÝ6 24 (VII-iii) 1254.800 1.191869 303.871 Ò× F6 9 (II-v) 1406.000 1.335486 500.838 Theory Versus Performance Practice 45

Table 12. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 3

1st position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÐÑ C3 3 (I-iii) 131.160 1.001738 3.005 ÐÒ F3 6 (II-ii) 174.570 1.333282 497.979 ÐÓ G3 12 (III-ii) 197.130 1.505585 708.389 ÐÔ Aî3 18 (V-ii) 212.300 1.621446 836.737 ÐÕ BÝ3 15 (IV-ii) 231.460 1.767781 986.328 ÐÖ C4 2 (I-ii) 262.250 1.001470 2.543 Ð× Dî4 20 (VI-ii) 285.400 1.089875 148.995 ÐØ EÝ4 23 (VII-ii) 311.140 1.188169 298.489 ÐÙ F4 10 (II-vi) 346.900 1.336185 501.743

2nd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÑÐ Eî4 25 (VIII-i) 320.500 1.223913 349.801 ÑÑ F4 5 (II-i) 348.970 1.332633 497.136 ÑÒ G4 11 (III-i) 394.930 1.508144 711.328 ÑÓ Aî4 17 (V-i) 427.390 1.632101 848.076 ÑÔ BÝ4 14 (IV-i) 462.800 1.767323 985.879 ÑÕ C5 1 (I-i) 523.730 1.000000 0.000 ÑÖ Dî5 19 (VI-i) 569.000 1.086438 143.526 Ñ× EÝ5 22 (VII-i) 617.650 1.179329 285.560 ÑØ F5 8 (II-iv) 698.550 1.333798 498.648

3rd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS ÑÙ Eî5 26 (VIII-ii) 638.520 1.219178 343.090 ÒÐ F5 7 (II-iii) 695.250 1.327497 490.450 ÒÑ G5 13 (III-iii) 791.640 1.511542 715.226 ÒÒ AÝ5 27 (IX-i) 824.560 1.574399 785.761 ÒÓ BÝ5 16 (IV-iii) 925.770 1.767647 986.197 ÒÔ C6 4 (I-iv) 1051.000 1.003380 5.84102 ÒÕ Dî6 21 (VI-iii) 1141.700 1.089970 149.146 ÒÖ EÝ6 24 (VII-iii) 1242.500 1.186203 295.621 Ò× F6 9 (II-v) 1396.800 1.333512 498.276 46 Theory Versus Performance Practice

An analysis of simultaneous sonorities based on stage one of the tuning process for the three tunings reveal octaves with a mean value of 1201.019 cents, a range between 1181.941 and 1218.810 cents, and a standard deviation of 7.771183. These results effectively pronounce the octave as being in the vicinity of the just perfect octave (2/1), although in the range from the double tritone (2025/1024, or 1180.447 cents) and acute or large octave (81/40, or 1221.506 cents) with a falsity of +1.494 and ß2.696 cents on each count. In comparison with the just perfect octave (2/1), the octave is within the range of ß18.059 and +18.810 cents.83

Table 13. Stage one of tuning process – octaves

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION ÐÑ C3 and C4 1195.693 1199.538 1197.814 1.953313 ÐÒ F3 and F4(2) 1199.157 1201.589 1200.430 1.220009 ÐÓ G3 and G4 1199.430 1208.833 1203.734 4.751527 ÐÔ Aî3 and Aî4 1204.005 1211.339 1207.821 3.675986 ÐÕ BÝ3 and BÝ4 1199.551 1207.298 1204.554 4.339307 ÐÖ C4 and C5 1197.457 1214.996 1208.078 9.337949 Ð× Dî4 and Dî5 1194.532 1214.343 1207.608 11.325798 ÐØ EÝ4 and EÝ5 1187.071 1201.563 1196.695 8.334857 ÐÙ Eî4 and Eî5 1193.289 1200.483 1198.049 4.122885 ÑÐ F4(2) and F5(3) 1192.715 1199.007 1195.012 3.472527 ÑÑ G4 and G5 1196.314 1203.897 1199.251 4.070141 ÑÒ BÝ4 and BÝ5 1197.044 1200.318 1198.098 1.923469 ÑÓ C5 and C6 1193.245 1205.841 1197.828 6.963106 ÑÔ Dî5 and Dî6 1181.941 1205.620 1189.783 13.715187 ÑÕ EÝ5 and EÝ6 1206.347 1218.810 1211.740 6.398768 ÑÖ F5(2) and F6 1199.628 1200.000 1199.802 0.187035

Simultaneous sonorities based on stage two of the tuning process for the three tunings on the other hand reveal perfect fifths with a mean value of 706.894 cents, a range between 694.718 and 716.381 cents, and a standard deviation of 6.537279. The primary perfect fifths, or those unique and sequentially superior are represented by F4 (2nd position) and C5, C4 and G4, and the mean values of 707.044 and 713.271 cents respectively. The range of these fifths being between 698.927 and 715.226 cents

4 80 indicates an inclination to represent the meantone perfect fifth (3/2× 81 , approximately 154/103, or 696.578 cents), just perfect fifth (3/2), and equal perfect fifth ( [12 2]7 , approximately 767/512, or 700.000 cents) with a falsity of +2.274, ß3.028, and ß1.073 cents at the lower end of the scale; while

83 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation ix. Theory Versus Performance Practice 47 the sléndro acute or large fifth ( [5 2]3 , approximately 97/64, or 720.000 cents), just perfect fifth (3/2), and equal perfect fifth ( [12 2]7 ) with a falsity of ß4.774, +13.271, and +15.226 cents at the higher end.84 The sléndro acute or large fifth is the identical interval found in five-tone equal temperament, and is produced by the ratio 1: (5 2)3 =1:1.515717.85

Table 14. Stage two of tuning process – perfect fifths

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION ÐÑ C3 and G3 704.027 712.644 707.351 4.633233 ÐÒ F3 and C4 694.718 704.565 699.396 4.941669 ÐÓ C4 and G4 708.785 716.381 713.271 3.980471 ÐÔ F4(2) and C5 702.864 712.313 707.044 4.817387 ÐÕ C5 and G5 698.927 715.226 704.444 9.338180 ÐÖ F5(3) and C6 706.550 715.391 709.860 4.820301

An analysis of simultaneous sonorities based on stage two of the tuning process for the three tunings reveal perfect fourths with a mean value of 497.436 cents, a range between 483.049 and 514.208 cents, and a standard deviation of 7.640715. The primary perfect fourths are represented by C4 and F5 (2nd position), and G4 and C5, and the mean values of 501.034 and 494.807 cents respectively. The range of these fourths being between 488.672 and 505.826 indicates an inclination to represent the sléndro grave or small fourth ( [5 2]2 , approximately 128/97, or 480.000 cents), just perfect fourth (4/3), and equal perfect fourth ( [12 2]5 , approximately 1024/767, or 500.000 cents) with a falsity of +8.672, ß9.373, and ß11.328 cents at the lower end of the scale; while the meantone perfect fourth

4 81 (4/3× 80 , approximately 103/77, or 503.422 cents), just perfect fourth (4/3), and equal perfect fourth ( [12 2]5 ) with a falsity of +2.380, +7.781, and +5.826 cents at the higher end.86 The sléndro grave or small fourth is the identical interval found in the five-tone equal temperament, and is produced by the ratio 1: (5 2)2 =1:1.319508. The sléndro acute or large fifth and grave or small fourth may also be referred to as the 5-et grave or small fourth and 5-et acute or large fifth, or alternatively as the quintal equal subfourth and quintal equal superfifth.87

84 Daniélou, Tableau Comparatif des Intervalles Musicaux 119-25. 85 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 455. 86 Daniélou, Tableau Comparatif des Intervalles Musicaux 118-24. 87 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 455. 48 Theory Versus Performance Practice

Table 15. Stage two of tuning process – perfect fourths

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION ÐÑ C3 and F3 494.973 503.495 498.418 4.488880 ÐÒ G3 and C4 483.049 494.186 490.463 6.420814 ÐÓ C4 and F4(2) 494.592 505.826 501.034 5.795747 ÐÔ C4 and F4(1) 499.200 514.208 508.238 7.960749 ÐÕ G4 and C5 488.672 498.615 494.807 5.365009 ÐÖ C5 and F5(3) 486.694 490.450 487.968 2.149959 Ð× C5 and F5(2) 498.648 501.249 500.319 1.450061 ÐØ G5 and C6 490.616 495.220 493.385 2.440269 ÐÙ C6 and F6 492.435 507.593 502.293 8.544883

Simultaneous sonorities based on stage two of the tuning process for the three tunings reveal tempered perfect fourths with a mean value of 501.373 cents, a range between 488.349 and 518.700 cents, and a standard deviation of 8.744854. The primary tempered fourths are represented by F4 (2nd position) and BÝ4, Aî4 and Dî4, BÝ4 and EÝ5, Eî4 and Aî4, and the mean values of 494.926, 509.466, 497.494, and 495.487 cents respectively. The range of these fourths being between 488.743 and 518.700 indicates an inclination to again (as in the primary perfect fourths) represent the sléndro grave or small fourth ( [5 2]2 ), just perfect fourth (4/3), and equal perfect fourth ( [12 2]5 ), but this time with a falsity of +8.098, ß9.302, and ß11.257 cents at the lower end of the scale, while the acute or large fourth (27/20, or 519.551 cents), just perfect fourth (4/3), and equal perfect fourth ( [12 2]5 ) with a falsity of ß0.852, +20.655, and +18.700 cents at the higher end.

Table 16. Stage two of tuning process – tempered perfect fourths

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION ÐÑ F3 and BÝ3 488.349 495.385 490.802 3.973019 ÐÒ Aî3 and Dî4 508.362 512.257 509.679 2.232698 ÐÓ BÝ3 and EÝ4 501.771 512.161 505.353 5.898947 ÐÔ Eî4 and Aî4 491.928 498.275 495.487 3.243106 ÐÕ F4(2) and BÝ4 488.743 501.094 494.926 6.175730 ÐÖ Aî4 and Dî5 495.451 518.700 509.466 12.340378 Ð× BÝ4 and EÝ5 496.279 499.681 497.494 1.897984 ÐØ F5(2) and BÝ5 495.746 499.267 498.011 1.965164 ÐÙ BÝ5 and EÝ6 505.825 518.158 511.136 6.342067

Theory Versus Performance Practice 49

An analysis of simultaneous sonorities based on stage two of the tuning process for the three tunings on the other hand reveal neutral thirds with a mean value of 350.057 cents, a range between 338.758 and 355.954 cents, and a standard deviation of 6.396299. The primary neutral third is represented by F4 (2nd position) and Aî4, and the mean value of 353.752 cents. The range of these thirds being between 350.940 and 355.954 indicates an inclination to represent seven equal quarter-tones (approximately 60/49, or 350.00 cents), the just major third (5/4, or 386.314 cents), and equal major third ( 3 2, approximately 63/50, or 400.000 cents)88 with a falsity of +0.940, ß35.374, and ß49.060 cents at the lower end of the scale; while an extreme grave or small major third (27/22, or 354.547 cents), just major third (5/4), and equal major third ( 3 2 ) with a falsity of +1.407, ß30.359, and ß44.046 cents at the higher end. Seven equal quarter-tones is the identical interval found in the twenty-four-tone equally- tempered division of the octave, and is produced by the ratio 1: (24 2)7 =1:224054.

Table 17. Stage two of tuning process – neutral thirds

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION ÐÑ F3 and Aî3 338.758 353.539 346.361 7.399249 ÐÒ F4(2) and Aî4 350.940 355.954 353.752 2.562172

Analysis of Variance

The fourth table presents an ‘analysis of variance’ with regards to the three tunings, and therefore provides a platform for the evaluation of the tuning of the seventy-two strings of the santñr for dastgàh-e segàh (on F) utilizing average and standard deviation criterion.89 In this case standard deviation has been derived from cents data, and frequencies normalized to A=440Hz, although in direct relation to ratio data, which has been obtained via the mathematical equation of 1200ïcentsÏ 24 2 =ratio.

88 Daniélou, Tableau Comparatif des Intervalles Musicaux 90-102. 89 “A computational procedure frequently used to analyze the data from an experimental study employs a statistical procedure known as the analysis of variance. For a single-factor experiment, this procedure uses a hypothesis test concerning equality of treatment means to determine if the factor has a statistically significant effect on the response variable. For experimental designs involving multiple factors, a test for the significance of each individual factor as well as interaction effects caused by one or more factors acting jointly can be made. A variety of numerical measures are used to summarize data. The proportion, or percentage, of data values in each category is the primary numerical measure for qualitative data. The mean, median, mode, percentiles, range, variance, and standard deviation are the most commonly used numerical measures for quantitative data. The mean, often called the average, is computed by adding all the data values for a variable and dividing the sum by the number of data values, while standard deviation is a measure of the variability (dispersion or spread) of any set of numerical values about their arithmetic mean. It is specifically defined as the square root of the arithmetic mean of the squared deviations.” For a further discussion, see Encyclopaedia Britannica, “Statistics,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM, Chicago: Encyclopaedia Britannica, 2001. 50 Theory Versus Performance Practice

Table 18. Tuning of the santñr for dastgàh-e segàh (on F) – analysis of variance

1st position

STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION ÐÑ C3 130.368 1.993204 1194.107 7.713846 ÐÒ F3 173.862 1.329089 492.526 5.999891 ÐÓ G3 196.163 1.499570 701.459 7.191309 ÐÔ Aî3 212.369 1.623461 838.887 1.889998 ÐÕ BÝ3 230.848 1.764720 983.327 2.620925 ÐÖ C4 260.408 1.990689 1191.922 9.337949 Ð× Dî4 285.069 1.089605 148.566 0.512523 ÐØ EÝ4 309.099 1.181457 288.680 8.496235 ÐÙ F4 349.261 1.334963 500.160 1.380069

2nd position

STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION ÑÐ Eî4 320.470 1.224917 351.221 1.248094 ÑÑ F4 347.810 1.329419 492.956 4.817387 ÑÒ G4 393.173 1.502808 705.193 5.365009 ÑÓ Aî4 426.662 1.630811 846.708 2.660578 ÑÔ BÝ4 462.912 1.769368 987.881 1.737114 ÑÕ C5 523.251 1.000000 0.000 0.000000 ÑÖ Dî5 572.642 1.094404 156.174 10.954514 Ñ× EÝ5 617.019 1.179203 285.375 0.273364 ÑØ F5 698.585 1.335086 500.319 1.450061

3rd position

STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION ÑÙ Eî5 640.217 1.223537 349.270 5.354195 ÒÐ F5 693.619 1.325595 487.968 2.149959 ÒÑ G5 786.006 1.502158 704.444 9.338180 ÒÒ AÝ5 823.584 1.573975 785.295 2.027411 ÒÓ BÝ5 924.807 1.767425 985.979 0.245081 ÒÔ C6 1048.400 1.001813 1197.828 6.963106 ÒÕ Dî6 1138.557 1.087964 145.957 2.771441 ÒÖ EÝ6 1242.435 1.187227 297.115 6.147445 Ò× F6 1397.011 1.334933 500.121 1.610794 Theory Versus Performance Practice 51

Utilizing the mean ratios from the ‘analysis of variance’ data, and considering solely the base pitch of second position C5, as well as all primary derivative pitches (second position F4, G4, B"4, Aù4, Dù5, E"5, Eù4, and third position A"5), it is possible to arrive at the ‘tuning characteristics’ personified by the three tunings, and therefore what may be stated as being a ‘performance practice’ tuning obtained via the intervallic analysis of Persian performer Qmars Piraglu’s instrument, which was tuned to dastgàh-e segàh by ear, and therefore not artificially influenced by tuning devices in order to adhere to strict theoretical schemes. Piraglu’s tuning methodology involved the obtainment of a fundamental C note calibrated approximately to A=440Hz, which would then serve as the reference for the generation of all required perfect fifths and fourths, as well as tempered fourths, with problem intervals such as A koron and A flat left to the discretion of the ear and the perceived musicality of performed extracts from associated gušes (individual pieces which make up the repertoire of a particular dastgàh).

Table 19. Dastgàh-e segàh (on F) – tuning characteristics

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE APPROXIMATE INTERVAL FREQUENCY RATIO AVERAGE NUMBER (HERTZ) (DECIMAL) (CENTS) ÐÑ C4 unison 261.626 1.000000 0.000 ÐÒ D4 just major tone (9th harmonic) 295.748 1.130424 212.237 ÐÓ EÝ4 Pythagorean minor third, or trihemitone 309.753 1.183957 292.339 ÐÔ Eî4 neutral third 320.939 1.226709 353.752 ÐÕ F4 just and Pythagorean perfect fourth 348.206 1.330933 494.926 ÐÖ G4 just and Pythagorean perfect fifth (3rd harmonic) 393.594 1.504416 707.044 Ð× Aî4 grave or small major sixth 430.750 1.646439 863.219 ÐØ BÝ4 Pythagorean minor seventh 464.127 1.774012 992.419 ÐÙ Bî4 grave or small major seventh 482.120 1.842785 1058.265

ÐÑ C5 octave 523.251 2.000000 1200.000

The abovementioned process consequently produces intervals approximating the just major tone (9/8), Pythagorean minor third (32/27), neutral third (11/9), just perfect fourth (4/3), just perfect fifth (3/2), grave or small major sixth (400/243), Pythagorean minor seventh (16/9), and grave or small major seventh (50/27). The findings proclaim the obvious Pythagorean connection with the resulting ‘mean’ measurements for the major second (equal to 212.237 cents, with a range between and 208.821 and 214.193 cents, and a standard deviation of 2.968984); minor third (equal to 292.339 cents, with a range between 288.626 and 299.361 cents, and a standard deviation of 6.084603); perfect fourth (equal to 494.926 cents, with a range between 488.743 and 501.094 cents, and a standard deviation of 52 Theory Versus Performance Practice

6.175730); perfect fifth (equal to 707.044 cents, with a range between 702.864 and 712.313 cents, and a standard deviation of 4.817387); and minor seventh (equal to 992.419 cents, with a range between 988.424 and 997.374 cents, and a standard deviation of 4.551339). The research interestingly also suggests a distinction between the neutral orientation of microtonal inflections such as Eù4 (equal to 353.752 cents, with a range between 350.940 and 355.954 cents, and a standard deviation of 2.562172); Aù5 (equal to 863.219 cents, with a range between 846.391 and 874.654 cents, and a standard deviation of 14.883326); and Bù4 (equal to 1058.265 cents, with a range between 1052.666 and 1064.027 cents, and a standard deviation of 5.682304); which reveal a ‘grave or small major’ intervallic orientation.

Tuning System Comparison

A tuning system comparison table incorporating data from the twenty-four equally-tempered quarter- tone scale of Vaziri, twenty-two-note Pythagorean scale of Barkešli, and Farhat’s theory of flexible intervals, or of the five primary intervals of performance practice; as well as the performance practice tuning of Piraglu, reveals a close link between the latter two. Farhat and Piraglu share similarities with all intervals (a falsity of +8.237, ß1.661, ß10.248, ß3.074, +5.044, +1.219, ß3.581, and ß7.735 cents on each count) but for the neutral third (Eî4), which has a closer association with the equally-tempered quarter-tone of Vaziri (a falsity of +3.752, as opposed to ß10.248). Interestingly, at 384.360 cents, Barkešli’s neutral third represents the Pythagorean diminished fourth (8192/6561), which is 34.360, 24.360, and 30.608 cents larger than the neutral thirds of Vaziri, Farhat, and Piraglu.

Table 20. Dastgàh-e segàh (on F) – tuning system comparison

DEGREE NOTE ALI NAQI VAZIRI MEHDI BARKEÊLI HORMOZ FARHAT QMARS PIRAGLU NUMBER (CENTS) (CENTS) (CENTS) (CENTS) ÐÑ C4 0.000 0.000 0.000 0.000 ÐÒ D4 200.000 203.910 204.000 212.237 ÐÓ EÝ4 300.000 294.135 294.000 292.339 ÐÔ Eî4 350.000 384.360 364.000 353.752 ÐÕ F4 500.000 498.045 498.000 494.926 ÐÖ G4 700.000 701.955 702.000 707.044 Ð× Aî4 850.000 882.405 862.000 863.219 ÐØ BÝ4 1000.000 996.090 996.000 992.419 ÐÙ Bî4 1050.000 1086.315 1066.000 1058.265

ÐÑ C5 1200.000 1200.000 1200.000 1200.000 Theory Versus Performance Practice 53

Utilizing Farhat’s theory of the division of the whole-tone and nomenclature for Persian intervals, it is then possible to conduct an analysis of all minor second (D and E"), small neutral tone (Eù and F, Bù and C), large neutral tone (D and Eù, G and Aù), and major second (C and D, E" and F, F and G, B" and C) intervals (encountered within the framework established via the base and primary derivative pitches in the performance practice tuning of Piraglu); and consequently produce a comparison table outlining range, as well as the average for each interval in the two supportive propositions.

Table 21. Dastgàh-e segàh (on F) – Hormoz Farhat’s and Qmars Piraglu’s division of the whole-tone

DEGREE INTERVAL HORMOZ FARHAT HORMOZ FARHAT QMARS PIRAGLU QMARS PIRAGLU NUMBER RANGE (CENTS) AVERAGE (CENTS) RANGE (CENTS) AVERAGE (CENTS) ÐÑ unison 0.000 0.000 0.000 0.000 ÐÒ minor second 90.000 90.000 74.433  85.664 80.102 ÐÓ small neutral tone 125.000  145.000 135.000 135.973  147.334 141.454 ÐÔ large neutral tone 150.000  170.000 160.000 136.747  162.655 148.845 ÐÕ major second 204.000 204.000 200.117  214.193 208.631

Performance Practice and Tuning

In conclusion, it must be stated that Farhat’s theory of flexible intervals or of the five primary intervals of performance practice certainly holds true in the final analysis, and especially in view of the fact that although general assumptions may be reached with regards to tuning practice, there is no doubt that standard deviation data is so conflicting in some instances (the three tunings producing an overall standard deviation average of 3.258156, with a range between 0.245081 and 10.954514; while octaves between 0.187035 and 13.715187; perfect fifths between 3.980471 and 9.338180; perfect fourths between 1.450061 and 7.960749; tempered fourths between 1.897984 and 12.340378; and neutral thirds between 2.562172 and 7.399249) that the results cannot be stated as being the axiom. Ayers’s own tuning research into the performance practice of Persian music – utilizing “thirteen improvised recorded examples from the mode áahàrgàh and two recorded examples from the mode êhñr” – declare the following proposition:

“The analysis of actual performance practice in Persian music raises the issue of accuracy in acoustic performance of any microtonal music. If pitch variation in a mode of as much as 30 cents, which is a sixth of a tone, can be acceptable, then what place do theoretical systems have in this music? If we are composing computer music, then we don’t have to depend on performers and we can tune the 54 Theory Versus Performance Practice

computer as accurately as we wish. If we chose a precise tuning, will it sound realistic if the pitches never vary? Will perfect music sound natural?”90

N. A. Jairazbhoy, and A. W. Stone, in their studies of intonation in present-day North Indian classical music found that, “It would appear that within each performance the intonation does vary, and that a variation of as much as fifteen cents, in seconds and thirds at least, could easily pass unnoticed. This is particularly noticeable in the series of notes taken from the upper register which suggests that there may be a tendency towards sharpening intervals in this register.” With regards to the notion of the universal systemization of a theoretical tuning, the following conclusions are reached:

“With this divergence between musicians (the maximum divergence noticed in the interval of the third was between Pannalal Gosh (439 cents) and Ustàd Umaro Khan (375 cents), a difference of 64 cents, or more than a quarter-tone), it would appear that intonation is a matter of personal choice, perhaps influenced by the teacher’s intonation, but not bound to it, and that any intonation within certain limits (perhaps within 25 or 30 cents in either side of the tempered intonation) can be acceptable. Under these circumstances it would seem pointless to consider applying the ancient 22 æruti system, or for that matter, any system of exact intonation to North Indian classical music.”91

In support of these findings, in Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation Boyle makes the appropriate observation that “sounds should be pitched according to the dictates of the ear. For this certain notes in the scale must be free to move about a comma (approximately 21.506 cents), which is perfectly possible on all but keyboard instruments.” It is further relevant to note that according to foreword contributor Kenneth Van Barthold, in the piano tuning practice of the latter part of the twentieth century, octaves began to be often “stretched for added brilliance,” and therefore “theoretically accurate equal temperament” has never existed in piano tuning, with stretched octaves often induced in the highest and lowest octaves.92 Theoretically accurate octaves or the ratio 2/1 is defined in musical acoustics by a “beat-free condition between all the partials of the upper musical tone and the even-numbered partials of the lower tone.” Stretched, as well as compressed octaves a common occurrence in Piraglu’s tuning of the santñr.

90 Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional Applications,” 197-98. 91 N. A. Jairazbhoy, and A. W. Stone, “Intonation in Present-Day North Indian Classical Music,” Bulletin of the School of Oriental and African Studies, University of London 26.1 (1963): 130-31. 92 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation ix. Theory Versus Performance Practice 55

The Piano Tuner’s Octave and Inharmonicity

The ‘piano tuner’s octave’ may be is stated as being an octave with a falsity of around three cents, or in other words, when (for example) “the fundamental component of C5 on a good piano is set about three cents higher than twice the fundamental component of C4.”93

Table 22. Octaves with a falsity of 3 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C5 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ) 1 261.626 – – – – – – – – – – – – 2 523.251 1 524.159 0.908 3 784.877 – – – – – – – – – – – – 4 1046.502 2 1048.317 1.815 5 1308.128 – – – – – – – – – – – – 6 1569.753 3 1572.476 2.723 7 1831.379 – – – – – – – – – – – – 8 2093.005 4 2096.635 3.630

In a chapter concerned with tunings and temperaments Boyle adds to the discussion of theoretically accurate octaves with the following comments:

“The tuner, who tunes to eliminate beats between the fundamental of the upper note and the second partial tone of the lower note, will naturally stretch the octave physically, and the musical ear, not surprisingly, seems to prefer them that way. This is simply a case of the ear asserting its right to judge – within the limits of choice allowed to it – when such an instrument is mostly nearly in tune with itself musically, and provides a good example of the danger of leaving out the ear and regarding any particular musical interval as being defined primarily and absolutely by a mathematical ratio.”

The present discussion coincides with the natural phenomenon of inharmonicity, which is a factor especially affecting plucked and struck strings (along with other musical sounds with a short decay), and one that displaces the upper partials of a vibrating string with the harmonic series.94 The upwards slope of inharmonicity on the pianoforte is encountered at around middle C in both directions. A reduction of

93 Benade, Fundamentals of Musical Acoustics 319. 94 “In a grand piano, the tension in each string is over 100 pounds, creating a total force on the frame of between 40,000 and 60,000 pounds. A large variation in tension between the lower and the higher strings could lead to warping of the piano frame, so that, in order to apply even tension throughout, the higher strings are shorter and smaller in diameter while the bass strings are constructed of heavy wire wound with additional thin wire. This construction makes the wires stiff, causing the overtones to be higher in frequency than the ideal harmonics and leading to the slight inharmonicity that plays an important part in the characteristic piano tone.” For a further discussion, see Encyclopaedia Britannica, “Sound,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM, Chicago: Encyclopaedia Britannica, 2001. 56 Theory Versus Performance Practice inharmonicity in a string may be brought about via an increase of physical length, and a decrease of diameter, which is the reasoning behind the double and triple stringing (unison twos and threes, or bichords and trichords) of the pianoforte. Physical length is therefore synonymous with flexibility, and directly responsible for the consequential decrease in inharmonicity, and hence, a more well-defined musical tone.95

The Tuning of Unisons

A survey incorporating both musically trained and untrained participants conducted in 1959 by Roger Kirk of the Baldwin Piano Company revealed the general preference for one or two cents deviation among the strings of a triple-string unison of a piano. Although the upper threshold could have been lessened for the musically trained, unisons with as much as eight cents deviation were accorded with wide acceptance. Unisons with a falsity of two cents are described by Arthur H. Benade as “reasonably smooth”, while ones with a falsity of eight cents as a “rather brighter sound, but is not yet the sort of jangle one gets with a spread of 15 to 20 cents.” It should be pointed out that in general performance practice it can be extremely difficult or sometimes even impossible to tune unisons to a “true zero-beat condition,” and one must also consider the important fact that theoretical unisons will produce reduced decay times.

Table 23. Unisons with a falsity of 2 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ) 1 261.626 1 261.928 0.302 2 523.251 2 523.856 0.605 3 784.877 3 785.784 0.907 4 1046.502 4 1047.712 1.210 5 1308.128 5 1309.640 1.512 6 1569.753 6 1571.568 1.814 7 1831.379 7 1833.496 2.117 8 2093.005 8 2095.424 2.419

95 “This reduction is directly proportional to the fourth power of the length and to the square of the diameter,” 1 6 explains Lloyd and Boyle. “Thus, increasing the length by a 5 th, i.e. from 1 to 5 , reduces the inharmonicity by 6 4 1 ( 5 ) =2.07, just over two times, i.e. more than halves its previous value. Again, decreasing the diameter by a 2 , i.e. from 1 1 2 1 1 to 2 , reduces its inharmonicity by ( 2 ) = 4 , i.e. quarters its previous values.” For a further discussion, see Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 166-67. Theory Versus Performance Practice 57

Table 24. Unisons with a falsity of 8 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ) 1 261.626 1 262.837 1.212 2 523.251 2 525.675 2.424 3 784.877 3 788.512 3.635 4 1046.502 4 1051.349 4.847 5 1308.128 5 1314.187 6.059 6 1569.753 6 1577.024 7.271 7 1831.379 7 1840.496 9.117 8 2093.005 8 2102.699 9.694

The Persian santär, essentially 72 strings organized into 18 sets of quadruple-string unisons, is an interesting case for comparison. In order to access deviation among the strings of a quadruple-string unison of a santär, analysis of a separate performance practice tuning by Piraglu restricted to the primary pitches of the second position strings is conducted. The results reveal unisons with a mean falsity of 4.442 cents, which is a pronouncement that coincides with a median perspective of Benade’s “reasonably smooth” and “rather brighter” unisons with falsities of two and eight cents respectively.96

Table 25. Unisons with a falsity of 4.442 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ) 1 261.626 1 262.298 0.672 2 523.251 2 524.595 1.344 3 784.877 3 786.893 2.016 4 1046.502 4 1049.191 2.689 5 1308.128 5 1311.489 3.361 6 1569.753 6 1573.786 4.033 7 1831.379 7 1836.084 4.705 8 2093.005 8 2098.382 5.377

The ‘unison’ experiment consists of the spectrum analysis of second position pitches Eî4, F4, G4, Aî4, BÝ4, C5, Dî5, and EÝ5, which have all been tuned in relationship to the first string of each set. In other words, in stage one of the tuning process of the santñr for dastgàh-e segàh (on F), second position F4 is acquired by tuning the first ‘outside’ F4 string to the first ‘outside’ C5 string (generating a perfect fifth). The tuning of F4 is then concluded with the generation of unisons by the three other strings of the set (2nd, 3rd, and 4th strings of the quadruple-string unison of the santär) in relationship to the first string. The

96 Benade, Fundamentals of Musical Acoustics 334-36. 58 Theory Versus Performance Practice analysis of the eight pitches surveyed conclude the characterization of unisons with a mean value of 4.442 cents, a range between ß9.381 and +9.130 cents, and a standard deviation of 2.416688. In consideration of beats, the results present a mean value of 1.201Hz, a range between ß1.280 and +1.283Hz, and a standard deviation of 0.749103.

Climate and Tuning

Specific climatic conditions no doubt play a significant part in tuning, and therefore the important factor of relative humidity must be taken into account.97 Piraglu makes a point of stating that he has found the climatic conditions of Melbourne, Australia, in comparison to Tehran, Iran, as “unsatisfactory” for the tuning of the santär. The questions therefore arise as to what are the desirable climatic conditions for the tuning of an instrument, and how can official meteorological statistical data for climatic dissimilarities between the two regions support Piraglu’s observations? The climate of Tehran, Iran (latitude: 35.41°N, longitude: 51.25°E, elevation: 1191m) features average minimum and maximum temperatures of 10.3°C and 22.8°C; record minimum and maximum temperatures ranging between ß21.0°C and 43.0°C (averaging ß1.0°C and 31.9°C); an average monthly total precipitation of 20.8mm; and average relative humidity ranging between 59.1% in the morning to 53.3% in the afternoon (averaging 56.2%, with a total range of 5.8%). Melbourne, Australia on the other hand (latitude: ß37.49°S, longitude: 144.58°E, elevation: 35m) features average minimum and maximum temperatures of 9.9°C and 19.8°C; record minimum and maximum temperatures ranging between ß3.0°C and 46.0°C (averaging 1.1°C and 34.8°C); an average monthly total precipitation of 54.2mm; and average relative humidity ranging between 68.7% in the morning to 55.8% in the afternoon (averaging 62.2%, with a total range of 12.9%).98 The tuning of a pianoforte for example, can be affected severely by significant changes in relative humidity over a one- or two-day period, with perpetual and excessive precipitation detrimental to the intonation of the instrument. The ideal level of humidity is 42%, or a level within the range of 35% and

97 According to the Encyclopaedia Britannica humidity is “the amount of water vapour in the air. It is the most variable characteristic of the atmosphere and constitutes a major factor in climate and weather. Atmospheric water vapour is an important factor in weather for several reasons. It regulates air temperature by absorbing thermal radiation both from the Sun and the Earth. Moreover, the higher the vapour content of the atmosphere, the more latent energy is available for the generation of storms. In addition, water vapour is the ultimate source of all forms of condensation and precipitation.” The article making the further point of importance that “care must be taken to distinguish between the relative humidity of the air and its moisture content or density, known as absolute humidity.” For a further discussion, see Encyclopaedia Britannica, “Humidity,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM (Chicago: Encyclopaedia Britannica, 2001) n. pag. 98 “BBC Weather Centre: World Weather,” BBC Home Page, 6 Apr. 2006, British Broadcasting Corporation, 6 Apr. 2006, Theory Versus Performance Practice 59

Table 26. Tuning of the santñr for dastgàh-e segàh (on F) – relationship of unisons to first strings

PITCH STRING FREQUENCY RATIO BEATS FALSITY (HERTZ) (DECIMAL) (HERTZ) (CENTS)

Eî4 Ñ 322.490 1.000000 0.000 +0.000 Ò 323.260 1.002388 0.770 +4.129 Ó 322.860 1.001147 0.370 +1.985 Ô 321.740 0.997674 0.750 ß4.031 F4 Ñ 351.100 1.000000 0.000 +0.000 Ò 351.600 1.001424 0.500 +2.464 Ó 351.830 1.002079 0.730 +3.596 Ô 350.360 0.997892 0.740 ß3.653 G4 Ñ 396.020 1.000000 0.000 +0.000 Ò 395.740 0.999293 0.280 ß1.224 Ó 394.630 0.996490 1.390 ß6.087

Ô 393.880 0.994596 2.140 ß9.381 Aî4 Ñ 428.820 1.000000 0.000 +0.000 Ò 429.530 1.001656 0.710 +2.864

Ó 429.410 1.001376 0.590 +2.380 Ô 430.750 1.004501 1.930 +7.774 BÝ4 Ñ 466.700 1.000000 0.000 +0.000

Ò 467.430 1.001564 0.730 +2.706 Ó 464.670 0.995650 2.030 ß7.547 Ô 464.760 0.995843 1.940 ß7.211 C5 Ñ 529.180 1.000000 0.000 +0.000 Ò 527.870 0.997524 1.310 ß4.291 Ó 527.300 0.996447 1.880 ß6.161 Ô 528.530 0.998772 0.650 ß2.128 Dî5 Ñ 572.060 1.000000 0.000 +0.000 Ò 574.090 1.003549 2.030 +6.133 Ó 572.470 1.000717 0.410 +1.240 Ô 573.470 1.002465 1.410 +4.262 EÝ5 Ñ 622.200 1.000000 0.000 +0.000

Ò 625.490 1.005288 3.290 +9.130 Ó 623.430 1.001977 1.230 +3.419 Ô 621.190 0.998377 1.010 ß2.813 60 Theory Versus Performance Practice

55%. High levels of humidity, 60% and over, produce an expansion of the soundboard, greater string tension, and therefore an increase in overall pitch. “Sticking keys, sluggish action, and rusting strings/tuning pins are other consequences of continued high humidity,” points out Martha Beth Lewis. Low levels of humidity, 34% and under, on the other hand produces a contraction of the soundboard, lesser string tension, and therefore a decrease in overall pitch. “Other effects of low humidity include rattling (loose) keys, slipping tuning pins, and cracks in the soundboard,” explains Lewis. Instruments constructed of a soundboard of reduced density and deficient of the structural support provided by the iron plate of the pianoforte are even more predisposed to the adverse effects of humidity.99 Meteorological statistical data reveal Melbourne as having no climactic period with the ideal level of relative humidity within the range of 35% and 55%, while Tehran, in striking contrast, a period of seven months between April and October. High levels of humidity, 60% and over, are prevalent in Melbourne for ten months of the year, between February and November, while in Tehran, only for five months of the year, between November and March. Neither region suffers from low levels of humidity of 34% and under. The average monthly total precipitation of Melbourne exceeds that of Tehran by 33.4mm, over two and a half times the level, and is therefore a significant divergent factor. All these factors substantiate Piraglu’s claims about the “ideal” conditions for the tuning of the santñr in Tehran.

Gušes of Dastgàh-e Segàh

A prominent radif associated with Mñsà Marñfi (1889-1965) has been utilized in the organization of modal material for the work, which subscribes to a collection of the following twenty-four gušes for dastgàh-e segàh:

i. Mogadameh ii. Daràmad-e Avva iii. Daràmad-e Dovvom iv. Daràmad-e Sevvom v. Piš Zangñleh vi. Zangñleh vii. Zangeh Šotor viii. Zàbol ix. Zàbol (Qesmat-e Dovvom) x. Zangñleh xi. Panjeh Mñye xii. Àvàz-e Mñye xiii. Forñd-e Mñye xiv. Bagiye-e Zàbol xv. Hesàr xvi. Hesàr (Qesmat-e Dovvom) xvii. Nagmeh xviii. Hesàr (Qesmat-e Sevvom) xix. Kerešmeh xx. Forñd-e Hesàr

99 Martha Beth Lewis, “Tuning Your Piano: Why Pianos Go Out of Tune,” Martha Beth Lewis’ Home Page, 1999, 8 Apr. 2006, . Theory Versus Performance Practice 61

xxi. Hozzàn xxii. Pas Hesàr xxiii. Moarbad xxiv. Moxàlef

Àzàdeh for santñr and tape will incorporate six of the most prominent elements of the radif of the dastgàh-e segàh belonging to the school of Marñfi. Pitch material from the main gušes, which include guše-ye zàbol, mñye, moxàlef and maqlub, as well as the daràmad and forñd have been accessed and categorized (according to Piraglu) to be then utilized in adherence to Western contemporary compositional practices, and therefore resulting in a work that has no intention of representing Persian classical music tradition.100 The selection of the six guše-ye of dastgàh-e segàh are represented within the compositional framework as individual demarcations (dividing the work methodically), with composer improvisation (sequentially exploring each guše) utilized to generate all the notated material of the work.

Ex. 6. Six of the most prominent elements of the radif of the dastgàh-e segàh

First tetrachord i. Daràmad-e segàh

ii. Guêe-ye zabçl

iii. Guêe-ye mñye

Second tetrachord iv. Guêe-ye moxàlef

100 Piraglu, “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties,” 41-43. 62 Theory Versus Performance Practice v. Guêe-ye maqlub

vi. Segàh forñd

A structural scheme based on ‘golden mean’ or ‘golden section’ proportions has been incorporated in the linear plan. The aesthetic notion of these proportions being a technique directly borrowed from art and architecture, with its conceptual basis stating that “if the proportion of ‘ab’ to ‘bc’ is the same as the proportion of ‘bc’ to the whole line, then ‘ac’ is segmented according to the golden mean.” The ratio ab  bc represented by this ideology is approximately 1:1.618 (or the relationship bc ac ), and manifests itself in the work at guêe-ye moxàlef with its introduction of melodic material based on the second tetrachord of dastgàh-e segàh (720 seconds Ï .618 = 444.96 seconds, or approximately 7:25).101

Table 27. Structural scheme

NUMBER EVENT CLOCK COUNTER DURATION GOLDEN (SECONDS) MEAN 1 daràmad-e segàh (first tetrachord) 0:00 0 2:50 – – – – 2 taknavàzi-e santñr 2:50 170 1:45 3 (3-1) 3 guêe-ye zabçl 4:35 275 1:05 2 (7-1) 4 taknavàzi-e santñr 5:40 340 0:40 5 (5-3) 5 guêe-ye mñye 6:20 380 0:40 4 (7-3) 6 taknavàzi-e santñr 7:00 420 0:25 6 (7-5) 7 guêe-ye moxàlef (second tetrachord) 7:25 445 1:45 1 (13-1) 8 taknavàzi-e santñr 9:10 550 1:05 8 (9-7) 9 guêe-ye maqlub 10:15 615 0:40 7 (13-7) 10 taknavàzi-e santñr 10:55 655 0:25 10 (11-9) 11 forñd-e segàh 11:20 680 0:25 9 (13-9) 12 taknavàzi-e santñr 11:45 705 0:15 11 (13-11) 13 fade out 12:00 720 1:00 – – – –

14 end 13:00 780 0:00 – – – –

101 Stefan Kostka, Materials and Techniques of Twentieth-Century Music, 2nd ed. (Upper Saddle River, NJ: Prentice- Hall, 1999) 150-51. Theory Versus Performance Practice 63

Sampling of the Santñr and Vocals

The following stage of the compositional process involves the assembly of the tape element of the work, which begins with the transferral of the complete set of santñr samples (the twenty-seven individual samples collected during the first recording session with Piraglu) into the memory of an Akai S3000XL Midi Stereo Digital Sampler. This will provide a platform for the retuning (to a hundredth of a cent) of each individual sample in the edit sample ‘pitch offset’ parameter window of the Akai S3000XL (to firstly adhere to equal temperament), in order to be then readjusted (in cent increments) in the edit program ‘tune’ parameter window (enabling the capture of the frequency ratios of Barkešli’s twenty-two-note division of the octave). The frequencies (in hertz), obtained via the spectrum analyzing module of Adobe Audition 2.0, once converted into cents will provide all the necessary data required to offset pitch. The alternative tuning system will service the accompanying sampled santñr – one of the two principal tape elements of the work – and the ‘call and response’ component that will effectuate the ‘theoretical’ and ‘performance practice’ tuning comparison proposed by Àzàdeh for santñr and tape. Performances of the ‘sampled santñr’ notated phrases of the work have been recorded on a midi sequencer (Roland MC-500 Mark II Micro Composer), utilizing the Akai S3000XL as a sound source, with the product finally digitally transferred to Sony Sound Forge 8.0 via optical cable (routed through the Fostex D-160 Digital Multitrack Recorder). A modified ‘white noise’ preset of the Waves X-Noise 5.2 DX plug-in (“an audio plug-in that intelligently learns from a section of noise, and then applies a broadband noise reduction to eliminate background noise from any source”) is utilized to remove unwanted white noise (threshold: ß20dB, reduction: 70%, [dynamics] attack: 30ms, release: 220ms; [high shelf] frequency: 1415Hz, gain: +0.0dB).

Table 28. Mehdi Barkešli’s twenty-two-note division of the octave tuning matrix ‘key of F’ (Program 01)

NOTE C – – Dî E" Eî F – – G A" Aî B" – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 ß20 ß06 ß16 ß02 +00 +02 ß08 ß18 ß04 +00

Stage two of the assembly requires the recording of the recitation of poetry by Piraglu (an original poem entitled Àzàdeh), as well as the individual syllables constituting the Farsi (Modern Persian) vocalizations of Àzàdeh,102 which include À, zà, and deh – the three syllables sung at two distinct pitches (348.834Hz

102 “The official language of Iran is Iranian or Modern Persian (Iranian/Indo-European), which is written in Arabic script.” For a further discussion, see Akira Nakanishi, Writing Systems of the World: Alphabets, Syllabaries, Pictograms (Rutland, VT: Charles E. Tuttle Co., 1980) 30. 64 Theory Versus Performance Practice and 174.417Hz, or F4 and F3). Àzàdeh includes the following four lines of text, which are presented below in Farsi (Arabic and Roman script), together with the English translation:

Kas nadànad darde bi darmàne mà,

Nobody knows the pain inside me,

Jore Leili dar dele por êñre mà.

The pain that emanates, being away from my beloved Àzàdeh.

Hazar kardim ze haráe xalgo donyà

Because my soul is with her, I am oblivious to everybody and everything

Kajàst Àzàdeh in mahpeykare mà?

Where is my beloved Àzàdeh, who reflects the beauty of moonlight?

Àzàdeh, Qmars Piraglu

Àzàdeh, Qmars Piraglu

All the samples are recorded on the Fostex D-160 Digital Multitrack Recorder, and then processed within Sony Sound Forge 8.0 utilizing various audio plug-ins: Waves L3 Multimaximizer 5.2 (“a multi-band peak limiting audio plug-in”)103 is utilized for general compression (left and right input: +0.0dB, threshold: ß2.3dB, out ceiling: ß0.3dB, release: 10.00ms); dB-L Audioware Mastering Limiter 1.05 (“an intelligent loudness maximizer and brickwall limiter audio plug-in with dithering”),104 for general limiting (gain: 3.0dB, release: 50ms, output: ß0.3dB, attack: hard, dither: off); while Sony Noise Reduction 2.0 (“a collection of four professional-level sound restoration audio plug-ins: vinyl restoration, noise reduction, clipped peak

103 Waves: The World’s Leading Developer of Audio Signal Processing Software, 2006, Waves Incorporated, 15 Dec. 2006, . 104 dB Audioware: Professional Audio Software, 2006, dB Audioware Limited, 15 Dec. 2006, . Theory Versus Performance Practice 65 restoration, and click and crackle removal”),105 to remove unwanted tape hiss (reduction type: mode 3, reduce noise by: 30.0dB, noise bias: +0.0dB, attack speed: 90ms, release speed: 50ms, windowing FFT size: 1024, windowing overlap: 67%). The two sets of three syllables are then arranged within the digital domain as a stereo configuration sounding a low F3-set panned left, left-right, and left-right-left; followed by a high F4-set panned right, right-left, and right-left-right – the time points selected in accordance to the ‘golden mean’ structural scheme and therefore acting as structural markers. An Antares Auto-tune 4.31 DX plug-in (“an audio plug-in that corrects intonation problems in vocals or solo instruments”)106 is utilized to tune the samples (input type: low male voice, key: F, scale: chromatic, remove: F! and E), while a Spin Audio 3D Delays 1.1 DX plug-in (“a multi-tap dimensional space delay audio plug-in that provides six independent stereo delay lines which can be freely positioned in 3D sound field”),107 to generate simultaneous quarter-note and eighth-note triplet delays (delay 1 [mode: ms, tempo: 120bpm, time: 375ms, feedback: 55%, filter: off, taps: ß6.0dB]; delay 2 [mode: ms, tempo: 120bpm, time: 125ms, feedback: 65%, filter: off, taps: ß6.0dB]; I/O [in: ß3.0dB, dry: +0.0dB, wet: ß4.0dB]). The singular 174.417Hz À component is further developed as an F pedal point or drone in the form of a 500ms and 250ms rhythmic pulse (at Ê=120 equal to simultaneous crotchets and quavers). A Waves Super Tap 5.0 DX plug-in (“an audio plug-in with six taps with up to six seconds of mono or true stereo delay, as well as independent Q10-style filtering, rotation [stereo panning], gain, and precision time control”)108 is utilized to generate a thirteen-minute rhythmic pulse ([tempo: 120bmp, ms: 500, modulation: off; gain: ß3.6dB]; delay 1 [gain: +0.0dB, rotation: ß45°, delay: 250ms, eq: off]; delay 2 [gain +0.0dB, rotation: ß45°, delay: 500ms, eq: off]; feedback [gain: 99%, rotation: 0°, delay: 500ms]), while a PSP Audioware Nitro 1.0.2 DX plug-in (“a multimode filter audio plug-in with many filter types derived from analog prototypes in addition to other useful processing blocks such as phaser, bit- crusher/downsampler, waveshaper, and interpolated delay blocks”),109 to generate ‘state variable low- pass’ filter sweeps throughout the sample (op 1 [frequency: 553Hz, resonance: 29.40%, level: +6.0dB]; op 2 [frequency: 790Hz, resonance: 68.97%, level: +6.0dB]; op 3 [attenuation: +0.0dB, level: maximum]; op 4 [attenuation: +0.0dB, level: maximum]; I/O [in: +0.0dB, mix: 25.00%, out: +1.0dB]).

105 Sony Media Software: Home for Vegas, Sound Forge and Acid, 2006, Sony Corporation of America, 15 Dec. 2006, . 106 Antares Audio Technologies, 2006, Antares Audio Technologies, 15 Dec. 2006, . 107 Spin Audio Software, 2006, Spin Audio Software, 15 Dec. 2006, . 108 Waves: The World’s Leading Developer of Audio Signal Processing Software, 2006, Waves Incorporated, 15 Dec. 2006, . 109 PSP Audioware: Audio Processors and Effects Plug-ins, 2006, PSP Audioware, 15 Dec. 2006, . 66 Theory Versus Performance Practice

A secondary thirteen-minute rhythmic pulse is then created from the sequential layering of a one- second sample (in order to duplicate sonic parameters generated by the Waves Super Tap 5.0 DX plug- in) of the primary rhythmic pulse, which is then subjected to thirteen digital fade-ins and therefore every ‘golden mean’ point of the structural scheme (a 500ms fade-out additionally executed to smooth out the junctions). The abovementioned PSP Audioware Nitro 1.0.2 DX plug-in setting with a slight modification of the effect level (mix: 75.00%) is also utilized to generate ‘state variable low-pass’ filter sweeps throughout the sample. The adoption of dissimilar effect levels (25.00% in the primary and 75.00% in secondary rhythmic pulses), and the fact that the superimposition of the two rhythmic pulses essentially generates a series of gradual digital cross-fades, adds up to an unbroken rhythmic pulse of continually transforming timbral colour. The sequential layering of a one-second sample of the secondary thirteen-minute rhythmic pulse is then employed to create a tertiary thirteen-minute rhythmic pulse. The PSP Audioware Nitro 1.0.2 DX plug-in is utilized to generate ‘stereo width balance’ spatial nuances (stereo field modifications) in the sample (op 1 [width: 100.00%, balance: left/100%, level: maximum]; op 2 [width: 100.00%, balance: right/100%, level: maximum]; op 3 [feedback: 50.00%, depth: 94.04%, level: maximum]; op 4 [pan left: left/26.6%, pan right: right/24.6%, level: +0.0dB]; I/O [in: +2.0dB, mix: 100.00%, out: ß8.0dB]). The resulting .wav file (similarly to the secondary rhythmic pulse) is then also subjected to thirteen digital fade- ins. The next step involves a DSound Stomp’n Fx DN-SG1 Noise Gate DX plug-in (“a dynamics effects processor audio plug-in whose function is to remove unwanted audio material below a certain threshold”),110 which is utilized to suppress the sustaining quality of the sample (tone level: ß32dB, attack: 170ms, release: 1880ms), while the Spin Audio 3D Delays 1.1 DX plug-in, to generate simultaneous ‘high cut’ filtered multiple-time delays (delay 1 [mode: ms, tempo: 120bpm, time: 1250ms, feedback: 3%]; delay 2 [mode: ms, tempo: 120bpm, time: 1000ms, feedback: 3%]; filter 1 and 2 [routing: out, type: low pass, frequency: 400Hz, gain: +0.0dB, q/filter slope steepness: 1.0]; delay 3 [mode: ms, tempo: 120bpm, time: 1500ms, feedback: 3%]; delay 4 [mode: ms, tempo: 120bpm, time: 1750ms, feedback: 3%]; filter 3 and 4 [routing: out, type: low pass, frequency: 200Hz, gain: +0.0dB, q/filter slope steepness: 1.0]; taps: ß1.9dB; I/O [in: ß1.0dB, dry: +0.0dB, wet: +0.0dB]). The final recordings are further subjected to digital sound processing modifications (namely equalization and reverberation) at the final stages of mixing and mastering, where all the tracks (including the principal solo santñr recording [recorded separately on the Fostex D-160 Digital Multitrack Recorder], as well as the accompanying ‘sampled’ santñr, primary pulse, secondary pulse, tertiary pulse, vocal samples, and recitation of poetry, which constitute the tape element of the work) are conclusively assembled. A separate mix omitting the principal solo santñr is also produced in order to reproduce the work in a live performance context.

110 DSound, 2005, DSound, 15 Dec. 2006, . 2. The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut Toy Piano

Equal Temperaments

According to American composer and theorist Harry Partch,111 there are two distinct classes of equal temperaments, with the first including “those which divide the already equal tone into further equal parts,” while the second including “those which compress one of the Pythagorean cycles into the 2/1 to obtain such divisions as nineteen equal degrees and fifty-three equal degrees.”112 The results of the former being third-tones (as in eighteen-tone equal temperament), quarter-tones (as in twenty-four-tone equal temperament), fifth-tones (as in thirty-tone equal temperament), sixth-tones (as in thirty-six-tone equal temperament), eighth-tones (as in forty-eight-tone equal temperament), twelfth-tones (as in seventy-two-tone equal temperament), and sixteenth-tones (as in ninety-six-tone equal temperament); while of the latter, namely nineteen-tone, thirty-one-tone, forty-three-tone, and fifty-three-tone equal temperaments. The superimposition of a second twelve-tone equally-tempered division of the octave a sixth of a tone, or 33.333 cents higher, as in thirty-six-tone equal temperament, is termed by Partch as a polypythagoreanism, due to its correlation to the Pythagorean concept of the juxtaposition of perfect fifths.113 The general accepted abbreviation for equal temperament, or tuning system of “logarithmically equal intervals” is ‘ET’, although some theorists subscribe to the lesser ambiguity of ‘EDO’, or equally- divided octave, which better defines the probability, or improbability of the ‘extended’ perimeter. Dan Streams makes the following observations: “Usually, but not always, equal temperaments assume octave- equivalence, of which the usual 12-edo is the most obvious example. For many theorists the preferred abbreviation or these types of temperaments is EDO, for which some other theorists substitute ED2; both of these specify that it is the 2:1 ratio which is to be equally divided.”114

111 “Visionary composer, theorist, and creator of musical instruments, Harry Partch (1901-1974) was a leading figure in the development of an indigenously American contemporary music. A pioneer in his explorations of new instruments and new tunings, Partch created multimedia theatre works that combine sight and sound in a compelling synthesis. He is acknowledged as a major inspiration to postwar experimental composers as diverse as Gyõrgy Ligeti, Lou Harrison, Philip Glass, and Laurie Anderson, and his book Genesis of a Music, first published in 1949, is now considered a classic.” For a further discussion, see Bob Gilmore, Harry Partch: A Biography (New Haven, CT: Yale U. Press, 1998) n. pag. 112 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 425. 113 Read, 20th-Century Microtonal Notation 13-145. 114 Dan Stearns, “Some Thoughts on an Alternative Definition of Equal Temperament,” Kronosonic, 2006, The International Society for Creative Guitar and String Music, 1 Aug. 2006, . 68 The Equally-Tempered Archetype

Studies of Microtonal Equal Temperaments

Mexican composer Julián Carrillo Trujillo (1875-1965) utilized quarter-tones, eighth-tones, and sixteenth- tones in his compositions; publishing his theories under the heading of Sonido Trece or the ‘Thirteenth Sound’ in a series of writings published in Mexico and the United States between 1927 and 1957 – the number thirteen merely representing a system beyond that of the traditional twelve tones, and essentially ninety-six-tone equal temperament ( 96 2 ). The pitch resources of such a system provide for 780 distinct intervals as opposed to the 12 of twelve-tone equal temperament. “The reason for stopping with this intervallic division was not dictated by theoretical considerations, but by a purely physical one: Carrillo felt that the human ear could not distinguish clearly beyond 1/16th of a tone,” explains Gerald R. Benjamin. Carrillo also introduced new or adapted instruments to produce quarter-, eighth-, and sixteenth-tones, and devised a numerical notation system based on “relative number notation.” The quarter-tone or “quarters of tone” system is represented thus: 0, 1/4, 2/4, 3/4, 5/4, 6/4, 7/4, 8/4, 9/4, 10/4, 11/4, 12/4, 13/4, 14/4, 15/4, 16/4, 17/4, 18/4, 19/4, 20/4, 21/4, 22/4, 23/4, and 0.115 Italian composer Ferruccio Busoni (1866-1957) on the other hand, subscribed to the theorem pronouncing the third-tone as a product of the division of the octave into six equal tones, and the further threefold subdivision of each equal tone, which results in an eighteen-tone scale and therefore eighteen- tone equal temperament ( 18 2 ). In his 1911 publication of Sketch of a New Aesthetic of Music Busoni states, “Tripartite tones are wholly independent intervals with a pronounced character, and not to be confounded with ill-tuned semitones. They form a refinement in chromatics based, as at present appears, on the whole-tone scale.” Other tripartite systems investigated by Busoni include sixth-tones, or the threefold subdivision of each equal semitone, which results in a thirty-six-tone scale and therefore thirty-six-tone equal temperament ( 36 2 ). Busoni devised a notation system for sixth-tones consisting of a six-line staff designating “open noteheads” to the spaces and “solid noteheads” to the lines, with each representing the tripartite divisions of diatonic and chromatic semitones respectively.116 If one were to entertain the notion that the “most important characteristic sought in a temperament is its ability, in principle, to support both traditional tonality and atonality with intervals of greater purity than are found in twelve-tone equal temperament,” Ramon Fuller’s A Study of Microtonal Equal Temperaments presents a set of sound guidelines. According to Fuller, octaves classify as the most important interval, because of their ability “to maintain audibly stable musical structures”, and therefore ‘pure’ or ‘just’ 2/1 ratios are of the outmost importance. Second in line come fifths, or 3/2s (the third harmonic), “because of their role in musical structure and the ear’s sensitivity to mistuning”, which should be represented within a temperament with a size ranging from 699.500 cents to 704.400 cents, or a just

115 Gerald R. Benjamin, “Julian Carrillo and ‘Sonido Trece’ (Dedicated to the Memory of Nabor Carrillo),” Anuario 3 (1967): 33-68. 116 Read, 20th-Century Microtonal Notation 95-96. The Equally-Tempered Archetype 69 perfect fifth (3/2) with a falsity of around two cents; the ideology representing a tempered cycle of fifths as opposed to a “mathematically exact” one, and therefore providing a link between the perfect fifth and the octave. Major and minor thirds (resembling the just major third, with a ratio of 5/4, or 386.314; and just minor third, with a ratio of 6/5, or 315.641 cents) become the next consideration. It is interesting to note that although twelve-tone equal temperament qualifies on the first two counts (representing the just perfect octave [2/1] correctly, while the just perfect fifth [3/2] with a falsity of ß1.955 cents), the common temperament features major and minor thirds with a falsity of +13.686 and ß15.641 cents respectively; the mean of the falsity equal to the value of 9.776 cents, and the maximum amount of error, 17.596 cents (represented by the equal minor seventh).

Table 29. Deviation of basic equally-tempered intervals from just intonation

COMPARATIVE TABLE JUST INTONATION EQUAL TEMPERAMENT INTERVAL RATIO CENTS RATIO CENTS FALSITY (CENTS) unison 1/1 0.000 1.000000 0.000 +0.000 diatonic semitone 16/15 111.731 1.059463 100.000 ß11.731 major tone 9/8 203.910 1.122462 200.000 ß3.910 minor third 6/5 315.641 1.189207 300.000 ß15.641 major third 5/4 386.314 1.259921 400.000 +13.686 perfect fourth 4/3 498.045 1.334840 500.000 +1.955 tritone 45/32 590.224 1.414214 600.000 +9.776 perfect fifth 3/2 701.955 1.498307 700.000 ß1.955 minor sixth 8/5 813.686 1.587401 800.000 ß13.686 major sixth 5/3 884.359 1.681793 900.000 +15.641 minor seventh 9/5 1017.596 1.781797 1000.000 ß17.596 major seventh 15/8 1088.269 1.887749 1100.000 +11.731 octave 2/1 1200.000 2.000000 1200.000 +0.000

Fuller also recommends that major thirds “should be no more than seven cents sharp in a good microtonal temperament,” adding that “to accommodate the ear’s bias, we will require a major third to be no more than four cents flat, for a total acceptable range from about 382.000 cents and 393.000 cents;” while allocating the rule for minor thirds that they be “no more than eight cents flat or four cents sharp, for a total acceptable range from 308.000 cents to 320.000 cents.” In conclusion, Fuller’s eight best temperaments (selected from all equal temperaments from 1 to 144) in ascending order include the following: fifty-three-tone, sixty-five-tone, eighty-seven-tone, ninety-nine-tone, one hundred and six-tone, one hundred and eighteen-tone, one hundred and thirty-tone, and one hundred and forty-tone equal temperaments (53-et, 65-et, 87-et, 99-et, 106-et, 118-et, 130-et, and 140-et).

70 The Equally-Tempered Archetype

Table 30. Fuller’s eight best equal temperaments

TEMPERAMENT MINOR THIRD MINOR THIRD MAJOR THIRD MAJOR THIRD PERFECT FIFTH PERFECT FIFTH PC NUMBER (CENTS) PC NUMBER (CENTS) PC NUMBER (CENTS) 53-et 14 316.981 17 384.906 31 701.887 65-et 17 313.846 21 387.692 38 701.538 87-et 23 317.241 28 386.207 51 703.448 99-et 26 315.152 32 387.879 58 703.030 106-et 28 316.981 34 384.906 62 701.887 118-et 31 315.254 38 386.441 69 701.695 130-et 34 313.846 42 387.692 76 701.538 140-et 37 317.143 45 385.714 82 702.857

In The Structure of Recognizable Diatonic Tunings Easley Blackwood117 presents the concept of equal temperaments and ‘recognizable diatonic tunings’ – “those in which the perfect fifths ultimately form a closed circle” – proposing that “in order for any array of notes to contain recognizable diatonic scales, it is both necessary and sufficient that the array should contain seven adjacent intervals that are the same size, and are perfect fifths within the range of recognizability.” The theorem in effect pronouncing that for a tuning to be capable of generating recognizable diatonic scales, its intervallic boundaries must adhere

4 3 to the formula: 7 << 5 ava , or 685.714 v << 720.000. The criteria for a “perfect fifth within the range of recognizability” therefore stipulates that the interval be not smaller than four sevenths of an octave, and not larger than three fifths of an octave, hence within the range of +18.045 cents and ß16.241 cents from a just perfect fifth (3/2). Tunings accommodating Blackwood’s ideals include 12, 17, 19, 22, 24, 26, 27, 29, 31, 33, 34, and 36 or more equal divisions of the octave.118 Dutch musicologist Rudolf Rasch 4  3 subscribes to the modification of the formula: 7 5 ava (‘less than’ replaced by ‘less than or equal to’), which has the capacity to indicate “which equal temperaments have no recognizable v, as a by- product of showing which have more than one.”119

117 American composer-theorist Easley Blackwood, a significant proponent of the equally-tempered modus operandi, subscribes to the notion that “extended microtonal systems based on equal-temperament tunings are as valid acoustically and musically, and as technically challenging as fractional divisions of the octave adhering to just or mean-tone intonational principles.” For a further discussion, see Read, 20th-Century Microtonal Notation 5. 118 Easley Blackwood, The Structure of Recognizable Diatonic Tunings (Princeton, NJ: Princeton U. Press, 1985) 221- 54. 119 Paul Rapoport, “The Structural Relationships of Fifths and Thirds in Equal Temperaments,” Journal of Music Theory 37.2 (Autumn, 1993): 359. The Equally-Tempered Archetype 71

Nicolas Mercator’s Fifty-Three-Tone Equally-Tempered Division of the Octave

According to Fuller, fifty-three-tone equal temperament ( 53 2 ) is “the most nearly ideal of all temperaments for working with pure intervals and just scales,” noting that due to the fact that it has a prime structure, there are no “embedded subtemperaments,” and “any one of the intervals of T53 can be arranged in a cycle that will generate the complete set of T53 pitch-classes.”120 Nicolas Mercator (1620-87) is acknowledged for discovering “that if the octave is divided into 53 equal intervals, 31 of them give a very perfect fifth and 17 a very good major third.”121 Daniel James Wolf offers the following criticism of fifty-three-tone equal temperament:

“While 53tet does provide excellent approximations of 5-limit intervals, it is problematic in at least two ways. For one, the temperament is awkward, if not unsuitable, for the performance of existing repertoire. If Western classical triadic tonality can be heard as premised upon having the best major third present in the tuning system equivalent to the (octave-equivalent) sum of four consecutive perfect fifths – a properly equally present in each of the major tuning or temperament environments used in common practice counterpoint and harmony (meantone, well-temperaments, 12tet) – then this premise is unfulfilled by 53tet. In 53tet, which might be thought of a scale of 53 modestly tempered syntonic commas, the best major third remains one scale step distant from the sum of four perfect fifths; as a consequence, realizations of existing repertoire may well tend to ‘drift’ in pitch from an initial tonic by the approximate comma interval of 1/53 octave. But, perhaps more critically in this speculative context, given the extravagant resources required to implement 53tet in notation or instruments, it does not offer significantly better and consistent representations of intervals beyond the 5-limit.”122

A contrasting ideology of the equally-tempered paradigm is presented by Dirk de Klerk in his study of equal temperaments (subscribing to thirty-four and forty-six equal divisions of the octave), which presents the following analytical commentary:

“If we ignore the seventh harmonic, the tunings with 34 and 46 divisions in the octave give results that are feasible as in the Mercator temperament (53-et). They are also a good deal more perfect than our 12- semitone system and also better than those of Wesley Woolhouse (19-et), Christian Huygens (31-et), and Von Janko (41-et). There is no point preferring Mercator’s system to 34 and 46 and Janko’s to 34. If we also include the seventh harmonic it appears that it is as feasible in the temperaments of Janko and Mercator and in 46 as it is in that of Huygens (and Fokker), whereas in the cases of 46 and Mercator the fifth and the minor third are better.”123

120 Ramon Fuller, “A Study of Microtonal Equal Temperaments,” Journal of Music Theory 35.1/2 (Spring-Autumn, 1991): 212-20. 121 Dirk de Klerk, “Equal Temperament,” Acta Musicologica 51.1 (Jan.-Jun., 1979): 140. 122 Daniel James Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 22.1-2 (2003): 4. 123 Klerk, “Equal Temperament,” 150. 2TeEulyTmee Archetype Equally-Tempered The 72 Table 31. Nicolas Mercator’s fifty-three-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ C¢ 53-et syntonic comma 53 2 1.013164 265.070 22.642 +23 ÐÓ C£ 53-et great diesis (53 2)2 1.026502 268.559 45.283 +45 ÐÔ CÚ 53-et grave or small just chromatic semitone, or minor half-tone (53 2)3 1.040015 272.094 67.925 ß32 ÐÕ D¹ 53-et Pythagorean limma (53 2)4 1.053705 275.676 90.566 ß09 ÐÖ D¸ 53-et just diatonic semitone, or major half-tone (53 2)5 1.067577 279.305 113.208 +13 Ð× DÝ 53-et great limma, acute or large half-tone (53 2)6 1.081630 282.897 135.849 +36 ÐØ D³ 53-et grave or small tone (53 2)7 1.095869 286.707 158.491 ß42 ÐÙ D² 53-et just minor tone (53 2)8 1.110295 290.482 181.132 ß19 ÑÐ D 53-et just major tone (53 2)9 1.124911 294.306 203.774 +04 ÑÑ D¢ 53-et acute or large tone (53 2)10 1.139720 298.180 226.415 +26 ÑÒ D£ 53-et supermajor second (53 2)11 1.154723 302.105 249.057 +49 ÑÓ DÚ 53-et augmented second (53 2)12 1.169924 306.082 271.698 ß28 ÑÔ E¸ 53-et Pythagorean minor third, or trihemitone (53 2)13 1.185325 310.111 294.340 ß06 ÑÕ EÝ 53-et just minor third (53 2)14 1.200929 314.194 316.981 +17 ÑÖ E³ 53-et neutral third (53 2)15 1.216738 318.330 339.623 +40 Ñ× E² 53-et grave or small major third (53 2)16 1.232756 322.520 362.264 ß38 ÑØ E 53-et just major third (53 2)17 1.248984 326.766 384.906 ß15 ÑÙ E¢ 53-et Pythagorean major third, or ditone (53 2)18 1.265426 331.068 407.547 +08 ÒÐ E£ 53-et acute or large major third (53 2)19 1.282084 335.426 430.189 +30 ÒÑ F³ 53-et subfourth (53 2)20 1.298961 339.841 452.830 ß47 ÒÒ F² 53-et grave or small fourth (53 2)21 1.316061 344.315 475.472 ß25 ÒÓ F 53-et just and Pythagorean perfect fourth (53 2)22 1.333386 348.848 498.113 ß02 ÒÔ F¢ 53-et acute or large fourth (53 2)23 1.350939 353.440 520.755 +21 ÒÕ F£ 53-et superfourth (53 2)24 1.368723 358.093 543.396 +43 ÒÖ FÚ 53-et grave or small augmented fourth (53 2)25 1.386741 362.807 566.038 ß34 DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) Ò× F¤ 53-et just tritone, or augmented fourth (53 2)26 1.404996 367.583 588.679 ß11 ÒØ F¥ 53-et acute or large tritone, or augmented fourth (53 2)27 1.423492 372.422 611.321 +11 ÒÙ GÝ 53-et acute or large diminished fifth (53 2)28 1.442231 377.324 633.962 +34 ÓÐ G³ 53-et subfifth (53 2)29 1.461216 382.292 656.604 ß43 ÓÑ G² 53-et grave or small fifth (53 2)30 1.480452 387.324 679.245 ß21 ÓÒ G 53-et just and Pythagorean perfect fifth (53 2)31 1.499941 392.423 701.887 +02 ÓÓ G¢ 53-et acute or large fifth (53 2)32 1.519686 397.589 724.528 +25 ÓÔ G£ 53-et superfifth (53 2)33 1.539692 402.823 747.170 +47 ÓÕ GÚ 53-et augmented fifth (53 2)34 1.559960 408.126 769.811 ß30 ÓÖ A¸ 53-et Pythagorean minor sixth (53 2)35 1.580496 413.498 792.453 ß08 Ó× AÝ 53-et just minor sixth (53 2)36 1.601302 418.942 815.094 +15 ÓØ A³ 53-et neutral sixth (53 2)37 1.622382 424.457 837.736 +42 ÓÙ A² 53-et grave or small major sixth (53 2)38 1.643739 430.044 860.377 ß40 ÔÐ A 53-et just major sixth (53 2)39 1.665377 435.705 883.019 ß17 ÔÑ A¢ 53-et Pythagorean major sixth (53 2)40 1.687301 441.441 905.660 +06 ÔÒ A£ 53-et acute or large major sixth (53 2)41 1.709512 447.252 928.302 +28 ÔÓ AÚ 53-et augmented sixth (53 2)42 1.732017 453.140 950.943 ß49 ÔÔ B¹ 53-et grave or small minor seventh (53 2)43 1.754817 459.105 973.585 ß26 ÔÕ B¸ 53-et Pythagorean minor seventh (53 2)44 1.777918 465.149 996.226 ß04

45 T ÔÖ BÝ 53-et acute or large minor seventh (53 2) 1.801323 471.272 1018.868 +19 73 Archetype Equally-Tempered he Ô× B³ 53-et neutral seventh (53 2)46 1.825036 477.476 1041.509 +42 ÔØ B² 53-et grave or small major seventh (53 2)47 1.849061 483.762 1064.151 ß36 ÔÙ B 53-et just diatonic major seventh (53 2)48 1.873402 490.130 1086.792 ß13 ÕÐ B¢ 53-et Pythagorean major seventh (53 2)49 1.898064 496.582 1109.434 +09 ÕÑ B£ 53-et acute or large major seventh (53 2)50 1.923050 503.119 1132.075 +32 ÕÒ C³ 53-et suboctave (53 2)51 1.948365 509.742 1154.717 ß45 ÕÓ C² 53-et grave or small octave (53 2)52 1.974014 516.452 1177.358 ß23

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 74 The Equally-Tempered Archetype

Pietro Aron’s Quarter-Comma Meantone Tempered Division of the Octave

Before the advent of twelve-tone equal temperament in the West, Pythagorean intonation was the predominant doctrine (“presumably from the Dark Ages to the 1200s”), although musicians eventually became dissatisfied with the extreme sharpness of the Pythagorean thirds (the just major third [5/4] with a falsity of +21.506 cents), and began to experiment by altering the fifths with an “indefinite amount of diminution called temperament,” or participate.124 Historically, the first theoretical account of temperament in the West (or quasi-Pythagorean intonation) comes during the Renaissance from Italian music theorist and composer Franchinus Gaffurius’s (1451-1522) Practica musica, which was published in Milan, in 1496. Meantone temperament, or the division of the just major third (5/4) into two mean tones

2 80 (equal to the meantone major tone, 9/8× 81 , or 193.157 cents), may be attributed to Italian music theorist and composer Pietro Aron (1489-1555), who in his 1523 treatise entitled Toscanello in musica described a method for tempering the fifths that would in time serve as the foundation for numerous systems of tempered intonation.125 It should be noted that as Mark Lindley rightly points out, “When meantone systems gradually went out of fashion on keyboard instruments after c. 1650, they yielded, not to equal temperament, but to a type of irregular system.” Irregular systems were prevalent between 1680 and c. 1800, and include ‘well temperaments’ prescribed by composers Andreas Werckmeister (1645-1706), Jean-Philippe Rameau (1683-1764), Francesco Antonio Vallotti (1697-1780), and Muzio Clementi (1752-1832); as well as by scientists and mathematicians Jacopo Francesco Riccati (1676- 1754), Johann Heinrich Lambert (1728-77), and Thomas Young (1773-1829). In A Venerable Temperament Rediscovered, Douglas Leedy provides an insightful account of the rationale behind the half-millennium domination of meantone temperament.126

“In the history of Western music, the keyboard temperament with the longest run was meantone, which was the nearly universal standard from 1350 or so until it was almost totally eclipsed by twelve-tone equal temperament in the first half of the nineteenth century. Meantone and the development (and ultimate success) of keyboard instruments are inextricably intertwined: because its sonorities are close to those of just tuning, meantone was able to emulate the sweetness of post-Pythagorean, pure triadic intonation

124 “In the visual arts, the mid- to late-15th century was a time of mathematically governed precision and elegance. Geometrically planned perspective was introduced into painting and intarsia, and the first great masters (Brunelleschi, Alberti, Francesco di Giorgio Martini) were architects who made remarkably sensitive use of very simple ratios in their buildings. It was at this time that musicians, evidently alert to the distinctive sound of the nearly pure thirds in a quasi- Pythagorean system, began to temper the fifths among the naturals so as to have unequivocally consonant thirds there as well.” For a further discussion, see Mark Lindley, Mathematical Models of Musical Scales: A New Approach (Bonn: Verlag für Systematische Musikwissenschaft, 1993) 134-38. 125 Barbour, Tuning and Temperament: A Historical Survey 25-28. 126 Lindley, Mathematical Models of Musical Scales: A New Approach 134-52. The Equally-Tempered Archetype 75

characteristics of vocal music from around the time of Dunstable – the earliest pieces were in fact transcriptions of vocal works.”

The author then continues the discussion with this evocative description of Meantone’s intervallic properties:

“What unfortunately cannot be conveyed in words is the warm and serene beauty of meantone’s consonant harmonies, its arresting dissonances, or the vivid colour, kaleidophonic variety, and expressive strength of its melodic intervals. These need to be heard and savoured over time, for time is necessary to accustom oneself to meantone’s richness and to becomes attuned to its subtlety and shading - much as one would need time to become accustomed to the replacement of a diet of uniform blandness with the variety of flavour, colour, and piquancy afforded by a Lucullan cuisine.”127

Pietro Aron’s quarter-comma meantone tempered division of the octave is based on the premise that the just major tone (9/8) and just minor tone (10/9), or the sum of the two intervals (the just major third [5/4]) is reduced to the mean of these two intervals via the cumulative diminution of just perfect fifths (3/2s) by the amount of a quarter of a syntonic comma. The syntonic comma, or comma of Didymus,128 is represented by the ratio 81/80, or 21.506 cents (equal to the subtraction of 10/9 from 9/8, or the

4 81 difference between the two tones), and a quarter of this measurement is equal to 80 , 1:1.003110, or 5.377 cents.129 81/80, and its reciprocal, 80/81, may also be theoretically referred to as the acute, or comma, and grave, or hypocomma.130 The consequence of this process (the diminution of just perfect fifths) also naturally leads to the augmentation of just perfect fourths (4/3s) by the amount of a quarter of a syntonic comma, and ultimately to ‘true’ just major thirds (5/4s). Lloyd and Boyle offer the following explanation: “The true major third (ratio 5/4) is the sum of a major and minor tone. Basic mean-tone temperament therefore made all the whole tones the mean of these two intervals, half a comma smaller than the major tone, half a comma larger than the minor tone, leaving each of the diatonic semitones (EF and BC) too sharp by quarter comma.”131 In order to generate a meantone scale, it is first necessary to construct a Pythagorean scale with twenty-seven distinct pitches, which are the necessary intervals to facilitate the general modulations of

127 Douglas Leedy, “A Venerable Temperament Rediscovered,” Perspectives of New Music 29.2 (Summer, 1991): 202-03. 128 “Didymus, academic philosopher of Nero’s time (ruled 54-68 A.D.) gave his name,” to the syntonic comma (81/80), “the difference between two between-degree relationships – 9/8 and 10/9 – in his diatonic tetrachord.” For a further discussion, see Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 368. 129 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 168. 130 Alexander J. Ellis, “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments wit Fixed Tones,” Proceedings of the Royal Society of London 13 (1863-64): 95. 131 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 168. 76 The Equally-Tempered Archetype

Western tonal music.132 The fifteen intervals generated by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!, B!, F#, C#, and G# (beginning with G (3/2) are then cumulatively reduced by the amount of a quarter of a syntonic comma, or by ß¼, ß½, ß¾, ß1, ß1¼, ß1½, ß1¾, ß2, ß2¼, ß2½, ß2¾, ß3, ß3¼, ß3½, and ß3¾ of a comma in each case; while the other eleven intervals, generated by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$, are cumulatively increased by the amount of a quarter of a syntonic comma, or by +¼, +½, +¾, +1, +1¼, +1½, +1¾, +2, +2¼, +2½, and +2¾ of a comma in each case. The abovementioned pattern demonstrates that by a subtraction of a syntonic comma (81/64×80/81=5/4), the Pythagorean major third (81/64) will be suitably reduced to a just major third (5/4). It may be therefore summarized that the calculation of the factor for each meantone interval firstly involves the calculation of the reciprocal, or multiplicative inverse of a series of one-quarter increments (¼, ½, ¾, 1, 1¼, 1½, 1¾, 2, 2¼, 2½, 2¾, 3, 3¼, 3½, and 3¾ equal to 4, 2, 1.333333, 1, 0.8, 0.666666, 0.571429, 0.5, 0.444444, 0.4, 0.363636, 0.307692, 0.285714, and 0.266666), which is then be utilized to divide the syntonic comma (81/80) into the appropriate quarter-comma additions or subtractions that are cumulatively applied to each Pythagorean ratio. For example: the meantone perfect fifth is equal to the first ascending Pythagorean fifth (3/2) minus one quarter of a comma, so 3/2 must be multiplied by the fourth root of 80/81 (four being

4 80 the reciprocal of 0.25, or ¼ï1=4), which is mathematically expressed as the factor 3/2× 81 , or the 4 80 decimal ratio 1.495349. The meantone perfect fifth (3/2× 81 ) at 696.578 cents therefore represents the just perfect fifth (3/2) tempered by the subtraction of a quarter of a syntonic comma (81/80), or

4 80 1:3/2× 81 =1:1.495349 (equivalent to 696.578+5.377=701.955); while the meantone perfect fourth 4 81 (4/3× 80 ) at 503.422 cents represents the just perfect fourth (4/3) tempered by the addition of a 4 81 quarter of a syntonic comma, or 1:4/3× 80 =1:1.337481 (equivalent to 503.422ß5.377=498.045 cents). The system reduced to twelve notes to the octave (being the tuning convention applied to standard keyboard instruments) produces the following series of pitches: C, C!, D, E", E', F, F!, G, G!, A, B", B', and C, or eight ascending fifths and three descending fifths modified by the quarter-comma meantone process. The selection of sharps and flats represent “the chromatically altered notes used in modal music,” and results in F!, G!, and C!, as just major thirds (5/4s) above D, E, and A respectively; while B", and E", as just major thirds (5/4s) below D and G. “The result of tuning the black notes in this way was to produce ‘wolves’ in the keys E" major and E major, and in all the major keys more remote, while the minor keys of C and E were faulty,” notes Lloyd and Boyle. Major keys represented well within this temperament include C, G, D, A, F, and B"; as well as minor keys G, D, and A,133 and hence “any keys

132 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 434. 133 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 102- 68. The Equally-Tempered Archetype 77 beyond the six which can be got with only three sharps and two flats are very seriously out of tune, and full of wolves.”134 The so-called ‘wolves’ are in reference to the A" major (G!, C, and E") and F minor triads (F, G!, and C), which are severely compromised in twelve-tone meantone temperament by the replacement of a major third, perfect fifth, and minor third with diminished fourth, diminished sixth, and augmented second intervals. The major third, perfect fifth, and minor third each at 427.373, 737.637, and 269.205 cents therefore represent just intervals with a falsity of +41.059, +35.683, and ß46.436 cents on each count. “That is why we find all the early music written in simple keys – they were the only ones available on keyboard instruments,” notes N. Lindsay Norden.135 The ‘wolf’ fifth (or meantone diminished sixth) at 737.637 cents, approximates the interval of a superfifth (192/125, or 743.014 cents), which is significantly higher than an acute or large fifth (243/160, or 723.014 cents), and way beyond any recognizable form of Fuller’s or Blackwood’s definition of a perfect fifth with a 720.000 cent upper limit. The interval further represents the eleventh descending Pythagorean fifth (or Pythagorean diminished sixth)

0.363636 81 tempered by an addition of two and three-quarter commas, or 262144/177147× 80 =1.531237. “The horrible effect was familiarly compared to the howling of ‘wolves’,” notes Helmholtz. “Similarly for B, D!, and F!, it was necessary to use B, E", and F!, E" being a great diesis (128/125, or 41.059 cents) too sharp, with similar excruciating effects.”136 It is interesting to note that according to Edward Dunne, and Mark McConnell, “the syntonic comma is actually a much greater problem in Western music than the Pythagorean comma. To encounter a Pythagorean comma, a piece would have to modulate through all twelve keys of the circle of fifth, but only a few modulations bring you to the syntonic comma.” Also, the fact that the syntonic comma (81/80) represents an equation involving three primes (81/80=1, or 2ô.5=3ô), just like 2x=3y (tripling fifths and doubling octaves), which “will never be a scale in which all the fifths, or a complete set of fifths and thirds, are correct,” the mathematical verity becomes apparent that “any method of constructing a twelve-tone scale by rational numbers is doomed to inconsistency.”137

134 James Swinburne, “The Ideal Scale: Its AEtiology, Lysis and SequelAE,” Proceedings of the Musical Association, 63rd sess. (1936-1937): 39-64. 135 N. Lindsay Norden, “A New Theory of Untempered Music: A Few Important Features with Special Reference to ‘A Capella’ Music,” The Musical Quarterly 22.2 (Apr., 1936): 221. 136 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 434-55. 137 Edward Dunne, and Mark McConnell, “Pianos and Continued Fractions,” Mathematics Magazine 72.2 (Apr., 1999): 107-09. 8TeEulyTmee Archetype Equally-Tempered The 78 Table 32. Pietro Aron’s quarter-comma meantone tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING

NUMBER (DECIMAL) (HERTZ)

ÐÑ C unison 1/1 1.000000 261.626 0.000 +00

(A) VII ß13 0.571429 80 ÐÒ CÚ meantone chromatic semitone, or minor half-tone 4 2187/2048× 81 1.044907 273.374 76.049 ß24

(D) V 1 1 0.8 81 ÐÓ DÝ meantone minor second 4 256/243× 80 1.069984 279.935 117.108 +17

(A) XIV ß3 1 0.285714 80 ÐÔ CÛ meantone double augmented octave 2 4782969/4194304× 81 1.091830 285.651 152.098 ß48

(A) IIß 1 2 80 ÐÕ D meantone major tone 2 9/8× 81 1.118034 292.506 193.157 ß07

(D) X 2 1 0.4 81 ÐÖ EÞ meantone diminished third 2 65536/59049× 80 1.144867 299.526 234.216 +34

(A) IXß2 1 0.444444 80 Ð× DÚ meantone augmented second 4 19683/16384× 81 1.168241 305.642 269.206 ß31

(D) III 3 3 1.333333 81 ÐØ EÝ meantone minor third 4 32/27× 80 1.196279 312.977 310.265 +10

ÐÙ E just major third (A) IV ß1 (5th harmonic) 81/64×80/81, or 5/4 1.250000 327.032 386.314 ß14

(D) VIII 2 0.5 81 ÑÐ FÝ meantone diminished fourth 8192/6561× 80 , or 32/25 1.280000 334.881 427.373 +27

(A) XI ß2 3 0.363636 80 ÑÑ EÚ meantone augmented third 4 177147/131072× 81 1.306133 341.718 462.363 ß38

(D) I  1 4 81 ÑÒ F meantone perfect fourth 4 4/3× 80 1.337481 349.919 503.422 +03

(A) VI ß11 0.666667 80 ÑÓ FÚ meantone tritone, or augmented fourth 2 729/512× 81 1.397542 365.633 579.471 ß21

(D) VI 11 0.666667 81 ÑÔ GÝ meantone diminished fifth 2 1024/729× 80 1.431084 374.408 620.529 +21

(A) XIII ß3 1 0.307692 80 ÑÕ FÛ meantone double augmented fourth 4 1594323/1048576× 81 1.460302 382.052 655.536 ß44 DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING

NUMBER (DECIMAL) (HERTZ)

(A) I ß 1 4 80 ÑÖ G meantone perfect fifth 4 3/2× 81 1.495349 391.221 696.578 ß03

(D) XI 2 3 0.363636 81 Ñ× AÞ meantone diminished sixth 4 262144/177147× 80 1.531237 400.611 737.637 +38

(A) VIII ß2 0.5 80 ÑØ GÚ meantone augmented fifth (25th harmonic) 6561/4096× 81 , or 25/16 1.562500 408.790 772.627 ß27

ÑÙ AÝ just minor sixth (D) IV 1 128/81×81/80, or 8/5 1.600000 418.601 813.686 +14

(A) XV ß3 3 0.266667 80 ÒÐ GÛ meantone double augmented fifth 4 14348907/8388608× 81 1.632667 427.147 848.676 ß51

(A) IIIß3 1.333333 80 ÒÑ A meantone major sixth 4 27/16× 81 1.671851 437.399 889.735 ß10

(D) IX 2 1 0.444444 81 ÒÒ BÞ meantone diminished seventh 4 32768/19683× 80 1.711975 447.896 930.794 +31

(A) X ß2 1 0.4 80 ÒÓ AÚ meantone augmented sixth 2 59049/32768× 81 1.746928 457.041 965.784 ß34

(D) II  1 2 81 ÒÔ BÝ meantone minor seventh 2 16/9× 80 1.788854 468.010 1006.843 +07 T eEulyTmee rhtp 79 Archetype Equally-Tempered he

(A) V ß1 1 0.8 80 ÒÕ B meantone major seventh 4 243/128× 81 1.869186 489.027 1082.892 ß17

(D) VII 13 0.571429 81 ÒÖ CÝ meantone diminished octave 4 4096/2187× 80 1.914046 500.763 1123.951 +24

(A) XII ß3 0.333333 80 Ò× BÚ meantone augmented seventh 531441/524288× 81 1.953125 510.987 1158.941 ß41

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 80 The Equally-Tempered Archetype

Joseph Sauveur’s Forty-Three-Tone Equally-Tempered Division of the Octave

There are a variety of other meantone temperaments that require special mention, and namely the 2/7 comma temperament of Italian music theorist and composer Gioseffo Zarlino (1517-90), 1/3 comma temperament of Spanish music theorist and organist Francisco de Salinas (1513-90), 1/5 comma temperament of Dutch organist Abraham Verheijen (fl. 1600), 2/9 comma temperament of Italian music theorist Lemme Rossi (1602-73), and 1/6 comma temperament of German organ builder and instrument maker Gottfried Silbermann (1683-1753). Additionally, 1/11 comma meantone temperament serves as an excellent approximation of twelve-tone equal temperament. 1/11of a syntonic comma (81/80) is equal to

11 81 11 80 80 , 1:1.001130, or 1.955 cents, and results in a fifth and fourth equal to 1:3/2× 81 =1:1.498307, or 11 81 700.000 cents, and 1:4/3× 80 =1:1.334840, or 500.000 cents respectively. Fifth-comma meantone temperament may be expressed as the mathematical equation that

5 15 produces the “first ratio for the fifth” as the “fifth root of 15:2,” or 2 . The fifth of a syntonic comma 5 81 (81/80) is equal to 80 , 1:1.002488, or 4.301 cents. The fifth-comma meantone perfect fifth 5 80 (3/2× 81 ) at 697.654 cents represents the just perfect fifth (3/2) tempered by the subtraction of a fifth 5 80 of a syntonic comma (81/80), or 1:3/2× 81 =1:1.496278 (equivalent to 697.654+4.301=701.955); 5 81 while the fifth-comma meantone perfect fourth (4/3× 80 ) at 502.346 cents represents the just perfect fourth (4/3) tempered by the addition of a fifth of a syntonic comma (81/80), or

5 81 1:4/3× 80 =1:1.336650 (equivalent to 502.346ß4.301=498.045 cents). The temperament (which incidentally approximates the forty-three-tone equally-tempered division of the octave [ 43 2 ]) features the “equal distortion of the fifths and the major thirds (equal to 390.615 cents); the former being one-fifth comma flat, while the latter, sharp by the same amount.”138 French acoustician Joseph Sauveur (1653-1716) proposed a forty-three-tone equally-tempered division of the octave (which closely approximates fifth-comma meantone temperament) in 1701, and consequently devised a highly complex notation system based on seven diatonic notes and syllables. Read explains: “Ut to si, were divided into six parts, which he termed mérides. Furthermore, each mérides was divided into seven eptamérides and these in turn into ten decamérides each, resulting in an octave comprised of 43 mérides, 301 eptamérides, and 3010 decamérides.”139 The fact that the common logarithm of two is equal to 0.301030, means that the decamérides “unit of musical measure” may be derived directly from this source. The 43-et perfect fifth ( [43 ]2 25 ) measures 697.674 cents; the 43-et perfect fourth ( [43 ]2 18 ), 502.326 cents; while the 43-et major third ( [43 ]2 14 ), 390.698 cents. These measurements represent fifth-comma meantone temperament intervals with a falsity of +0.020, ß0.020, and +0.083 cents on each count.140

138 Barbour, Tuning and Temperament: A Historical Survey 31-83. 139 Read, 20th-Century Microtonal Notation 133-34. 140 Barbour, Tuning and Temperament: A Historical Survey 122. Table 33. Joseph Sauveur’s forty-three-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ D$ 43-et diminished second 43 2 1.016250 265.877 27.907 +28 ÐÓ B# 43-et double augmented seventh (43 2)2 1.032765 270.198 55.814 ß44 ÐÔ C! 43-et chromatic semitone, or minor half-tone (43 2)3 1.049547 274.588 83.721 ß16 ÐÕ D" 43-et minor second (43 2) 4 1.066603 279.051 111.628 +12 ÐÖ E& 43-et double diminished third (43 2)5 1.083936 283.585 139.535 +40 Ð× C# 43-et double augmented octave (43 2)6 1.101550 288.194 167.442 ß33 ÐØ D 43-et major tone (43 2)7 1.119450 292.877 195.349 ß05 ÐÙ E$ 43-et diminished third (43 2)8 1.137642 297.636 223.256 +23 ÑÐ C% 43-et triple augmented octave (43 2)9 1.156129 302.473 251.163 ß49 ÑÑ D! 43-et augmented second (43 2)10 1.174916 307.388 279.070 ß21 ÑÒ E" 43-et minor third (43 2)11 1.194009 312.383 306.977 +07 ÑÓ F$ 43-et double diminished fourth (43 2)12 1.213412 317.460 334.884 +35 T eEulyTmee rhtp 81 Archetype Equally-Tempered he ÑÔ D# 43-et double augmented second (43 2)13 1.233131 322.618 362.791 ß37 ÑÕ E 43-et major third (43 2)14 1.253169 327.861 390.698 ß09 ÑÖ F" 43-et diminished fourth (43 2)15 1.273534 333.189 418.605 +19 Ñ× G& 43-et triple diminished fifth (43 2)16 1.294229 338.603 446.512 +47 ÑØ E! 43-et augmented third (43 2)17 1.315261 344.106 474.419 ß26 ÑÙ F 43-et perfect fourth (43 2)18 1.336634 349.698 502.326 +02 ÒÐ G$ 43-et double diminished fifth (43 2)19 1.358355 355.380 530.233 +30 ÒÑ E# 43-et double augmented third (43 2)20 1.380429 361.155 558.140 ß42 2TeEulyTmee Archetype Equally-Tempered The 82 DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÒÒ F! 43-et tritone, or augmented fourth (43 2)21 1.402861 367.024 586.047 ß14 ÒÓ G" 43-et diminished fifth (43 2)22 1.425658 372.989 613.953 +14 ÒÔ A& 43-et double diminished sixth (43 2)23 1.448825 379.050 641.860 +42 ÒÕ F# 43-et double augmented fourth (43 2)24 1.472369 385.209 669.767 ß30 ÒÖ G 43-et perfect fifth (43 2)25 1.496296 391.469 697.674 ß02 Ò× A$ 43-et diminished sixth (43 2)26 1.520611 397.831 725.581 +26 ÒØ F% 43-et triple augmented fourth (43 2)27 1.545321 404.296 753.488 ß47 ÒÙ G! 43-et augmented fifth (43 2)28 1.570433 410.866 781.395 ß19 ÓÐ A" 43-et minor sixth (43 2)29 1.595953 417.542 809.302 +09 ÓÑ B& 43-et double diminished seventh (43 2)30 1.621888 424.327 837.209 +37 ÓÒ G# 43-et double augmented fifth (43 2)31 1.648244 431.223 865.116 ß35 ÓÓ A 43-et major sixth (43 2)32 1.675029 438.230 893.023 ß07 ÓÔ B$ 43-et diminished seventh (43 2)33 1.702249 445.352 920.930 +21 ÓÕ C& 43-et triple diminished octave (43 2)34 1.729911 452.589 948.837 +49 ÓÖ A! 43-et augmented sixth (43 2)35 1.758022 459.944 976.744 ß23 Ó× B" 43-et minor seventh (43 2)36 1.786591 467.418 1004.651 +05 ÓØ C$ 43-et double diminished octave (43 2)37 1.815624 475.014 1032.558 +33 ÓÙ A# 43-et double augmented sixth (43 2)38 1.845128 482.733 1060.465 ß40 ÔÐ B 43-et major seventh (43 2)39 1.875112 490.577 1088.372 ß12 ÔÑ C" 43-et diminished octave (43 2) 40 1.905583 498.549 1116.279 +16 ÔÒ D& 43-et double diminished second (43 2) 41 1.936549 506.651 1144.186 +44 ÔÓ B! 43-et augmented seventh (43 2) 42 1.968019 514.884 1172.093 ß28

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 The Equally-Tempered Archetype 83

Origins of Equal Temperament

Equal temperament, or the division of the octave into equal parts may be accredited to Prince Chu Tsai- yü, “a sixth-generation descendant of Hung Hsi, the fourth emperor of the Ming dynasty,” who in his 1584 treatise entitled A New Account of the Science of the Pitch Pipes presented the string lengths for twelve- tone equal temperament.141 According to Kuttner:

“Lü Hsüeh Hsin Shuo (A New Account of the Science of the Pitch Pipes) contains a complete nine-digit monochord of equal temperament with the octave ratio 10:5 for what appears to be a calculation of string lengths; for the lengths of pitch-pipes there monochords based on the octave ratios 100:50 and 90:45, with four decimals, as well as tabulations of pipe diameters and circumferences on the base 100:50 with two decimals.”

The roots of two for the string lengths in a monochord are then published in Chu Tsai-yü’s 1595-96 follow-up treatise entitled Lü Lü Ching I, and although correct to nine places, have been calculated with the absence of logarithms. It should be noted that although “the twelfth root of two is numerically implied as quantitative definition of the semitone in equal temperament,” in Tsai-yü’s second theoretical work, “it is never stated explicitly as a mathematical expression.”142 Partch offers the following commentary on the nature of Prince Chu Tsai-yü’s accomplishment:

“Of the prince’s accomplishment a contemporary modern theorist reminds us that ‘the computation would have to begin, for certain tones, with numbers containing 108 zeros, of which the 12th root would have to be extracted, as (Marin) Mersenne did, by taking the square root twice and then the cube root. This lengthy and laborious procedure was followed without error.’”143

The nine-digit string lengths of Tsai-yü’s monochord depicted in the following table are not a result of rounded off values of a ten-digit calculation, and therefore a reduction to lower terms, but the true values of a calculation based on the hundredth millionth.144

141 Stuart Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization (New York: Vintage, 2003) 163-66. 142 Chu Tsai-yü’s 1595-96 treatise entitled Lü Lü Ching I “contains an enormously detailed mathematical investigation of all conceivable parameters involved in the definition of pitches in equal temperament tuning, including string lengths and pitch-pipe dimensions, such as tube lengths, inner and outer diameters, circular surface areas, bore and volume of pipes through three octave ranges in 9- and 10-digit tabulations,” states Fritz A. Kuttner. For a further discussion, see Fritz A. Kuttner, “Prince Chu Tsai-Yu’s Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory,” Ethnomusicology 19.2 (May, 1975): 166-67. 143 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 380-81. 144 Barbour, Tuning and Temperament: A Historical Survey 77-78. 84 The Equally-Tempered Archetype

Table 34. Chu Tsai-yü’s monochord

DEGREE NOTE LENGTHS RATIO CENTS FALSITY NUMBER (DECIMAL) (CENTS) ÐÑ C4 1,000,000,000 1.000000 0.000 +0.000 ÐÒ C!4 943,874,312 1.059463 100.000 +0.000 ÐÓ D4 890,898,718 1.122462 200.000 +0.000 ÐÔ D!4 840,896,415 1.189207 300.000 +0.000 ÐÕ E4 793,700,525 1.259921 400.000 +0.000 ÐÖ F4 749,153,538 1.334840 500.000 +0.000 Ð× F!4 707,106,781 1.414214 600.000 +0.000 ÐØ G4 667,419,927 1.498307 700.000 +0.000 ÐÙ G!4 629,960,524 1.587401 800.000 +0.000 ÑÐ A4 594,603,557 1.681793 900.000 +0.000 ÑÑ A!4 561,231,024 1.781797 1000.000 +0.000 ÑÒ B4 529,731,547 1.887749 1100.000 +0.000

ÐÑ C5 500,000,000 2.000000 1200.000 +0.000

The mathematical solution to the dilemma of the cycle of fifths, or the irresolvable succession of just perfect fifths or 3/2s rests on the unequivocal formula, equivalent to the frequency ratio 1.059463094, or the numerical approximation of the proportion 749:500 (derived from 750:500, which is equal to the ratio 3/2, and refined in chapter one of Lü Lü Ching I to the proportion 749.153.538:500.000.000). The explanation for the infinite nature of a 3/2 series is that octaves (or the ratio 2/1) are based on multiples of two, while fifths (or the ratio 3/2), on multiples of three; and because two and three are prime numbers (divisible only by themselves and one), presented is the mathematical verity that only powers of identical prime numbers can be equal.145 Calculus illustrates that a series of twelve 3/2s yields the following values in cents: 0.000, 113.685, 203.910, 317.595, 407.820, 521.505, 611.730, 701.955, 815.640, 905.865, 1019.550, 1109.775, and 1223.460. The intonational anomaly generated by the twelfth ascending fifth, which has a ratio of 531441/524288 (1:1.013643) and measures 23.460 cents, is generally referred to as the Pythagorean comma. In other words, “the sum of twelve 3/2s (3ñò/2ñù) is greater than seven octaves (2÷/1ñ) by a Pythagorean comma.”146 In striking contrast, a series of twelve 749:500s yields: 0.000, 97.516, 199.290, 296.806, 398.581, 496.097, 597.871, 699.645, 797.161, 898.935, 996.451, 1098.226, and 1195.742; while a series of twelve 749.153.538:500.000.000s yields twelve equal semitones.

145 Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization 40-170. 146 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 34. The Equally-Tempered Archetype 85

In the West, French monk, mathematician and physicist Marin Mersenne (1588-1648) is not only acknowledged for “the discovery of the overtone series in the natural sounds of the trumpet,” but also for being “the first Westerner to give the correct mathematical solution for equal temperament.”147 Mersenne’s most accurate geometrical approximations are characterised by Barbour as follows:

“By the familiar Euclidean method he found the mean proportional between a line and its double, subtracted the original line from the mean, and then subtracted this difference from the doubled line. The length thus found was the larger of the desired means – that is, the string length for the major third.”

In mathematical terms, Mersenne’s solution is depicted via the equation (3ß 2 2 ):2, which is equal to the string length of 0.7928930, or the decimal ratio 1.261204 for the major third. The division of the major third into four equal semitones or “mean proportionals” produce a semitone equal to 100.440 cents in this region, while the subsequent division of the remaining major sixth (the distance between the major third and the octave) into eight equal semitones produce a dissimilar semitone equal to 99.780 cents. The mean falsity of the semitones in Mersenne’s geometrical approximation for equal temperament, therefore have a value of 0.881 cents, with a maximum error of 1.762 cents.

Table 35. Marin Mersenne’s geometrical approximation for equal temperament

DEGREE NOTE LENGTHS RATIO CENTS FALSITY NUMBER (CENTS) ÐÑ C4 1.000000 1.000000 0.000 +0.000 ÐÒ C!4 0.943634 1.059733 100.440 +0.440 ÐÓ D4 0.890446 1.123033 200.881 +0.881 ÐÔ D!4 0.844670 1.190115 301.321 +1.321 ÐÕ E4 0.792893 1.261204 401.762 +1.762 ÐÖ F4 0.748487 1.336029 501.542 +1.542 Ð× F!4 0.706567 1.415293 601.321 +1.321 ÐØ G4 0.666996 1.499260 701.101 +1.101 ÐÙ G!4 0.629640 1.588209 800.881 +0.881 ÑÐ A4 0.594377 1.682435 900.661 +0.661 ÑÑ A!4 0.561088 1.782251 1000.440 +0.440 ÑÒ B4 0.529664 1.887989 1100.220 +0.220

ÐÑ C5 0.500000 2.000000 1200.000 +0.000

147 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 381-82. 86 The Equally-Tempered Archetype

It is then, circa 1596, that Flemish mathematician and inventor Simon Stevin (1548-1620) recognizes that the solution to equal temperament lies in finding “eleven mean proportional parts between two and one,”148 and hence produces “an essay containing the mathematical formulation of equal temperament as 12 2 for the first time in Western musical theory.” The essay – rediscovered and edited in 1884 by Dutch mathematician Dr. David Bierens de Haan (1822-95), to be then published for the very first time – presents the calculation of a monochord defining “twelve semitone values, correct to four decimal places, as the 12 successive powers of the twelfth root of two.”149 The method adopted by Stevin involves the calculation of degree numbers 7, 4, and 5 (or F!, D!, and E) – each subordinate to cubic and quartic levels of mathematical complexity – with the next step requiring the strategy of proportion, or “the rule of three”, and therefore the division of the fifth degree (7937) by the fourth (8408) to produce the second degree (9440). “This method is much easier than to extract the roots for each individual note, which runs into difficulties with the roots of the prime numbers, as for the notes 2, 6, 8, and 12 (C!, F, G, and B), where the 12th root itself must be extracted,” explains Barbour. The method delivers a mean falsity of 0.199 cents, with a maximum error of 0.391 cents.

Table 36. Simon Stevin’s monochord

DEGREE NOTE LENGTHS RATIO CENTS FALSITY NUMBER (CENTS) ÐÑ C4 10000 1.000000 0.000 +0.000 ÐÒ C!4 9440 1.059322 99.769 ß0.231 ÐÓ D4 8911 1.122209 199.609 ß0.391 ÐÔ D!4 8408 1.189343 300.199 +0.199 ÐÕ E4 7937 1.259922 400.001 +0.001 ÐÖ F4 7493 1.334579 499.662 ß0.338 Ð× F!4 7071 1.414227 600.017 +0.017 ÐØ G4 6675 1.498127 699.792 ß0.208 ÐÙ G!4 6301 1.587050 799.617 ß0.383 ÑÐ A4 5945 1.682086 900.302 +0.302 ÑÑ A!4 5612 1.781896 1000.096 +0.096 ÑÒ B4 5298 1.887505 1099.776 ß0.224

ÐÑ C5 5000 2.000000 1200.000 +0.000

148 Barbour, Tuning and Temperament: A Historical Survey 54-55. 149 Kuttner, “Prince Chu Tsai-Yu’s Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory,” Ethnomusicology 167-68. The Equally-Tempered Archetype 87

It must be noted that in spite of the geometrical and numerical approximations of Mersenne, Stevin, and other European theorists of the sixteenth and seventeenth centuries, it is not until 1630 that the first printed numerical solution to equal temperament based on the theory of logarithmic computation appears; German mathematician Johann Faulhaber (1580-1635) establishing lengths derived from the division of a monochord with a length of the required 20,000 units in order to generate a series legitimately expressing twelve equally-tempered semitones. The mean falsity of 0.100 cents (with a maximum error of 0.181 cents) depicted in Faulhaber’s monochord is a direct result of the expected error obtained via the utilization of logarithmic tables.150

Table 37. Johann Faulhaber’s monochord

DEGREE NOTE LENGTHS RATIO CENTS FALSITY NUMBER (CENTS) ÐÑ C4 20000 1.000000 0.000 +0.000 ÐÒ C!4 18877 1.059490 100.045 +0.045 ÐÓ D4 17817 1.122523 200.095 +0.095 ÐÔ D!4 16817 1.189273 300.096 +0.096 ÐÕ E4 15874 1.259922 400.001 +0.001 ÐÖ F4 14982 1.334935 500.124 +0.124 Ð× F!4 14141 1.414327 600.139 +0.139 ÐØ G4 13347 1.498464 700.181 +0.181 ÐÙ G!4 12598 1.587554 800.166 +0.166 ÑÐ A4 11891 1.681944 900.156 +0.156 ÑÑ A!4 11224 1.781896 1000.096 +0.096 ÑÒ B4 10594 1.887861 1100.103 +0.103

ÐÑ C5 10000 2.000000 1200.000 +0.000

The Twelve-Tone Equally-Tempered Division of the Octave

The twelve-tone equally-tempered division of the octave is the division of the octave into twelve equal intervals, technically referred to as tempered half-tones; the frequency ratio of each semitone therefore mathematically representing the twelfth root of two, or in different terms, the distance between any two tones representing twelve times the logarithm on the base of two of the frequency ratio.151 The ratio of

150 Barbour, Tuning and Temperament: A Historical Survey 54-78. 151 Olson, Music, Physics and Engineering 46-47. 88 The Equally-Tempered Archetype the equally-tempered semitone may be expressed in mathematical terms as 1: 12 2 =1:1.059463094,152 or the fraction approximations: 1024/967 (2ñð/967) and 512/483 (2ù×3×7×23). The correct measurements for the string lengths (based on 2,000,000 units) of the equally-tempered monochord are depicted in the following table.153

Table 38. The equally-tempered monochord

DEGREE NOTE LENGTHS RATIO CENTS FALSITY NUMBER (CENTS) ÐÑ C4 2000000 1.000000 0.000 +0.000 ÐÒ C!4 1887749 1.059463 100.045 +0.000 ÐÓ D4 1781797 1.122462 200.095 +0.000 ÐÔ D!4 1681793 1.189207 300.096 +0.000 ÐÕ E4 1587401 1.259921 400.001 +0.000 ÐÖ F4 1498307 1.334840 500.124 +0.000 Ð× F!4 1414214 1.414214 600.139 +0.000 ÐØ G4 1334840 1.498307 700.181 +0.000 ÐÙ G!4 1259921 1.587401 800.166 +0.000 ÑÐ A4 1189207 1.681793 900.156 +0.000 ÑÑ A!4 1122462 1.781797 1000.096 +0.000 ÑÒ B4 1059463 1.887749 1100.103 +0.000

ÐÑ C5 1000000 2.000000 1200.000 +0.000

The methodology involved in the construction of a twelve-tone equally-tempered scale requires the establishment of twelve degrees via the equation f= 12 2 ; the equation producing the figure 1.059463094 (the frequency ratio of one tempered semitone), which when multiplied by the powers of 2, 3, 4, etc., generates the ratios for the remainder of the scale degrees. The multiplication by the power of 9 ( [12 2]9 ) generates the frequency ratio of A=440Hz (1.68179283), or equal major sixth ( [4 2]3 ), while the mathematical equation of 440ï1.68179283 presents the relative frequency of middle C (261.6255654Hz), or 1/1. Once data is available for all frequency ratios it simply becomes a matter of the multiplication of every ratio by 261.6255654Hz (the frequency of middle C) in order to obtain the frequency values in hertz (Hz) of every scale degree. The mathematical formula for cents is then utilized to generate the cent values of every scale degree, although unnecessary in equal temperament because of the obvious outcomes.

152 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 431. 153 Daniélou, Tableau Comparatif des Intervalles Musicaux 28. The Equally-Tempered Archetype 89

The acoustical realities of equal temperament are that although equally-tempered fifths approximate just perfect fifths, the system delivers major thirds and minor thirds tempered by seven and eight times as much as fifths respectively. “Lustrous and calm in their pure form, they were now slightly rough and somewhat bland,” notes Stuart Isacoff with regards to the intonational transformation of just major thirds (5/4s) into equal major thirds ( 3 2).154 The root position twelve-tone equally-tempered major triad is represented by the ratio 1.000000:1.259921:1.498307, and the pitches C, E¢, and G; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß13.686 and +1.955 cents on each count. G4 (391.995Hz), or the equal perfect fifth ( [12 ]2 7 ), presents 0.886 beats between the third harmonic of C4 (784.877Hz) and the second harmonic of G4 (783.991Hz), and 1.772 beats between the sixth harmonic of C4 (1569.753Hz) and the fourth harmonic of G4 (1567.982Hz); while E¢4 (329.628Hz), or the equal major third ( 3 2 ), 10.382 beats between the fifth harmonic of C4 (1308.128Hz) and the fourth harmonic of E¢4 (1318.510Hz).

Table 39. The beating characteristics of the twelve-tone equally-tempered major triad

C4 FREQUENCY E¢4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 329.628 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 391.995 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 659.255 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 783.991 0.886 – – – – – – – – 3 988.883 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1175.986 – – – – 5 1308.128 4 1318.510 10.382 – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1567.982 1.772 – – – – – – – – 5 1648.138 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1977.765 – – – – 5 1959.977 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The root position twelve-tone equally-tempered minor triad is represented by the ratio 1.000000: 1.189207:1.498307, and the pitches C, E¸, and G; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß15.641 and +1.955 cents on each count. G4, or the equal perfect

154 Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization 118. 90 The Equally-Tempered Archetype fifth ( [12 2]7 ), presents 0.886 beats between the third harmonic of C4 and the second harmonic of G4, and 1.772 beats between the sixth harmonic of C4 and the fourth harmonic of G4; while E¸4 (311.127Hz), or the equal minor third ( 4 2 ), 14.118 beats between the sixth harmonic of G4 and the fifth harmonic of EÝ4 (1555.635Hz).

Table 40. The beating characteristics of the twelve-tone equally-tempered minor triad

C4 FREQUENCY E¸4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 311.127 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 391.995 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 622.254 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 783.991 0.886 – – – – – – – – 3 933.381 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1175.986 – – – –– – – – – – – – – – 4 1244.508 – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1555.635 14.118 4 1567.982 1.772 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1866.762 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1959.977 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The following series of tables depict the beating characteristics of the thirds, sixths, fifths, and fourths of twelve-tone equal temperament with reference to the tuning scheme of the pianoforte. The ‘tuning beats’ column represents the approximation of beats per second that a piano tuner must consider. The tuning methodology begins with the tuning of the first fifth, or F3 and C4 (middle C), with 0.591 (or 0.6) beats; and is followed by the tuning of the first fourth, G3 and C4, with 0.886 (or 0.9) beats. “Leave the F slightly sharp, so that it sounds as a very slow wow – wow trailing off to nothing,” explains Ian McCombie. “This interval should beat just over one beat in two seconds. It is the only note in the tempered scale which is sharp. The G is left slightly flat, and should beat about beat per second flat.” The process continues, to include a selection of thirds, sixths, fifths, and fourths, as well as relevant octave (2/1) relationships.155

155 Ian McCombie, The Piano Handbook (London: David & Charles, 1980) 89-95. The Equally-Tempered Archetype 91

Table 41. Piano tuning – twelve-tone equal temperament (thirds)

NUMBER LOWER NOTE 5TH PARTIAL (HERTZ) UPPER NOTE 4TH PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS ÐÑ F3 873.071 A3 880.000 6.929 7.0 ÐÒ FÚ3 924.986 AÚ3 932.328 7.341 7.5 ÐÓ G3 979.989 B3 987.767 7.778 8.0 ÐÔ GÚ3 1038.262 C4 1046.502 8.241 8.5 ÐÕ A3 1100.000 CÚ4 1108.731 8.731 9.0 ÐÖ AÚ3 1165.409 D4 1174.659 9.250 9.5 Ð× B3 1234.708 DÚ4 1244.508 9.800 10.0 ÐØ C4 1308.128 E4 1318.510 10.382 10.5

Table 42. Piano tuning – twelve-tone equal temperament (sixths)

NUMBER LOWER NOTE 5TH PARTIAL (HERTZ) UPPER NOTE 3RD PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS ÐÑ F3 873.071 D4 880.994 7.924 8.0 ÐÒ FÚ3 924.986 DÚ4 933.381 8.395 8.5 ÐÓ G3 979.989 E4 988.883 8.894 9.0 ÐÔ GÚ3 1038.262 F4 1047.685 9.423 9.5

Table 43. Piano tuning – twelve-tone equal temperament (fifths)

NUMBER LOWER NOTE 3RD PARTIAL (HERTZ) UPPER NOTE 2ND PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS ÐÑ F3 523.842 C4 523.251 0.591 0.6 ÐÒ G3 587.993 D4 587.330 0.664 0.7 ÐÓ A3 660.000 E4 659.255 0.745 0.8 ÐÔ FÚ3 554.992 CÚ4 554.365 0.626 0.6 ÐÕ GÚ3 622.957 DÚ4 622.254 0.703 0.7 ÐÖ AÚ3 699.246 F4 698.456 0.789 0.8

Table 44. Piano tuning – twelve-tone equal temperament (fourths)

NUMBER LOWER NOTE 4TH PARTIAL (HERTZ) UPPER NOTE 3RD PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS ÐÑ G3 783.991 C4 784.877 0.886 0.9 ÐÒ A3 880.000 D4 880.994 0.994 1.0 ÐÓ B3 987.767 E4 988.883 1.116 1.1 ÐÔ FÚ3 739.989 B3 740.825 0.836 0.8 ÐÕ GÚ3 830.609 CÚ4 831.548 0.938 1.0 ÐÖ AÚ3 932.328 DÚ4 933.381 1.053 1.0

92 The Equally-Tempered Archetype

Exposiciones for Sampled Microtonal Schoenhut Toy Piano

Exposiciones for Sampled Microtonal Schoenhut Toy Piano is an ‘acousmatic’156 work that attempts to explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut model 6625, 25-key toy piano and a complex polyrhythmic scheme. All equal temperaments between one and twenty-four – essentially functioning as tuning modulations – as well as all polyrhythms (divisible only by 1 and including their inversions) between the ranges of 2 and 15 are presented. In other words, polyrhythmic ratios 3:2Ê (2:3Ê), 5ç:2Ê (2É:5Ê), 4:3Ê (3:4), 5:3Ê (3:5), and so on – fifty-seven polyrhythmic sets in total, with the last set represented by 15:14Ê (14:15Ê) – alongside two complementary scales (Indonesian pélog and sléndro forms with primary and secondary scale tones, as well as primary and secondary auxiliary tones) shaped via microtonal inflections produced by sequential tuning modulations featuring the first twenty-four equally-tempered divisions of the octave. The work was especially composed for ‘The Extensible Toy Piano Project’ – a toy piano festival and symposium hosted by Clark University, Department of Visual and Performing Arts, in Worcester, Massachusetts, USA between Friday and Saturday, November 4 and 5, 2005.

“It has a deceptively simple mechanism – plastic hammers hitting steel rods. Yet, the toy piano produces a rich and quirky sound palette. John Cage brought the instrument from a treasured plaything to a bona fide musical instrument with his Suite for Toy Piano (1948). Our aim is to bring the instrument into the 21st Century.” 157

A Brief History of the Toy Piano

The history of the toy piano (Fr. piano jouet; Ger. Spielzeugklavier; It. pianino-giocattolo) begins as Troiger’s ‘Stahlklavier’ (Dessau, 1792) and Franz Schuster’s ‘Adiaphonon’ (Vienna, c. 1818), to be later developed in Philadelphia, USA, in 1872, where German immigrant Albert Schoenhut ultimately conceives

156 According to Francis Dhomont, “The term Acousmatic Music (or Art) designates works that have been composed for loudspeakers, to be heard in the home – on radio or on CD/tape – or in concert, through the use of equipment (digital or analog) that allows the projection of sound in 3-dimensional space.” For a further discussion see, Francis Dhomont, “Acousmatic Update,” Contact! 8.2 Spring, 1995, CEC – Communauté Électroacoustique Canadienne / Canadian Electroacoustic Community, 27 Jan. 2006, . A further description of the term ‘acousmatic’ is offered in an interview with François Bayle by Sandra Desantos: “In acousmatic music, one may recognize the sound sources, but one also notices that they are out of their usual context. In the acousmatic approach, the listener is expected to reconstruct an explanation for a series of sound events, even if this explanation is provisional.” For a further discussion see, Sandra Desantos, “Acousmatic Morphology: An Interview with François Bayle,” Computer Music Journal 21.3 (Fall, 1997): 17. 157 The Extensible Toy Piano Project, ed. David Claman and Matt Malsky, 1 Jan. 2005, Clark U., Worcester, MA, 21 August 2005, . The Equally-Tempered Archetype 93 the child’s toy that in time will also capture the imagination of the modern composer. The instrument is usually made out of wood or plastic, and is dimensionally less than fifty centimetres in width, with a range between two diatonic and three chromatic octaves. It has a simple sounding mechanism (similar to that of the full-sized keyboard glockenspiel) consisting of plastic hammers operated via a keyboard, which strike fixed metal plates or steel rods. Traditionally, toy pianos were modelled on uprights, but following the 1950s grand piano varieties were commonplace.

Fig. 3. Matt Malsky, Schoenhut Model 6625: 25-Key Toy Piano.158

Contemporary works that have incorporated the toy piano include John Cage’s Suite for Toy Piano (1948) and George Crumb’s Ancient Voices of Children (1970), as well as other works by Renaud Gagneux, Mauricio Kagel, Louis Roquin, Zygmunt Krauze, and Leonid Aleksandrovich, among many. Internationally acclaimed concert pianist Margaret Leng Tan made her debut on the toy piano in 1993 at New York’s Lincoln Centre, and went on to introduce the model 6625, 25-key Schoenhut Traditional Spinet to Carnegie Hall in 1997; also releasing a compact disc entitled The Art of the Toy Piano in that same year – a collection of works by Stephen Montague, John Lennon and Paul McCartney, Toby Twining, Jed Distler, Philip Glass, David Lang, Julia Wolfe, Ludwig van Beethoven, Guy Klucevsek, Raphael Mostel, and Erik Satie.159

158 Matt Malsky, “Schoenhut Model 6625: 25-Key Toy Piano,” Feb. 2005, The Extensible Toy Piano Project, ed. David Claman and Matt Malsky, 1 Mar. 2005, Clark U., Worcester, MA, 21 Aug. 2005, . 159 Hugh Davies, “Toy Piano,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 615. 94 The Equally-Tempered Archetype

The Schoenhut Toy Piano Sample

The Schoenhut model 6625, 25-key toy piano samples (recorded in dead studio space [96kHz/24bit] by engineer John Shirley at Clark University, Department of Visual and Performing Arts, in Worcester, MA, USA, utilizing two Neumann TLM 103s [positioned front-L and back-R] and a Nuendo recorder) include three sets of twenty-five (two-octave chromatic span) forte (  ), mezzo forte (  ), and piano (  ) samples, as well as one corresponding set of keyboard release clicks. All these sounds multi-sampled on an Akai S3000XL Midi Stereo Digital Sampler – tuned firstly to standard A=440Hz twelve-tone equal temperament, modified within thirty velocity cross-faded patches, and then operated via a midi sequencer; a gong-like detuned middle C (octave down) sounding the tonal centre, as well as marking the downbeat, while another severely gated alternative providing the rhythmic pulse. The final Schoenhut Toy Piano sample may therefore be characterized as comprising of six basic sample elements (three dynamic ranges, keyboard clicks, gong, and counter).160 The musical rationale of the gong and counter has a close association with that of the gong and kempli in the gamelan music of Indonesia. Michael Tenzer explains the significance of the gong in gamelan music:

“The most important function of the gongs in gamelan music is to mark structural points in a composition. The number of gongs employed for this purpose depends on the ensemble. Gong kebyar uses one or two gong ageng (the largest and deepest) for the beginnings and endings of melodies and other strong accents. If two, they are used in alternation and never together. Other divisions are the responsibility of the medium-sized kempur, the small, chiming kemong, and the nearly ever-present kempli, whose clear, dry sound taps out the steady beat on which all musicians depend when the rhythms get tricky. Other types of gamelans make use of some of these gongs, plus others like the tiny kelenang, the flat-bossed kajar and bebende, or a resonant version of the kempli called tawa-tawa.”161

The elements then undergoing further processing within a digital audio editor PC software environment (Sony Sound Forge 8.0). A modified ‘white noise’ preset of the Waves X-Noise 5.2 plug-in is utilized to remove unwanted noise from all the samples (threshold: ß20.0dB, reduction: 70%; [dynamics] attack: 30ms, release: 220ms; [high shelf] frequency: 1415Hz, gain: +0.0dB), while the DSound Stomp’n Fx DN-

160 The Schoenhut model 6625, 25-key toy piano features are as follows: “25-key two-octave designer spinet; age range: 3 and up; chromatically tuned with lovely chime like notes produced by little hammers striking precision ground, German steel music rods; play-by-colour teaching method makes playing simple and fun; removable colour strip fits behind the keys to guide small fingers from chord to chord; helps to build your child’s confidence and develop basic playing skills; comes with a songbook which contains a collection of familiar tunes; available in mahogany, white, or red finish; dimensions: 19¾” highÏ10¼” deepÏ17” wide; matching bench: 9¼” highÏ6” deepÏ10” wide; weighs 17 lbs.” For a further discussion, see Schoenhut Toy Piano Company: Over 130 Years, 2005, Schoenhut Toy Piano Company, 21 Aug. 2005, .

161 Michael Tenzer, Balinese Music (Singapore: Periplus Editions, 1991) 37. The Equally-Tempered Archetype 95

SG1 Noise Gate DX plug-in, to gate any remaining unwanted sonic material from the keyboard clicks and counter samples (tone level: ß45dB, attack: 0.0ms, release: 185ms).

Sound Diffusion

The optimum method of performance for Exposiciones for Sampled Microtonal Schoenhut Toy Piano is 7-channel sound diffusion, with a scheme designed around the speaker system available in the performance space of the Razzo Recital Hall at Clark University, Department of Visual and Performing Arts, in Worcester, MA, USA, although the work may be performed utilizing a simple two-channel system (CD player). The system incorporates seven EAW speakers in various dimensions: three on-stage large soffit- mounted speakers (left front = 1, centre front = 2, right front = 3); plus four other smaller ‘left and right side sets’ soffit-mounted speakers (right side = 4, left side = 7); as well as left and right rear sets (left rear = 6, right rear = 5). Sound diffusion organized in such a manner as to sonically delineate the various elements of the toy piano samples: toy piano (1 = 75.0%, 3 = 100.0%, 4 = 12.5%, 7 = 50.0%); key clicks (1 = 100.0%, 3 = 75.0%, 4 = 50.0%, 7 = 12.5%); gong (2 = 100.0%); downbeat counter (4 = 50.0%, 5 = 100.0%, 6 = 100.0%, 7 = 50.0%); and upbeat counter (1 = 50.0%, 3 = 50.0%, 4 = 100.0%, 5 = 50.0%, 6 = 50.0%, 7 = 100.0%).

Fig. 4. Sound diffusion matrix (Razzo Recital Hall, Clark University, Worcester, MA, USA)

96 The Equally-Tempered Archetype

Polyrhythmic Theory

The New Harvard Dictionary of Music defines a polyrhythm as being “the simultaneous use of two or more rhythms that are not readily perceived as deriving from one another or as simple manifestations of the same meter; sometimes also cross-rhythm.” The most common examples of this phenomena is illustrated via the juxtaposing of triple and duple subdivisions of the beat, or alternatively of 3/4 and 6/8 meters. In classical music, the simple polyrhythmic technique is termed ‘hemiola’.162 According to Kevin Barrett, a better understanding of polyrhythms can be achieved via their conceptualization as rhythmic ratios, and therefore “three notes of equal value over a pulse of two notes” interpreted as a “ratio of three notes to two notes, or 3:2;” or the inverted alternative of “two notes of equal value over a pulse of three notes” interpreted as a “ratio of two notes to three notes, or 2:3.” Barrett establishes the criteria for a true polyrhythm as being when the two rhythms in question do not have a common divisor other than one, and therefore excluding ratios such as 4:2 and 2:4.163 American guitarist Steve Vai provides a thorough discussion on the technique required to perform complex polyrhythmic ratios in Little Black Dots: Tempo Metal (an online article discussing his 1978 transcriptions for American composer, guitarist, singer, film director, and satirist, Frank Zappa [1940- 93]).164 In the article, 5ç:2Ê (transcribed as five quavers in the time of two crotchets) is utilized as an example to demonstrate that the “first number (5) shows the number of beats to be superimposed over the space provided,” while the “second number (2) designates the number of beats upon which the first number is to be superimposed.” It is understood therefore, that by finding a common denominator for the two (in this case the number ten, and therefore sixteenth notes), and “subdividing and putting five units of measurements on both sides of the beat, you can now see that the second beat will fall on the upstroke of the third eighth note of the quintuplet.” The following example demonstrates how the 5:2Ê polyrhythmic subdivision, which has the common denominator of ten, is subdivided into five groups of semiquavers in order to delineate crotchet pulses.

162 Randel, ed., The New Harvard Dictionary of Music 646. 163 Kevin Barrett, “Understanding Polyrhythms,” Funkster’s Groove Theory, 23 Mar. 2004, 18 Jan. 2006, . 164 “With more than 60 albums to his credit, composer, arranger, guitarist, and bandleader Frank Zappa demonstrated a mastery of pop idioms ranging from jazz to rock of every conceivable variety, penned electronic and orchestral works, parlayed controversial satire, and testified in Congress against censorship. As astute an entrepreneur as he was a musician, he was impatient with any division between popular and high art; he combined scatological humour with political wit, required of his players (Little Feat founder Lowell George, guitarists Adrian Belew and Steve Vai, and drummer Terry Bozzio among them) an intimidating skill, and displayed consistent innovation in instrumental and studio technology.” For a further discussion, see Patricia Romanowski, Holly George-Warren, and Jon Pareles, ed., The New Rolling Stone Encyclopedia of Rock & Roll (New York: Fireside, 1995) 1111. The Equally-Tempered Archetype 97

Ex. 7. 5:2Ê Polyrhythmic subdivision

The second example demonstrates how the 7ç:3Ê polyrhythmic subdivision (transcribed as seven quavers in the time of three crotches), which has the common denominator of twenty-one, is subdivided into seven groups of semiquaver triplets in order to delineate crotchet pulses.165

Ex. 8. 7ç:3Ê Polyrhythmic subdivision

The organizational framework of the ‘polyrhythm 1-16 set’ scheme utilized in the work is based on the proposition of including all the fifty-seven polyrhythmic sets available between the ranges of two and fifteen, and therefore one-hundred-and-fourteen unique polyrhythms. All the polyrhythms have been organized in ascending order of complexity, as well as ‘golden mean’ arch form defined polyrhythmic density, with equal temperaments one to four allocated one polyrhythmic set each (3:2Ê [2:3Ê]; 5ç:2Ê [2É:5Ê]; 4:3Ê [3:4]; 5:3Ê [3:5]); five to eight, two (7ç:2Ê [2É:7Ê] and 5:4Ê [4:5Ê]; 7ç:3Ê [3É:7Ê] and 9è:2Ê [2È:9Ê]; 8ç:3Ê [3É:8Ê] and 7:4Ê [4:7Ê]; 6:5Ê [5:6Ê] and 7:5Ê [5:7Ê]; nine to twelve, three (11è:2Ê [2È:11Ê], 10ç:3Ê [3É:10Ê], and 9ç:4Ê [4É:9Ê]; 8:5Ê [5:8Ê], 7:6Ê [6:7Ê], and 11ç:3Ê [3É:11Ê]; 9:5Ê [5:9Ê], 13è:2Ê [2È:13Ê], and 11ç:4Ê [4É:11Ê]; 8:7Ê [7:8Ê], 13è:3Ê [3È:13Ê], and 11ç:5Ê [5É:11Ê]); thirteen to sixteen, four (9:7Ê [7:9Ê], 15è:2Ê [2È:15Ê], 14è:3Ê [3È:14Ê], and 13ç:4Ê [4É:13Ê]; 12ç:5Ê [5É:12Ê], 11:6Ê [6:11Ê], 10:7Ê [7:10Ê], and 9:8Ê [8:9Ê]; 13ç:5Ê [5É:13Ê], 11:7Ê [7:11Ê], 15ç:4Ê [4É:15Ê], and 14ç:5Ê [5É:14Ê]; 13ç:6Ê [6É:13Ê], 12:7Ê [7:12Ê], 11:8Ê [8:11Ê], and 10:9Ê [9:10Ê]); seventeen to nineteen, three (13:7Ê [7:13Ê], 11:9Ê [9:11Ê], and 13:8Ê [8:13Ê]; 11:10Ê [10:11Ê], 15ç:7Ê [7É:15Ê], and 13:9Ê [9:13Ê]; 15:8Ê [8:15Ê], 14:9Ê [9:14Ê], and 13:10Ê [10:13Ê]); twenty to twenty-two, two (12:11Ê [11:12Ê] and 13:11Ê [11:13Ê]; 14:11Ê [11:14Ê] and 13:12Ê [12:13Ê]; 15:11Ê [11:15Ê] and 14:13Ê [13:14Ê]); while the final two, twenty-three to twenty-four, one each (15:13Ê [13:15Ê]; 15:14Ê [14:15Ê]). A further aspect of the tripartite organizational framework is the lower limit imposed to the units of time: 2, or half note (1-et to 9-et); 4, or quarter note (10-et to 17-et); and 8, or eighth note (18-et to 24-et).

165 Steve Vai, “Little Black Dots: Tempo Metal,” The Official Steve Vai Website, 1983, 18 Jan. 2006, 8TeEulyTmee Archetype Equally-Tempered The 98 Fig. 5. Polyrhythm 1-16 set

1-et 3:2Ê (2:3Ê) 2-et 5ç:2Ê (2É:5Ê) 3-et 4:3Ê (3:4) 4-et 5:3Ê (3:5) 5-et 7ç:2Ê (2É:7Ê) 5:4Ê (4:5Ê) 6-et 7ç:3Ê (3É:7Ê) 9è:2Ê (2È:9Ê) 7-et 8ç:3Ê (3É:8Ê) 7:4Ê (4:7Ê) 8-et 6:5Ê (5:6Ê) 7:5Ê (5:7Ê) 9-et 11è:2Ê (2È:11Ê) 10ç:3Ê (3É:10Ê) 9ç:4Ê (4É:9Ê) 10-et 8:5Ê (5:8Ê) 7:6Ê (6:7Ê) 11ç:3Ê (3É:11Ê) 11-et 9:5Ê (5:9Ê) 13è:2Ê (2È:13Ê) 11ç:4Ê (4É:11Ê) 12-et 8:7Ê (7:8Ê) 13è:3Ê (3È:13Ê) 11ç:5Ê (5É:11Ê) 13-et 9:7Ê (7:9Ê) 15è:2Ê (2È:15Ê) 14è:3Ê (3È:14Ê) 13ç:4Ê (4É:13Ê) 14-et 12ç:5Ê (5É:12Ê) 11:6Ê (6:11Ê) 10:7Ê (7:10Ê) 9:8Ê (8:9Ê) 15-et 13ç:5Ê (5É:13Ê) 11:7Ê (7:11Ê) 15ç:4Ê (4É:15Ê) 14ç:5Ê (5É:14Ê) 16-et 13ç:6Ê (6É:13Ê) 12:7Ê (7:12Ê) 11:8Ê (8:11Ê) 10:9Ê (9:10Ê) 17-et 13:7Ê (7:13Ê) 11:9Ê (9:11Ê) 13:8Ê (8:13Ê) 18-et 11:10Ê (10:11Ê) 15ç:7Ê (7É:15Ê) 13:9Ê (9:13Ê) 19-et 15:8Ê (8:15Ê) 14:9Ê (9:14Ê) 13:10Ê (10:13Ê) 20-et 12:11Ê (11:12Ê) 13:11Ê (11:13Ê) 21-et 14:11Ê (11:14Ê) 13:12Ê (12:13Ê) 22-et 15:11Ê (11:15Ê) 14:13Ê (13:14Ê) 23-et 15:13Ê (13:15Ê) 24-et 15:14Ê (14:15Ê) The Equally-Tempered Archetype 99

Alain Daniélou’s Scale of Proportions

The intervallic structure of the Indonesian pélog and sléndro pentatonic scales utilized in the work have been defined by the ratios of just intonation, or the ‘scale of proportions’ (the harmonic division of the octave), as presented by Daniélou in Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. The ‘harmonic division of the octave’ is based on the harmonic series, and is a division of the octave into fifty-three distinct intervals. It is a scale of just intonation, where the intervals are called ‘pure’ or ‘just’ because there are no beats between the notes or their harmonics. The quarter- tone (three-quarter-tone) is a result of the further division of the disjunctions of this scale (just diatonic semitone, or major half-tone [16/15]), which altogether presents a total of sixty-six unique intervals (the octave included). In Indian musical theory this system is referred to as the ‘sixty-six ærutis’ – a theoretical extension to the system of the ‘twenty-two ærutis’ of contemporary performance practice.166 It must be further noted that the ‘scale of proportions’ (the modal or harmonic division of the octave) is based on the ‘modal’ musical model, and therefore in principal serves a selection of intervals with primary relationships to a fixed tonic. Daniélou explains the significance of the ‘modal’ musical application of intervals:

“Indian deæå music is essentially modal, which means that the intervals on which the musical structure is built are calculated in relation to a permanent tonic. This does not mean that the relations between sounds other than the tonic are not considered, but that each note will be established first according to its relation to the fixed tonic and not, as in the case of the cycle of fifths, by any permutation of the basic note. The modal structure can thus be compared to the proportional division of a straight line rather than to the periodic movement of a spiral. According to the symbolism of numbers, the proportional divisions are connected with certain ideas, forms, and emotions.”

With regards to the nature of harmonic science, Daniélou adds the following:

“The object of harmonic science is to classify these proportions according to their symbolism and the feelings, images, or symbols they express. Only on this basis can modes be logically constructed and their expression precisely defined. All the notes obtained in the harmonic system are distinct from those of the cyclic system, which is based on different data. Yet though the notes are theoretically distinct and their sequence follows completely different rules, in practice they lead to a similar division of the octave into fifty-three intervals.”167

166 “Æruti means ‘to hear’ or ‘that which is heard’. Musically, it points to the interval, between notes, which can be just perceived auditorily. Musically viable pitches in an octave are literally infinite: this was recognized and explicitly stated so in our ancient texts. But for practical reasons twenty-two have been enumerated and distributed within the span Sa to Sa’ (C to C’). Further, like the seven notes, they have been given names and divided into five classes based, purportedly, on their aesthetic connotations.” For a further discussion, see B. Chaitanya Deva, Indian Music (New Delhi: Indian Council for Cultural Relations, 1974) 29-30. 167 Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 91-145. 0 h qal-eprdArchetype Equally-Tempered The 100 Table 45. Alain Daniélou’s scale of proportions (the harmonic division of the octave)

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ C¢ syntonic comma 81/80 1.012500 264.896 21.506 +22 ÐÓ C£ great diesis 128/125 1.024000 267.905 41.059 +41 CË / Dì Greek enharmonic or septimal quarter-tone 31/30 1.033333 270.346 56.767 +57 ÐÔ CÚ grave or small just chromatic semitone, or minor half-tone 25/24 1.041667 272.527 70.672 ß29 ÐÕ D¹ Pythagorean limma 256/243 1.053498 275.622 90.225 ß10 ÐÖ D¸ just diatonic semitone, or major half-tone 16/15 1.066667 279.067 111.731 +12 Ð× DÝ great limma, acute or large half-tone 27/25 1.080000 282.556 133.238 +33 CÍ / Dí three-quarter-tone 135/124 1.088710 284.834 147.143 +47 ÐØ D³ grave or small tone 800/729 1.097394 287.106 160.897 ß39 ÐÙ D² just minor tone 10/9 1.111111 290.695 182.404 ß18 ÑÐ D just major tone (9th harmonic) 9/8 1.125000 294.329 203.910 +04 ÑÑ D¢ acute or large tone 256/225 1.137778 297.672 223.463 +23 ÑÒ D£ supermajor second 59049/51300 1.151053 301.145 243.545 +44 DË / Eì five quarter-tones 93/80 1.162500 304.140 260.677 +61 ÑÓ DÚ augmented second (75th harmonic) 75/64 1.171875 306.592 274.582 ß25 ÑÔ E¸ Pythagorean minor third, or trihemitone 32/27 1.185185 310.075 294.135 ß06 ÑÕ EÝ just minor third 6/5 1.200000 313.951 315.641 +16 DÍ / Eí seven quarter-tones 75/62 1.209677 316.483 329.547 +30 ÑÖ E³ neutral third 8000/6561 1.219326 319.007 343.301 ß57 Ñ× E² grave or small major third 100/81 1.234568 322.995 364.807 ß35 ÑØ E just major third (5th harmonic) 5/4 1.250000 327.032 386.314 ß14 ÑÙ E¢ Pythagorean major third, or ditone (81st harmonic) 81/64 1.265625 331.120 407.820 +08 ÒÐ E£ acute or large major third 32/25 1.280000 334.881 427.373 +27 EË / Fí nine quarter-tones 31/24 1.291667 337.933 443.081 +43 ÒÑ F³ subfourth 125/96 1.302083 340.658 456.986 ß43 ÒÒ F² grave or small fourth 320/243 1.316872 344.527 476.539 ß23 ÒÓ F just and Pythagorean perfect fourth 4/3 1.333333 348.834 498.045 ß02 ÒÔ F¢ acute or large fourth 27/20 1.350000 353.195 519.551 +20 ÒÕ F£ superfourth 512/375 1.365333 357.206 539.104 +39 FË / Gì eleven quarter-tones 62/45 1.377778 360.462 554.812 +55 ÒÖ FÚ grave or small augmented fourth 25/18 1.388889 363.369 568.717 ß31 DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) Ò× F¤ just tritone, or augmented fourth (45th harmonic) 45/32 1.406250 367.911 590.224 ß10 ÒØ F¥ acute or large tritone, or augmented fourth 64/45 1.422222 372.090 609.776 +10 ÒÙ GÝ acute or large diminished fifth 36/25 1.440000 376.741 631.283 +31 FÍ / Gí thirteen quarter-tones 90/62 1.451613 379.779 645.188 +45 ÓÐ G³ subfifth 375/256 1.464844 383.241 660.896 ß39 ÓÑ G² grave or small fifth 40/27 1.481481 387.593 680.449 ß20 ÓÒ G just and Pythagorean perfect fifth (3rd harmonic) 3/2 1.500000 392.438 701.955 +02 ÓÓ G¢ acute or large fifth 243/160 1.518750 397.344 723.014 +23 ÓÔ G£ superfifth 192/125 1.536000 401.857 743.014 +43 GË / Aì fifteen quarter-tones 31/20 1.550000 405.520 758.722 +59 ÓÕ GÚ augmented fifth (25th harmonic) 25/16 1.562500 408.790 772.627 ß27 ÓÖ A¸ Pythagorean minor sixth 128/81 1.580247 413.433 792.180 ß08 Ó× AÝ just minor sixth 8/5 1.600000 418.601 813.686 +14 GÍ / Aí seventeen quarter-tones 50/31 1.612903 421.977 827.592 +28 ÓØ A³ neutral sixth 81/50 1.620000 423.833 835.193 ß75 ÓÙ A² grave or small major sixth 400/243 1.646091 430.659 862.852 ß37 ÔÐ A just major sixth 5/3 1.666667 436.043 884.359 ß16 ÔÑ A¢ Pythagorean major sixth (27th harmonic) 27/16 1.687500 441.493 905.865 +06 ÔÒ A£ acute or large major sixth 128/75 1.706667 446.508 925.418 +25 AË / Bì nineteen quarter-tones 31/18 1.722222 450.577 941.126 +41 ÔÓ AÚ augmented sixth 125/72 1.736111 454.211 955.031 ß45 ÔÔ B¹ grave or small minor seventh 225/128 1.757813 459.889 976.537 ß23 ÔÕ B¸ Pythagorean minor seventh 16/9 1.777778 465.112 996.090 ß04 T ÔÖ BÝ acute or large minor seventh 9/5 1.800000 470.926 1017.596 +18 101 Archetype Equally-Tempered he AÍ / Bí twenty-one quarter-tones (29th harmonic) 29/16 1.812500 474.196 1029.577 +30 Ô× B³ neutral seventh 4000/2187 1.828989 478.510 1045.256 ß55 ÔØ B² grave or small major seventh 50/27 1.851852 484.492 1066.762 ß33 ÔÙ B just diatonic major seventh (15th harmonic) 15/8 1.875000 490.548 1088.269 ß12 ÕÐ B¢ Pythagorean major seventh 243/128 1.898438 496.680 1109.775 +10 ÕÑ B£ acute or large major seventh 48/25 1.920000 502.321 1129.328 +29 BË / Cí twenty-three quarter-tones 60/31 1.935484 506.372 1143.233 +43 ÕÒ C³ suboctave (125th harmonic) 125/64 1.953125 510.987 1158.941 ß41 ÕÓ C² grave or small octave 160/81 1.975309 516.791 1178.494 ß22

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 102 The Equally-Tempered Archetype

Notation for the Twenty-Four Equal Temperaments

The notation adopted in Exposiciones for Sampled Microtonal Schoenhut Toy Piano (inspired by Daniélou’s work) is highly illustrative of the affects of each individual equal temperament on the two pentatonic scales, and their consequential intervallic deviation from just intonation. It is based on approximations of the harmonic division of the octave, to the closest syntonic comma (81/80), and Daniélou’s subdivision of the whole-tone, or just major tone (9/8).

Table 46. Alain Daniélou’s subdivision of the whole-tone

NOTE INTERVAL RATIO (FRACTION) CENTS C unison 1/1 0.000 C¢ syntonic comma 81/80 21.506 C£ great diesis 128/125 41.059 CË / Dì Greek enharmonic or septimal quarter-tone 31/30 56.767 CÚ grave or small just chromatic semitone 25/24 70.672 D¹ Pythagorean limma 256/243 90.225 D¸ just diatonic semitone 16/15 111.731 DÝ great limma 27/25 133.238 CÍ / Dí three-quarter-tone 135/124 147.143 D³ small tone 800/729 160.897 D² just minor tone 10/9 182.404 D just major tone (9th harmonic) 9/8 203.910

The system of accidentals utilizes twenty-three unique symbols – three identical to conventional sharp (Ú), flat (Û), and natural (Ö) accidentals (raising, lowering or neutralizing a tone by 25/24, or 70.672 cents), with an additional four derived from standard quarter-tone notation in twentieth century contemporary music practice; the latter representative of the division of the ‘unequal’ major half-tone, or just diatonic semitone (16/15), and not of the ‘equal’ quarter-tone (1:1.029302237, or 50.000 cents) derived from twenty-four-tone equal temperament ( 24 2 ). The quarter-tone symbols include (Ë) and (í), raising or lowering a tone by 31/30 (Greek enharmonic or septimal quarter-tone), or 56.767 cents; and (Í) and (ì), raising or lowering a tone by 135/124 (three-quarter-tone), or 147.143 cents. Additional symbols include (¢, ¦, ¤, ¨) and (², ¶, ´, ¸), raising or lowering an unaltered, naturalized, sharpened, or flattened tone by 81/80 (one syntonic comma), or 21.506 cents; and (£, §, ¥, ©) and (³, ·, μ, ¹), raising or lowering an unaltered, naturalized, sharpened, or flattened tone by 128/125 (approximately two syntonic commas, or one great diesis), or 41.059 cents. It should be noted that two syntonic commas is equal to the ratio 6561/6400, and 43.013 cents. The Equally-Tempered Archetype 103

Paul Rapoport has devised a viable alternative for notating equal temperaments based on the expansion of Pythagorean notation, limited to the fifth harmonic. Nevertheless, this approach was not adopted in the work as it would not have provided a platform for comparative analysis of the twenty-four equal temperaments. In spite of this very fact, the system of notation was extensively explored and in view of its virtues, deserves a mention. In simple terms, Rapoport’s system of notation is based on firstly acknowledging the octave, or the symbol a, and establishing the number of units and size of an individual unit (in 12-et, twelve units equal to a hundred cents per unit, or 1200÷12=100). The next task then becomes to identify three key elements: the fifth, or v closest to just (in 12-et equal to seven units, or 700 cents); major third, or t closed to just (in 12-et equal to four units, or 400 cents); and the Pythagorean major third, or %, generated via the formula %=4vß2a (in 12-et also equal to four units, or 400 cents). It is then simply a matter of further establishing the existence of the following intervals, and setting a unit where appropriate:

i. syntonic comma (81/80), or k (%ßt, or 4vßtß2a) ii. Pythagorean comma (531441/524288), or p (3%ßa, or 12vß7a) iii. great diesis (128/125, or 41.059 cents), or d (aß3t, or 3kßp) iv. skhisma (32805/32768, or 1.955 cents), or s (2%+tßa, or pßk) v. diaskhisma (2048/2025, or 19.553 cents), or q (aß[%ß2t] or 2kßp) vi. Pythagorean and just major tone (9/8), or Uw (2vßa) vii. Pythagorean limma (256/243), or Uh (3aß5v) viii. Pythagorean apotome (2187/2048), or ! (7vß4a) ix. just minor third (6/5), or jm3 (vßt) x. Pythagorean minor third, or trihemitone (32/27, or 294.135 cents), or Um3 (vß%) xi. just diatonic semitone (16/15), or jh (aßvßt) xii. grave or small just chromatic semitone (25/24), or jc (2tßv)

In 12-et, only Uw, Uh, !, jm3, Um3, jh, and jc present a positive result, with 2, 1, 1, 3, 3, 1, and 1 units respectively; and therefore the Pythagorean apotome (2187/2048), or !, is selected as the appropriate notational device. The paper concludes that the “most important comma for notation is k (syntonic comma), followed in order by d (great diesis), q (diaskhisma), and s (skhisma).” Rapoport’s system of notation certainly has its merits when dealing with one single equal temperament at a time, although in the context of tuning modulations, presented is the dilemma of having to decipher seventy-two unique symbols for chromas such as the syntonic comma (81/80), Pythagorean comma (531441/524288), great 104 The Equally-Tempered Archetype diesis (128/125), skhisma (32805/32768), and diaskhisma (2048/2025), along with conventional quarter- sharp and flat, sharp and flat, and three-quarter-sharp and flat symbols.168 The ‘notation for the twenty-four equal temperaments’ table presents all the possible pitch allocations within the octave, and represents the scheme utilized in the work for the notation of all equal temperament between one and twenty-four. The system has been adopted for practicality, although it is able to approximate the intervals to the closest syntonic comma, and therefore useful when making comparisons between one equal temperament and another.

Sléndro and Pélog Scales

Sléndro and pélog scales represent the two genera (the feminine and masculine genus) of Javanese music. Saléndro or sléndro, derived from Æailendra, being the name of the Indian dynasty in regional rule from the latter part of the eighth century through to the latter part of the tenth century.169 The tunings of sléndro and pélog scales from East and Central Java are based on two very distinct non-equidistant pentatonic and heptatonic tuning systems or laras, and accordingly are represented within the Javanese orchestra or gamelan via different sets of instruments.170 A gamelan orchestra or ensemble is typically made up of distinct combinations of gongs, metallophones, xylophones, drums, bowed and plucked strings, bamboo flutes, small cymbals and singers, with the participation of between “three or four musicians” and “twenty-five instrumentalists and ten to fifteen singers.”171 A complete gamelan consists of a double set of instruments; each tuned to sléndro and pélog, and called a gamelan gedhé. The gamelan situated at the Sri Wedhari theatre auditorium in Solo, Java – belonging to the Central Javanese gamelan ensemble tradition of Karawitan – serves as an illustration of the two tuning systems, although it

168 Paul Rapoport, “The Notation of Equal Temperaments,” Xenharmonikôn: An Informal Journal of Experimental Music 16 (Autumn, 1995): 61-84. 169 Laurence Picken, “The Music of Far Eastern Asia: 2. Other Countries,” Ancient and Oriental Music, ed. Egon Wellesz (London: Oxford U. Press, 1957) 166-67. 170 “Sléndro is a five-tone scale that very roughly approaches equal-size intervals. The intervals vary within a given scale and across orchestras, but the underlying tuning concept can be considered as a rough five-tone equal temperament,” explains Braun, while “pélog is a seven-tone scale, whose underlying tuning concept is less obvious. One possibility is that the tuning approaches a nine-tone equal temperament, using both single steps (133 cents) and double steps (267 cents) from an imagined nine-tone equal temperament scale, which is a particular element of the historical tradition of Javanese Music.” Martin Braun, “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic Musical Scale,” Neuroscience of Music, 11 Mar. 2006, Sweden, 10 Apr. 2006, . 171 Benjamin Brinner, Knowing Music, Making Music: Javanese Gamelan and the Theory of Musical Competence and Interaction (Chicago: U. of Chicago Press, 1995) XVII-XX. The Equally-Tempered Archetype 105

Table 47. Notation for the twenty-four equal temperaments

NOTATION COMPASS OR RANGE NOTATION COMPASS OR RANGE (CENTS) (CENTS) C 1189.247 10.752 F¥ 600.000 620.528 C¢ 10.753 31.282 GÝ 620.529 638.234 C£ 31.283 48.912 FÍ / Gí 638.235 653.041 CË / Dì 48.913 63.719 G³ 653.042 670.671 CÚ 63.720 80.448 G² 670.672 691.201 D¹ 80.449 100.977 G 691.202 712.483 D¸ 100.978 122.483 G¢ 712.484 733.013 DÝ 122.484 140.189 G£ 733.014 750.867 CÍ / Dí 140.190 154.019 GË / Aì 750.868 765.674 D³ 154.020 171.650 GÚ 765.675 782.403 D² 171.651 193.156 A¸ 782.404 802.932 D 193.157 213.685 AÝ 802.933 820.638 D¢ 213.686 233.503 GÍ / Aí 820.639 831.391 D£ 233.504 252.110 A³ 831.392 849.021 DË / Eì 252.111 267.629 A² 849.022 873.605 DÚ 267.630 284.358 A 873.606 895.111 E¸ 284.359 304.887 A¢ 895.112 915.640 EÝ 304.888 322.593 A£ 915.641 933.271 DÍ / Eí 322.594 336.423 AË / Bì 933.272 948.077 E³ 336.424 354.053 AÚ 948.078 965.783 E² 354.054 375.560 B¹ 965.784 986.313 E 375.561 397.066 B¸ 986.314 1006.842 E¢ 397.067 417.595 BÝ 1006.843 1023.586 E£ 417.596 435.226 AÍ / Bí 1023.587 1037.416 EË / Fí 435.227 450.032 B³ 1037.417 1056.008 F³ 450.033 466.761 B² 1056.009 1077.515 F² 466.762 487.291 B 1077.516 1099.021 F 487.292 508.797 B¢ 1099.022 1119.550 F¢ 508.798 529.327 B£ 1119.551 1136.279 F£ 529.328 546.957 BË / Cí 1136.280 1151.086 FË / Gì 546.958 561.764 C³ 1151.087 1168.716 FÚ 561.765 579.470 C² 1168.717 1189.246 F¤ 579.471 599.999 C 1189.247 10.752 106 The Equally-Tempered Archetype must be noted that absolute pitch, as well as fixed theoretically defined tuning systems, should be considered anomalies, and hence intonation traditions unique to but one gamelan. The roman numerals above the pitches represent a form of the Central Javanese cipher notation system of Kepatihan, which is based on the principles of the undotted numeral representative of the central octave, while dots above and below, delineating the high and low octaves respectively. The names accorded to each pitch are based on a system of interchangeable sléndro and pélog repertoire. It must be further noted that in spite of the heptatonic structure of pélog, the tuning system is essentially pentatonic in traditional compositions, with additional ‘auxiliary’ pitches serving as temporary substitute neighbour tones.172

Ex. 9. Gamelan gedhé sléndro and pélog tunings (Sri Wedhari theatre auditorium, Solo, Central Java)

In order to ascertain the size of the intervals of this particular gamelan gedhé, the following example represents a transposition to the key of C of the actual tunings.

Ex. 10. Gamelan gedhé sléndro and pélog tunings in the key of C

The following presentation represents a harmonically tuned ‘just intonation’ reinterpretation of the sléndro and pélog scales; the intervallic delineations facilitating the display of the harmonic characteristics. Additional auxiliary tones have been added to the scales in order to generate a heptatonic scale

172 Marc Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory (Berkeley: U. of California Press, 2004) XV-41. The Equally-Tempered Archetype 107 structure from the two essentially pentatonic forms. The sléndro tuning may be simplified as consisting of the intervals: major second, perfect fourth, perfect fifth, and minor seventh; and in the key of C consisting of the pitches: C, D, F, G, and B"; while pélog: minor second, minor third, perfect fifth, and minor sixth; and in the key of C consisting of the pitches: C, D", E", G and A".

Ex. 11. The harmonic characteristics of the sléndro and pélog scales i. The sléndro scale

ii. The pélog scale

One-Tone Equal Temperament

One-tone equal temperament is generated by the factor 1 2 , and produces the intervals of the octave (with the simple frequency ratio of 2/1) and the unison (1/1), or what David D. Doty describes as “the ‘non-interval’ between two tones with exactly the same frequency.” 1/1, or the unison represents relative pitch, and not absolute pitch in general terms, although in this particular context, the frequency relative to A=440Hz (standard pitch), and equal to 261.626Hz. The octave is a twofold multiplication of 1/1, hence equal to the frequency of 523.251Hz. The octave may also be stated as being the primary interval 108 The Equally-Tempered Archetype of the prime number two, and “the only interval just intonation and equal temperament have in common.”173 Sethares proclaims the octave as “the most consonant interval after the unison.” 2/1 and 1/1 are of course represented by the first two partials of the harmonic series.174 “The proportion of the whole to its half or of the half to the whole is so natural that it is the first to be understood,” states French composer and theorist Jean-Philippe Rameau; acknowledging Italian music theorist and composer Gioseffo Zarlino (1517-90) for recognizing the octave as “the mother, the source, and the origin of all intervals,” because, “by the division of its two terms all other harmonious chords are generated.”175 The first three bars of the work highlight the singular application of both intervals.

Ex. 12. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 1-3

The following table depicts the tonal resources of one-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 48. The one-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

173 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 36. 174 Sethares, Tuning, Timbre, Spectrum, Scale 1-3. 175 Jean-Philippe Rameau, Treatise on Harmony, trans. Philip Gossett (New York: Dover Publications, 1971) 8-10. The Equally-Tempered Archetype 109

Ex. 13. 1-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 49. 1-tone equal temperament tuning matrix no. 1 (Program 01)

NOTE C – – – – – – – – – – – – – – – – – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Two-Tone Equal Temperament

Two-tone equal temperament is generated by the factor 2 2 , and introduces the equal tritone, or the sum of three equal whole-tones, which according to Richard Bobbitt was “proscribed in early polyphonic music as diabolus in musica (the devil in music),” and “remains even today a ferment among the family of intervals due to its peculiar characteristics.”176 Interesting among these characteristics is the fact that the interval of a tritone does not appear in the first sixteen partials of the harmonic series, with the just tritone, or augmented fourth (45/32, or 590.224 cents) first identified as the forty-fifth harmonic; the alternative cyclic tritone (3ô÷/2÷ô, or 591.885 cents), as the forty-seventh ascending fifth in the ‘scale of fifths’, or cyclic division of the octave; while the simpler Pythagorean tritone (729/512, or 611.730 cents), as the sixth ascending fifth. The equal tritone, or augmented fourth ( 2 2 ) is produced by the

176 According to Hans Tischler, “The diabolus in musica, however often it was exorcised by theorists, was nevertheless as familiar to musicians as was Satan to the people of the age – and of much later ages, witness Luther and the belief in witches. The avoidance of the tritone was hardly so generally practiced that it can be used as an unfailing guide to musica ficta.” For a further discussion, see Hans Tischler, “Musica Ficta in the Thirteenth Century,” Music & Letters 54.1 (Jan., 1973): 48. 110 The Equally-Tempered Archetype division of the octave into two equal parts (or two tones equal to 600.000 cents each), and approximated with the frequency ratio 181/128.177 It is absolutely symmetrical, and is therefore non- invertible.178 The notation (F¥) is reflective of the equal tritone fitting within the range of 600.000 and 620.528 cents, and may therefore be prescribed as approximating the acute or large tritone, or augmented fourth (64/45, or 609.776) with a falsity of ß9.776 cents. It should nevertheless be noted that theoretically it additionally represents the just tritone (45/32) with a falsity of +9.776 cents. The pélog scale form scheme of the composition is able to accommodate the interval to produce a two- tone symmetrical scale form, as well as a simultaneous sonority with an intervallic value of 600.000 cents.

Ex. 14. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 4

The following table depicts the tonal resources of two-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 50. The two-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

177 Daniélou, Tableau Comparatif des Intervalles Musicaux 144-45. 178 Richard Bobbitt, “The Physical Basis of Intervallic Quality and its Application to the Problem of Dissonance,” Journal of Music Theory 3.2 (Nov., 1959): 190-92. The Equally-Tempered Archetype 111

Ex. 15. 2-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 51. 2-tone equal temperament tuning matrix no. 1 (Program 02)

NOTE C – – – – – – – – – – F¥ – – – – – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Three-Tone Equal Temperament

Three-tone equal temperament is generated by the factor 3 2 , and introduces the equal major third (approximately 63/50, or 400.000 cents) and its inversion, the equal minor sixth ( [3 2]2 , approximately 100/63, or 800.000 cents).179 Relative to the just major thirds (5/4s) and just minor sixths (8/5s, or 813.686 cents) of just intonation, three-tone equal temperament produces major thirds ( 3 2 ) and minor sixths ( [3 2]2 ) with a falsity of +13.686 and ß13.686 cents on each count. The tuning modulation, although unable to generate a recognizable sléndro or pélog scale form, is nevertheless able to deliver the first theoretical triad, with the enharmonic reinterpretation of the pitches C, E¢, and A¸ generating an augmented triad (C, E¢, and G¤). The triad is nevertheless not included in the work for the very reason that formulated aesthetic guideless predicate a clear intention to highlight principal tones (of the sléndro or pélog scale forms generated), with an attempt to exclude secondary ‘auxiliary’ tones from the melodic framework as much as possible. Simultaneous sonorities introduced in the work with this temperament include the dyads C and A¸, and E¢ and C, which represent the interval of an equal minor sixth ( [).3 2]2

179 Daniélou, Tableau Comparatif des Intervalles Musicaux 102-03. 112 The Equally-Tempered Archetype

Ex. 16. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 7

The following table depicts the tonal resources of three-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 52. The three-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÐÓ A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 17. 3-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL. The Equally-Tempered Archetype 113

Table 53. 3-tone equal temperament tuning matrix no. 1 (Program 03)

NOTE C – – – – – – E¢ – – – – – – A¸ – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Four-Tone Equal Temperament

Four-tone equal temperament ( 4 2 ) presents a series of four equal tones 300.000 cents in size (approximately 44/37),180 and therefore the juxtaposition of four equal minor thirds, or two sets of two- tone equally-tempered intervals interlocked an equal minor third ( 4 2 ) apart. In comparison with the just minor third (6/5), the equal minor third ( 4 2 ) has a falsity of ß15.641 cents, while its inversion, the equal major sixth ( [4 2]3 ), a falsity of +15.641 cents with the just major sixth (5/3, or 884.359 cents). Within the combined sléndro and pélog scale form schemes, the temperament presents the pitches of the equally-tempered diminished seventh tetrad, with the pitches C, E¸, F¥, and A¢ enharmonically reinterpreted as C, E¸, G¸, and A¢. The only simultaneous sonority introduced in the work with this temperament is the dyad E¸ and C, which represents the interval of an equal major sixth ( [4 2]3 ).

Ex. 18. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 9

The following table depicts the tonal resources of four-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

180 Daniélou, Tableau Comparatif des Intervalles Musicaux 78. 114 The Equally-Tempered Archetype

Table 54. The four-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 + 00 ÐÒ E¸ equal or Pythagorean minor third, or trihemitone 1.189207 311.127 300.000 + 00 ÐÓ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 + 00 ÐÔ A¢ equal or Pythagorean major sixth 1.681793 440.000 900.000 + 00

ÐÑ C octave 2.000000 523.251 1200.000 + 00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 19. 4-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 55. 4-tone equal temperament tuning matrix no. 1 (Program 04)

NOTE C – – – – E¸ – – – – F¥ – – – – A¢ – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

The Equally-Tempered Archetype 115

Five-Tone Equal Temperament

Five-tone equal temperament is generated by the factor 5 2 , and with its five equal tones 240.00 cents in size, which represent the 5-et supermajor second (approximately 54/47),181 is not only the first equally- tempered division of the octave able to produce intervals outside of the domain of twelve-tone equal temperament, but also the first able to adequately represent at least one of the Javanese scale forms. The complete sléndro form is outlined via the pitches C, D£, F², G¢, and A!, and subsequently presents the opportunity to evaluate the common notion proposed by theorists such as Martin Braun, proclaiming that “the tuning concept (of sléndro) can be considered as a roughly five-tone equal temperament.”182 The sléndro tuning of the gamelan gedhé at the Sri Wedhari theatre auditorium in Solo, Central Java, analyzed by Marc Perlman presents a platform for comparison with the following series of unequal intervals: 238.000, 237.000, 252.000, 223.000, and 250.000 cents. The mean of this example (and the division of the octave into five equal tones) is represented by the figure of 240.000 cents (with a standard deviation value of 11.683321); proving the theory as somewhat significant, although analysis of deviation values (ß2.000, ß3.000, +12.000, ß17.000, and +10.000 cents in each case) illustrate a disproportionate relationship, and make the theoretical position no doubt inconclusive for the very reasons that Perlman makes clear with the following statement: “There is no absolute pitch, or even a standardized intonation, for these two laras; each fine gamelan may have its own distinctive realization of them. The sample tone measurements are therefore only illustrative, not definitive.”183 Simultaneous sonorities introduced in the work with this temperament include the dyads D£ and A!, F² and C, and C and G¢, which at 720.000 cents, represents the 5-et acute or large fifth ( [5 2]3 ). The interval is at the upper limits of Blackwood’s criteria for a “perfect fifth within the range of recognizability”, and represents the just perfect fifth (3/2) with a falsity +18.045 cents.184

Ex. 20. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 12

181 Daniélou, Tableau Comparatif des Intervalles Musicaux 64. 182 Braun, “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic Musical Scale,” Neuroscience of Music n. pag. 183 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41. 184 Blackwood, The Structure of Recognizable Diatonic Tunings 197. 116 The Equally-Tempered Archetype

The following table depicts the tonal resources of five-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 56. The five-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D£ 5-et supermajor second 1.148698 300.529 240.000 +40 ÐÓ F² 5-et grave or small fourth 1.319508 345.217 480.000 ß20 ÐÔ G¢ 5-et acute or large fifth 1.515717 396.550 720.000 +20 ÐÕ AÚ 5-et augmented sixth 1.741101 455.517 960.000 ß40

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 21. 5-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

The Equally-Tempered Archetype 117

Table 57. 5-tone equal temperament tuning matrix no. 1 (Program 05)

NOTE C – – D£ – – – – F² – – G¢ – – – – A! – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +40 +00 +00 ß20 +00 +20 +00 +00 ß40 +00

Six-Tone Equal Temperament

Six-tone equal temperament ( 6 2 ), with its return to intervals within the domain of twelve-tone equal temperament, presents the essential scalar material to produce the equally-tempered whole-tone hexatonic scale, featuring six equal whole-tones 200.00 cents in size (approximately 55/49).185 The temperament may alternatively be expressed as the juxtaposition of three sets of two-tone equally- tempered intervals, or two sets of three-tone equally-tempered intervals interlocked an equal major tone ( 6 2 ) apart. It must be noted that in comparison with the just major tone (9/8) and the just minor tone (10/9, or 182.404 cents), the equal major tone ( 6 2 ) has a falsity of +3.910 and ß17.596 cents on each count. Charles Villiers Stanford makes the following statement in opposition to the very notion of an equally-tempered whole-tone and the scale derived via its juxtaposition – a series of six equally- tempered whole-tones encompassing the span of an octave:

“It is physically impossible for a scale of whole-tones to reach a half at the octave (9/8, or 203.910×3 = 611.730 cents); a scale of whole greater tones would arrive at an octave which is too sharp (9/8, or 203.910×6 = 1223.460 cents); and any combination of the greater and lesser tones will be found equally impossible at the octave. The only way to reach the octave by whole-tones is to make each and every one of the intervals out of tune as the pianoforte is.”186

“The whole-tone hexatonic scale offers a limited basis for extended musical expression,” notes Vincent Persichetti, and due to the fact that “when the scale is mirrored there is no change except in register.” The scale also offers only two possible transpositions (minor seconds apart), with no prospects for modality and hence the generation of unique modes. Persichetti concludes that “its intervallic make-up deprives the scale of the fundamental intervals, the perfect fourth and fifth, and of the leading tone,” and that “a real feeling of tonality, therefore, must be established by harmony outside the whole-tone category.”187 The temperament also introduces the equal minor seventh ( [6 2]5 , approximately 98/55, or

185 Daniélou, Tableau Comparatif des Intervalles Musicaux 102. 186 Charles Villiers Stanford, “On Some Recent Tendencies in Composition,” Proceedings of the Musical Association, 47th Sess. (1920): 40. 187 Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice 54-55. 118 The Equally-Tempered Archetype

1000.000 cents),188 which represents the acute or large minor seventh (9/5, or 1017.596 cents) with a falsity of ß17.596 cents. Simultaneous sonorities appearing in the work include the equal major third ( 3 2 ) and major sixth ( [4 2]3 ), which are depicted via the Pythagorean approximations of A¸ and C, and C and A¸.

Ex. 22. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 16

The following table depicts the tonal resources of six-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 58. The six-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D equal or just major tone 1.122462 293.665 200.000 +00 ÐÓ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÐÔ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÐÕ A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÐÖ B¸ equal or Pythagorean minor seventh 1.781797 466.164 1000.000 +00

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

188 Daniélou, Tableau Comparatif des Intervalles Musicaux 55. The Equally-Tempered Archetype 119

Ex. 23. 6-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 59. 6-tone equal temperament tuning matrix no. 1 (Program 06)

NOTE C – – D – – E¢ – – F¥ – – A¸ – – B¸ – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Seven-Tone Equal Temperament

Seven-tone equal temperament is generated by the factor 7 2 , and with its seven equal tones 171.429 cents in size, which represent the 7-et grave or small tone (approximately 56295/50989),189 make it an intonation generally associated with the system of tuning in the traditional music of Siam (Thailand). The equidistant claim is speculative amongst ethnomusicologists, with scientific analysis revealing a much more complex system of intervallic diversity.190 Terry E. Miller and Sam-ang Sam make the following observations:

“In the case of Khmer music, there has never been a strong claim for equidistance, and in fact Khmer tuning is clearly non-equidistant. Certain pitch levels are considered useable, others not. In sum, then, Thai tuning is functionally equidistant while Khmer is only apparently non-equidistant but variable.”191

189 Daniélou, Tableau Comparatif des Intervalles Musicaux 48. 190 Picken, “The Music of Far Eastern Asia: 2. Other Countries,” Ancient and Oriental Music 163. 191 Terry E. Miller, and Sam-ang Sam, “The Classical Musics of Cambodia and Thailand: A Study of Distinctions,” Ethnomusicology 39.2 (Spring-Summer, 1995): 237-38. 120 The Equally-Tempered Archetype

The temperament introduces some unique simultaneous sonorities to the work, which include the dyads F¢ and AÍ, or the 7-et acute or large fourth ( [7 2]3 , approximately 689/512, or 514.286 cents); C and G², and D³ and A², or the 7-et grave or small fifth ( [7 2]4 , approximately 1024/689, or 685.714 cents); and E³ and C, or the 7-et grave or small major sixth ( [7 2]5 , approximately 105/64, or 857.143 cents).192 At 685.714 cents, the 7-et grave or small fifth ( [7 2]4 ) is at the lower limits of Blackwood’s criteria for a “perfect fifth within the range of recognizability,” and represents the just perfect fifth (3/2) with a falsity of ß16.241 cents. The 7-et grave or small major sixth ( [7 2]5 ) and 7-et acute or large fourth ( [7 2]3 ) represent deviations of just intervals (5/3 and 4/3) by ß27.216 and +16.241 cents on each count.193

Ex. 24. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 18

The following table depicts the tonal resources of seven-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 60. The seven-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D³ 7-et grave or small tone 1.104090 288.858 171.429 ß29 ÐÓ E³ 7-et neutral third 1.219014 318.925 342.857 ß57 ÐÔ F¢ 7-et acute or large fourth 1.345900 352.122 514.286 +14 ÐÕ G² 7-et grave or small fifth 1.485994 388.774 685.714 ß14 ÐÖ A² 7-et grave or small major sixth 1.640671 429.241 857.143 ß43 Ð× AÍ / Bí 7-et twenty-one quarter-tones 1.811447 473.921 1028.571 +29

ÐÑ C octave 2.000000 523.251 1200.000 +00

192 Daniélou, Tableau Comparatif des Intervalles Musicaux 89-127. 193 Blackwood, The Structure of Recognizable Diatonic Tunings 197. The Equally-Tempered Archetype 121

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 25. 7-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 61. 7-tone equal temperament tuning matrix no. 1 (Program 07)

NOTE C – – D³ E³ – – F¢ – – G² – – A² AÍ – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 ß29 +43 +00 +14 +00 ß14 +00 ß43 +29 +00

Eight-Tone Equal Temperament

Eight-tone equal temperament ( 8 2 ) presents a series of eight equal tones 150.000 cents in size (approximately 1024/939), and therefore the juxtaposition of two sets of four-tone equally-tempered intervals, or four sets of two-tone equally-tempered intervals interlocked an equal three-quarter-tone ( 8 2 ) apart. The exercise produces the equally-tempered three-quarter-tone octatonic scale, which features eight equal three-quarter-tones, and hence, the addition of the equal three-quarter-tone ( 8 2 ), nine quarter-tones ( [8 2]3 , approximately 83/64, or 450.000 cents), fifteen quarter-tones ( [,8 2]5 approximately 128/83, or 750.000 cents), and twenty-one quarter-tones ( [8 2]7 , approximately 939/512, or 1050.000 cents) to the vocabulary of equal intervals.194 The simultaneous sonorities of E¸ and F¥, CÍ and Fí, Fí and G£, and C and E¸ highlight the utilization of the equal minor third ( 4 2 ) in the

194 Daniélou, Tableau Comparatif des Intervalles Musicaux 42-113. 122 The Equally-Tempered Archetype work; while C and Fí, E¸ and G£, and F¥ and B³, the nine equal quarter-tones ([8 2]3 ), which is one of these very distinct equal quarter-tone intervals.

Ex. 26. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 22-24

The following table depicts the tonal resources of eight-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 62. The eight-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CÍ / Dí equal three-quarter-tone 1.090508 285.305 150.000 +50 ÐÓ E¸ equal or Pythagorean minor third, or trihemitone 1.189207 311.127 300.000 +00 ÐÔ EË / Fí nine equal quarter-tones 1.296840 339.286 450.000 ß50 ÐÕ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÐÖ G£ fifteen equal quarter-tones 1.542211 403.482 750.000 +50 Ð× A¢ equal or Pythagorean major sixth 1.681793 440.000 900.000 +00 ÐØ B³ twenty-one equal quarter-tones 1.834008 479.823 1050.000 ß50

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 27. 8-tone equal temperament

The Equally-Tempered Archetype 123

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 63. 8-tone equal temperament tuning matrix no. 1 (Program 08)

NOTE C CÍ – – E¸ – – Fí F¥ G£ – – A¢ – – B³

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +50 +00 +00 +00 ß50 +00 +50 +00 +00 +00 ß50

Nine-Tone Equal Temperament

Nine-tone equal temperament ( 9 2 ), with its twofold tripartite division of the octave, presents a subdivision of the equal major third ( 3 2 ) into three equal tones 133.333 cents in size, which represent the 9-et great limma (approximately 553/512). The twofold tripartite division of the octave is in reference to the juxtaposition of three sets of three-tone equally-tempered intervals interlocked an equal 9-et great limma ( 9 2 ) apart, which delivers not only the first triad in the work, but also the first complete exposition of the pélog scale with the pitches C, DÝ, Eì, F£, G³, A¸, and B². The triad consists of the pitches C, Eì, and G³, and presents a minor third 266.667 cents in size ( [9 2]2 , 9-et five quarter tones, or approximately 7/6) and a perfect fifth 666.667 cents in size ( [9 2]5 , 9-et subfifth, or approximately 147/100).195 Eì, although equal to 9-et five quarter tones ( [9 2]2 ), functions harmonically as a minor third, and therefore in association with the other pitches forms a harmonic structure that may be regarded as being essentially a nine-tone equally-tempered minor triad. In comparison with the just minor triad (equal to the proportions 10:12:15), the nine-tone equally-tempered minor triad features two acutely dissonant intervals, which form the complex ratio 1.000000:1.166529:1.469734. The fundamental C4 at standard pitch (A=440Hz) may be utilized to illustrate that the third harmonic of C4 does not correspond with the second harmonic of G³4 (769.040Hz), or the 9-et subfifth ( [9 2]5 ). A just perfect fifth with a 3/2 proportion would have presented identical frequencies, and hence a ‘zero beating’ condition between

195 Daniélou, Tableau Comparatif des Intervalles Musicaux 38-102. 124 The Equally-Tempered Archetype the two intervals. In striking contrast, nine-tone equal temperament presents 15.836 beats, with 31.673 beats between the sixth harmonic of C4 and the fourth harmonic of G³4 (1538.080Hz). Eì4 (305.194Hz), or 9-et five quarter tones ( [9 2]2 ), presents 43.784 beats between the sixth harmonic of C4 and the fifth harmonic of Eì4 (1525.969Hz). The level of dissonance is clearly within the sonic realm of ‘roughness’, and therefore outside of the 20-25Hz threshold that Doty defines as a distinctly audible rate of beating.196

Table 64. The beating characteristics of the nine-tone equally-tempered minor triad

C4 FREQUENCY Eì4 FREQUENCY BEATS G³4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 305.194 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 384.520 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 610.388 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 769.040 15.836 – – – – – – – – 3 915.581 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1153.560 – – – – – – – – – – – – 4 1220.775 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1525.969 43.784 4 1538.080 31.673 – – – – – – – – 6 1831.163 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1922.601 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The pélog scale is outlined with the pitches C, DÝ, Eì, F£, G³, A¸, and B², and equal to 0.000, 133.333, 266.667, 533.333, 666.667, 800.000, and 1066.667 cents. The pélog tuning of the gamelan gedhé analyzed by Perlman presents the following series of unequal intervals: 116.000, 165.000, 269.000, 119.000, 100.000, 192.000, and 239.000 cents. The mean of this example (and the division of the octave into seven equal tones) is represented by the figure of 171.429 cents (with a standard deviation

196 “Beats can be perceived clearly when the difference is less than 20-25Hz, but as the difference increases beyond this point the beats blend together, giving rise to a general sensation of roughness. This roughness gradually decreases as the difference increases, persisting until the difference exceeds the critical band, which, for most of the audio range falls between a whole-tone and a minor third,” explains Doty. For a further discussion, see Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 20-22. The Equally-Tempered Archetype 125 value of 65.158415), and therefore with some certainty, it may be stated that with the falsities of ß17.333, +14.333, +16.667, +2.333, ß31.000, and ß106.667 cents on each count, the intervals of the gamelan gedhé only remotely resemble those of nine-tone equal temperament. The inclusion of A£ in the pélog system – although far from being representative of any legitimate form of a minor seventh – replaces the last value of contention (B²) with a falsity of +27.667 cents. In consideration of the pentatonic outline, or principal tones of the pélog scale (the pitches C, DÝ, Eì, G³, and A¸) – just like the previous sléndro example – the mean value of 240.00 cents (with a standard deviation of 157.310203) may be ascertained, but with an even greater degree of inequality.197 Simultaneous sonorities include the triad G³, C, and G³; and the inversion F£, C, and F£; which may be represented by the 9-et superfourth ( [9 2]4 , or approximately 200/147) and 9-et subfifth ( [).9 2]5 198 The two intervals represent an extremely dissonant just perfect fourth (4/3) and just perfect fifth (3/2) with a falsity +35.288 and ß35.288 on each count.

Ex. 28. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 27

Other simultaneous sonorities include the dyads Eì and G³, B² and Eì, and G³ and B²; all equal to the interval of an equal major third ( 3 2 ); with a passage spelling F£ and G³, G³ and A£, and a melodic descent to A¸, marking the intervals of the 9-et great limma ( 9 2 ), 9-et five quarter tones ( [),9 2]2 and 9- et great limma ( 9 2 ) respectively.

Ex. 29. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 28-29

197 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41. 198 Daniélou, Tableau Comparatif des Intervalles Musicaux 130-131. 126 The Equally-Tempered Archetype

The following table depicts the tonal resources of nine-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 65. The nine-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ DÝ 9-et great limma, or large half-tone 1.080060 282.571 133.333 +33 ÐÓ DË / Eì 9-et five quarter-tones 1.166529 305.194 266.667 ß33 ÐÔ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÐÕ F£ 9-et superfourth 1.360790 356.017 533.333 +33 ÐÖ G³ 9-et subfifth 1.469734 384.520 666.667 ß33 Ð× A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÐØ AË / Bì 9-et nineteen quarter-tones 1.714488 448.554 933.333 +33 ÐÙ B² 9-et grave or small major seventh 1.851749 484.465 1066.667 ß33

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 30. 9-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

The Equally-Tempered Archetype 127

Table 66. 9-tone equal temperament tuning matrix no. 1 (Program 09)

NOTE C DÝ – – Eì E¢ F£ – – G³ A¸ AË – – B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +33 +00 ß33 +00 +33 +00 ß33 +00 +33 +00 ß33

Ten-Tone Equal Temperament

Ten-tone equal temperament ( 10 2 ) presents a subdivision of the 5-et supermajor second ( 5 2 ) into two equal tones 120.000 cents in size, which represent the 10-et just diatonic semitone (approximately 15/14). The temperament may be alternatively expressed as the juxtaposition of two sets of five-tone equally-tempered intervals, or five sets of two-tone equally-tempered intervals interlocked a 10-et just diatonic semitone ( 10 2 ) apart. The intonation scheme therefore produces the identical pitches for sléndro as five-tone equal temperament, although additional pitches generate possible auxiliary pitches of an extended pentatonic form. Sethares makes the following observations: “The 10-tet tuning has no fifth, no third, no major seconds, and no dominant sevenths. The only interval common to both 10-tet and 12-tet (other than the octave) is the 600-cent interval normally called the tritone, augmented fourth, or diminished fifth.”199 A variety of simultaneous sonorities are explored in the work within the scope of this temperament, which include the dyads E² and G¢, and F² and A³, or the 10-et grave or small major third ( [,10 ]2 3 approximately 16/13, or 360.000 cents), and C and AÚ, or the 5-et augmented sixth ( 5 ]2[ 4 , approximately 47/27, or 960.000 cents).200 The two intervals represent the just major third (5/4) and the acute or large minor seventh (9/5) with a falsity of ß26.314 and ß57.596 cents on each count.

Ex. 31. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 32

199 Sethares, Tuning, Timbre, Spectrum, Scale 291. 200 Daniélou, Tableau Comparatif des Intervalles Musicaux 34-92. 128 The Equally-Tempered Archetype

Other simultaneous sonorities include the dyads D£ and A³, or the equal tritone ( 2 2 ), and C and G¢, or the 5-et acute or large fifth ( [).5 2]3

Ex. 32. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 34

The following table depicts the tonal resources of ten-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 67. The ten-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D¸ 10-et just diatonic semitone, or major half-tone 1.071773 280.403 120.000 +20 ÐÓ D£ 5-et supermajor second 1.148698 300.529 240.000 +40 ÐÔ E² 10-et grave or small major third 1.231144 322.099 360.000 ß40 ÐÕ F² 5-et grave or small fourth 1.319508 345.217 480.000 ß20 ÐÖ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 Ð× G¢ 5-et acute or large fifth 1.515717 396.550 720.000 +20 ÐØ A³ 10-et neutral sixth 1.624505 425.012 840.000 ß60 ÐÙ AÚ 5-et augmented sixth 1.741101 455.517 960.000 ß40 ÑÐ B 10-et just diatonic major seventh 1.866066 488.211 1080.000 ß20

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 33. 10-tone equal temperament

The Equally-Tempered Archetype 129

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 68. 10-tone equal temperament tuning matrix no. 1 (Program 10)

NOTE C D¸ D£ – – E² F² F¥ G¢ A³ – – AÚ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +20 +40 +00 ß40 ß20 +00 +20 +40 +00 ß40 ß20

Eleven-Tone Equal Temperament

Eleven-tone equal temperament is generated by the factor 11 2 , and presents eleven equal intervals 109.091 cents in size, which represent the 11-et just diatonic semitone (approximately 82/77). An implied triad makes its appearance in the composition in bar 38. This triad (essentially an eleven-tone equally-tempered major triad) consists of the pitches C, EË, and G³, and presents a major third 436.364 cents in size ( [11 2]4 , 11-et nine quarter-tones, or approximately 659/512) and a perfect fifth 654.545 cents in size ( [11 2]6 , 11-et subfifth, or approximately 54/37); hence an extremely dissonant harmonic construct.201 EË, although equal to 11-et nine quarter-tones ( [11 2]4 ), functions harmonically as a major third. In comparison with the just major triad (equal to the proportions 4:5:6), the eleven-tone equally- tempered major triad forms the complex ratio 1.000000:1.286665:1.459480. The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of +50.050 and ß47.410 cents, which is approximately an equal quarter-tone ( 24 2 ) deviation on each count. G³4 (381.837Hz), or the 11-et subfifth ( [11 2]6 ), presents 21.202 beats between the third harmonic of C4 and the second harmonic of G³4 (763.675Hz), and 42.404 beats between the sixth harmonic of C4 and the fourth harmonic of G³4 (1527.349Hz); while EË4 (336.624Hz), or 11-et nine quarter-tones ( [11 2]4 ), 38.370 beats between the fifth harmonic of C4 and the fourth harmonic of EË4 (1346.498Hz).

201 Daniélou, Tableau Comparatif des Intervalles Musicaux 30-135. 130 The Equally-Tempered Archetype

Table 69. The beating characteristics of the eleven-tone equally-tempered major triad

C4 FREQUENCY EË4 FREQUENCY BEATS G³4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 336.624 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 381.837 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 673.249 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 763.675 21.202 – – – – – – – – 3 1009.873 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1145.512 – – – – 5 1308.128 4 1346.498 38.370 – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1527.349 42.404 – – – – – – – – 5 1683.122 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2019.747 – – – – 5 1909.187 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Simultaneous sonorities introduced in the work with this temperament include the dyads F£ and G³, or the 11-et just diatonic semitone ( 11 2), and B¹ and C, or the 11-et acute or large tone ( [11 2]2 , approximately 245/216, or 218.182 cents).202

Ex. 34. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 41

Other simultaneous sonorities include the dyads D¢ and F£, EË and Aì, and G³ and B¹, or 11-et seven quarter-tones ( [11 2]3 , approximately 29/24, or 327.273 cents), and C and EË, and B¹ and D¢, or 11-et nine quarter-tones ( [).11 2]4 203

202 Daniélou, Tableau Comparatif des Intervalles Musicaux 58. 203 Daniélou, Tableau Comparatif des Intervalles Musicaux 84. The Equally-Tempered Archetype 131

Ex. 35. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 42

The following table depicts the tonal resources of eleven-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 70. The eleven-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D¸ 11-et just diatonic semitone, or major half-tone 1.065041 278.642 109.091 +09 ÐÓ D¢ 11-et acute or large tone 1.134313 296.765 218.182 +18 ÐÔ DÍ / Eí 11-et seven quarter-tones 1.208089 316.067 327.273 +27 ÐÕ EË / Fí 11-et nine quarter-tones 1.286665 336.624 436.364 +36 ÐÖ F£ 11-et superfourth 1.370351 358.519 545.455 +45 Ð× G³ 11-et subfifth 1.459480 381.837 654.545 ß45 ÐØ GË / Aì 11-et fifteen quarter-tones 1.554406 406.672 763.636 ß36 ÐÙ A² 11-et grave or small major sixth 1.655507 433.123 872.727 ß27 ÑÐ B¹ 11-et grave or small minor seventh 1.763183 461.294 981.818 ß18 ÑÑ B 11-et just diatonic major seventh 1.877862 491.297 1090.909 ß09

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 36. 11-tone equal temperament

132 The Equally-Tempered Archetype

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 71. 11-tone equal temperament tuning matrix no. 1 (Program 11)

NOTE C D¸ D¢ DÍ EË F£ – – G³ Aì A² B¹ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +09 +18 +27 +36 +45 +00 ß45 ß36 ß27 ß18 ß09

Twelve-Tone Equal Temperament

Twelve-tone equal temperament ( 12 2 ), with its division of the octave into twelve equally-tempered semitones 100.000 cents in size, marks the central structural climax of the composition. The temperament may alternatively be expressed as the juxtaposition of two sets of six-tone equally- tempered intervals, three sets of four-tone equally-tempered intervals, four sets of three-tone equally- tempered intervals, or six sets of two-tone equally-tempered intervals interlocked an equal semitone ( 12 2 ) apart. Philippe De Vitry (1291-1361) has the following to say about the semitone: “The semitone, as Bernardus said, is the sugar and spice of all music, and without it song is corrupted, altered, and destroyed,” while theorizing the interval of the semitone as “the interval between two unisons, which in the human voice is incapable of, and will admit of, division or the interpolation of a middle sound.”204 With regards to the rationale behind the adoption of twelve-tone equal temperament as the intonation standard of Western music, Blackwood offers the following explanation: “There is no doubt in my mind that of all equal tunings, twelve is the most versatile and most expressive. It’s no accident that we have twelve instead of thirteen.”205

204 Philippe de Vitry, “Philippe de Vitry’s ‘Ars Nova’: A Translation,” Journal of Music Theory 5.2 (Winter, 1961): 10. 205 Douglas Keislar, Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 29.1 (Winter, 1991): 180. The Equally-Tempered Archetype 133

The first simultaneous sonority of interest makes its appearance in the first beat of bar 44, which incorporates the pitches C, G, F, and C, and may be theoretically explained as two equal perfect fourths ( [12 2]5 ) juxtaposed an equal minor seventh apart ( [6 2]5 ), or C(add 11/omit 3). Consequential sonorities B¢ and F, E¢ and B¸, and F¥ and C (F¥ alternatively expressed via its enharmonic equivalent of G¸), all equal to 600.000 cents, or the equal tritone ( 2 2 ), while E¢ and C, and C and B¸, the equal minor sixth ( [3 2]2 ) and equal minor seventh ( [6 2]5 ).

Ex. 37. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 44-45

Other sonorities include the dyads F and G¸, and B¢ and C, which represent the equal semitone ( 12 2 ). In comparison with the just diatonic semitone (16/15), the equal semitone ( 12 2 ) has a falsity of ß11.731 cents, while its inversion, the equal major seventh ( [12 2]11 , approximately 967/512, or 1100.000 cents),206 a falsity of +11.731 cents with the just diatonic major seventh (15/8, or 884.359 cents) Twelve-tone equal temperament is the first temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognizability,”207 and produces the just perfect fifth (3/2) and just perfect fourth (4/3) with a falsity of +1.955 and ß1955 cents on each count.

Ex. 38. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 46

The following table depicts the tonal resources of twelve-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

206 Daniélou, Tableau Comparatif des Intervalles Musicaux 29. 207 Blackwood, The Structure of Recognizable Diatonic Tunings 197. 134 The Equally-Tempered Archetype

Table 72. The twelve-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D¹ equal semitone, or Pythagorean limma 1.059463 277.183 100.000 +00 ÐÓ D equal or just major tone 1.122462 293.665 200.000 +00 ÐÔ E¸ equal or Pythagorean minor third, or trihemitone 1.189207 311.127 300.000 +00 ÐÕ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÐÖ F equal, or just and Pythagorean perfect fourth 1.334840 349.228 500.000 +00 Ð× F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÐØ G equal, or just and Pythagorean perfect fifth 1.498307 391.995 700.000 +00 ÐÙ A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÑÐ A¢ equal or Pythagorean major sixth 1.681793 440.000 900.000 +00 ÑÑ B¸ equal or Pythagorean minor seventh 1.781797 466.164 1000.000 +00 ÑÒ B¢ equal or Pythagorean major seventh 1.887749 493.883 1100.000 +00

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 39. 12-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

The Equally-Tempered Archetype 135

Table 73. 12-tone equal temperament tuning matrix no. 1 (Program 12)

NOTE C D¹ D E¸ E¢ F F¥ G A¸ A¢ B¸ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Thirteen-Tone Equal Temperament

Thirteen-tone equal temperament is generated by the factor 13 2 , and presents thirteen equal intervals 92.308 cents in size, which represent the 13-et Pythagorean limma (approximately 77/73), and approximate the Pythagorean limma (256/243) with a falsity of +2.083 cents. The thirteen-tone equally- tempered division of the octave now for the first time presents more than one possibility for the approximation of a particular tone. In this case, that tone being the fifth, with the possible representation as G£, 13-et thirteen quarter-tones ( [13 2]7 , approximately 61/42, or 646.154 cents), or the alternative of Gí, the 13-et superfifth ( [13 2]8 , approximately 72/47, or 738.462 cents).208 Throughout the work, the criteria for accessing whether this second tone is incorporated into the scalar scheme as a primary or secondary ‘auxiliary’ tone will be dependent on the accuracy of the approximation. In thirteen-tone equal temperament Gí ( [13 2]7 ), hence accorded as an auxiliary tone, due to the fact that G£ ( [13 2]8 ) is 36.507 cents higher than the just perfect fifth (3/2), while Gí ([13 2]7 ), 55.801 cents lower, or with a falsity exceeding that of the primary fifth G£ ([13 2]8 ) by 19.294 cents. In the case of more than two possibilities for the approximation of one tone, only the closest two will be taken into account. Melodic development in bar 51 of the work exploits the intervallic nuances of the two microtonal inflections of the fifth via the intonational reinterpretation of pitch recapitulation.

Ex. 40. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 51

208 Daniélou, Tableau Comparatif des Intervalles Musicaux 26-137. 136 The Equally-Tempered Archetype

American composer Easley Blackwood (1933-) composed Twelve Microtonal Etudes for Electronic Music Media in 1979 and 1980 as part of a research project supported by the National Endowment for the Humanities in association with Webster College in St. Louis, USA. The twelve-movement work explored the tonal and modal characteristic of all the equal divisions of the octave between thirteen and twenty-four, stimulating the devising of appropriate notational practices, and culminating in the compact disc release of Microtonal Compositions by Easley Blackwood, which feature Blackwood on the polyfusion synthesizer. It is therefore most appropriate to hereon incorporate some of the comments expressed by the composer in relation to each temperament. According to Blackwood, the thirteen- tone equally-tempered division of the octave is “the most alien tuning of all; so dissonant that no three- note combination sounds like major or minor triad.”209 Two new triads are introduced to the work via this temperament. The first triad (essentially a thirteen-tone equally-tempered major triad) consists of the pitches C, E², and G£, and presents the 13-et grave or small major third ( [13 2]4 , approximately 26/21, or 369.231 cents) and the 13-et superfifth ( [13 2]8 ). The second triad consists of the pitches C, DÚ, and Gí, and presents the 13-et augmented second ( [13 2]3 , approximately 601/512, or 276.923 cents) and 13-et thirteen quarter-tones ( [13 2]7 ). Enharmonically reinterpreted as C, E¹, and Gí, the latter triad serves as an implied thirteen-tone equally-tempered minor triad.210

Ex. 41. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 55

The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß17.083 and +36.507 cents, or approximate equal sixth-tone ( 36 2 , approximately 261/256, or 33.333 cents) and third-tone ( 18 2 , 133/128, or 66.667 cents) deviations on each count.211 G£4 (400.802Hz), or the 13-et superfifth ( [13 2]8 ), presents 16.726 beats between the third harmonic of C4 and the second harmonic of G£4 (801.603Hz), and 33.453 beats between the sixth harmonic of C4 and the fourth harmonic of G£4 (1603.206Hz); while E²4 (323.821Hz), or the 13-et grave or small major third ( [),13 2]4

209 Easley Blackwood, liner notes, Microtonal Compositions by Easley Blackwood, perf. Easley Blackwood (polyfusion synthesizer), and Jeffrey Kust (guitar), rec. 16 Sep. 1990, Cedille, 1994, CDR 90000 018, n. pag. 210 Daniélou, Tableau Comparatif des Intervalles Musicaux 74-94. 211 Daniélou, Tableau Comparatif des Intervalles Musicaux 8-18. The Equally-Tempered Archetype 137

12.844 beats between the fifth harmonic of C4 and the fourth harmonic of E²4 (1295.283Hz). This triad is certainly more consonant than the eleven-tone equally-tempered major triad previously presented (C, EË, and G³), with a major third 436.364 cents in size a perfect fifth 654.545 cents in size, although when compared with the theoretical major triad of twelve-tone equal temperament (C, E¢, and G), which presents an equal major third ( 3 2 ) and perfect fifth ( [12 2]7 ) 400.000 and 700.000 cents in size, it is quite dissonant in character.

Table 74. The beating characteristics of the thirteen-tone equally-tempered major triad

C4 FREQUENCY E²4 FREQUENCY BEATS G£4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 323.821 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 400.802 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 647.642 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 801.603 16.726 – – – – – – – – 3 971.463 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1202.405 – – – – 5 1308.128 4 1295.283 12.844 – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1603.206 33.453 – – – – – – – – 5 1619.104 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1942.925 – – – – 5 2004.008 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The two microtonal inflections of 3/2 are also explored as simultaneous sonorities, with the dyads C and Gí, and Gì and C, or 13-et thirteen quarter-tones ( [),13 2]7 and C and G£, or the 13-et superfifth ( [).13 2]8

Ex. 42. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 50

138 The Equally-Tempered Archetype

Other sonorities include the dyads G£ and B¸, or the 13-et augmented second ( [13 2]3 ); Gí and B¸, or the 13-et grave or small major third ( [);13 2]4 E² and GÍ, or the 13-et subfourth ( [13 2]5 , approximately 47/36, or 461.538 cents); DÚ and GÍ, or 13-et eleven quarter-tones ( [13 2]6 , approximately 705/512, or 553.846 cents); D² and A£, or the 13-et superfifth ( [13 2]8 ), E² and C, or the 13-et seventeen quarter- tones ( [13 2]9 , approximately 21/13, or 830.769 cents); and DÚ and C, or the 13-et acute or large major sixth ( [13 2]10 , approximately 1024/601, or 923.077 cents).212 The following table depicts the tonal resources of thirteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 75. The thirteen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D¹ 13-et Pythagorean limma 1.054766 275.954 92.308 ß08 ÐÓ D² 13-et just minor tone 1.112531 291.067 184.616 ß15 ÐÔ DÚ 13-et augmented second 1.173460 307.007 276.923 ß23 ÐÕ E² 13-et grave or small major third 1.237726 323.821 369.231 ß31 ÐÖ F³ 13-et subfourth 1.305512 341.555 461.538 ß38 Ð× FË / Gì 13-et eleven quarter-tones 1.377009 360.261 553.846 ß46 ÐØ FÍ / Gí 13-et thirteen quarter-tones 1.452423 379.991 646.154 ß54 ÐÙ G£ 13-et superfifth 1.531966 400.802 738.462 +38 ÑÐ GÍ / Aí 13-et seventeen quarter-tones 1.615866 422.752 830.769 +31 ÑÑ A£ 13-et acute or large major sixth 1.704361 445.904 923.077 +23 ÑÒ B¸ 13-et Pythagorean minor seventh 1.797702 470.325 1015.385 +15 ÑÓ B¢ 13-et Pythagorean major seventh 1.896155 496.083 1107.692 +08

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 43. 13-tone equal temperament

212 Daniélou, Tableau Comparatif des Intervalles Musicaux 75-34. The Equally-Tempered Archetype 139

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 76. 13-tone equal temperament tuning matrix no. 1 (Program 13)

NOTE C D¸ D² DÚ E² F³ Gì G£ GÍ A£ B¸ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß08 ß15 ß23 ß31 ß38 ß46 +38 +31 +23 +15 +08

Table 77. 13-tone equal temperament tuning matrix no. 2 (Program 14)

NOTE – – – – – – – – – – – – Gí – – – – – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +46 +00 +00 +00 +00 +00

Fourteen-Tone Equal Temperament

Fourteen-tone equal temperament ( 14 2 ) presents a subdivision of the 7-et grave or small tone ( 7 2) into two equal tones 85.714 cents in size, which represent the 14-et Pythagorean limma (approximately 269/256). The temperament may alternatively be expressed as the juxtaposition of two sets of seven- tone equally-tempered intervals, or seven sets of two-tone equally-tempered intervals interlocked a 14-et Pythagorean limma ( 14 2 ) apart. Within the sléndro scalar scheme two options become available for the approximation of the major second, being D³ or DË, or the 7-et grave or small tone ( 7 2 ) and 14-et five quarter-tones ( [14 2]3 , approximately 297/256, or 257.143 cents); two for the major third, being E³ or E£, or the 7-et neutral third ( [7 2]2 , approximately 128/105, or 342.857 cents) and 14-et acute or large major third ( [14 2]5 , approximately 16807/13122, or 428.571 cents); as well as two options for the augmented sixth, being AÚ or AÍ, or the 14-et augmented sixth ( [14 2]11 , approximately 512/297, or 942.857 cents) and 7-et twenty-one quarter-tones ( [7 2]6 , approximately 50989/28147, or 1028.571 140 The Equally-Tempered Archetype cents).213 The pélog scalar scheme accommodates only the latter set of intervals. Bar 58 of the work presents D³ and DË as an intonational reinterpretation of a melodic sequence, while bar 59 alternatively presents C and D³, and C and DË as intonational reinterpretations of a simultaneous sonority.

Ex. 44. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 58-59

A further example of an intonational reinterpretation of a melodic sequence is bar 61, which highlights AÚ and AÍ.

Ex. 45. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 61

In spite of E£ ([14 2]5 ), or the 14-et acute or large major third being technically closer to the just major third (5/4) – E³ ([7 2]2 ), or the 7-et neutral third and E£ ([14 2]5 ) represent 5/4 with a falsity of ß43.457 and +42.258 on each count – E³ ([7 2]2 ) is relegated with the role of primary tone (and not the expected secondary ‘auxiliary’ tone) for simple aesthetic considerations; hereby presenting an exception to the rule where the accorded status of a tone is dependent on the accuracy of the approximation to just intervals. It must be nevertheless noted that the 1.199 cents difference between the tones is negligible, and therefore not a point of theoretical contention. Blackwood offers the following description of fourteen-tone equal temperament: “This very discordant tuning offers two highly contrasting modal arrangements. One is the division of an octave into

213 Daniélou, Tableau Comparatif des Intervalles Musicaux 24-108. The Equally-Tempered Archetype 141 seven equal parts, sounding like a diatonic scale with no distinctions of major and minor. The other is a combination of two differently tuned diminished seventh chords.”214 One new triad (essentially a fourteen-tone equally-tempered major triad) is introduced to the work via this temperament, which consists of the pitches C, E³, and G², and presents a major third 342.857 cents in size ( [7 2]2 , or 7-et neutral third) and a perfect fifth 685.714 cents in size ( [7 2]4 , or 7-et grave or small fifth). The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß43.457 and ß16.241 cents on each count. G²4 (388.774Hz), or 7-et grave or small fifth ( [7 2]4 ), presents 7.329 beats between the third harmonic of C4 and the second harmonic of G²4 (777.548Hz), and 14.658 beats between the sixth harmonic of C4 and the fourth harmonic of G²4 (1555.096Hz); while E³4 (318.925Hz), or the 7-et neutral third ( [7 2]2 ), 32.427 beats between the fifth harmonic of C4 and the fourth harmonic of E³4 (1275.700Hz).

Table 78. The beating characteristics of the fourteen-tone equally-tempered major triad

C4 FREQUENCY E³4 FREQUENCY BEATS G²4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 318.925 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 388.774 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 637.850 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 777.548 7.329 – – – – – – – – 3 956.776 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1166.322 – – – – 5 1308.128 4 1275.700 32.427 – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1555.096 14.658 – – – – – – – – 5 1594.626 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1913.551 – – – – 5 1943.870 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The concept of exploiting the intervallic nuances of two microtonal inflections is developed further in fourteen-tone equal temperament, with its incorporation into simultaneous sonorities as a structural compositional device. Bar 62 features a repeat of the harmonic sequence of C and GÚ, or the 14-et augmented fifth ( [14 2]9 , approximately 26244/16807, or 771.429 cents); AÚ and G², or the 14-et

214 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 142 The Equally-Tempered Archetype augmented sixth ( [14 2]11 ); A² and F¢, or the 7-et grave or small major sixth ( [7 2]5 ); and G² and E£, and F¢ and DË, or the or 14-et augmented sixth ( [14 2]11 ); with the two final dyads replaced by G² and E³, and F¢ and D³, or the 7-et grave or small major sixth ( [7 2]5 ). The recapitulation of the harmonic sequence emphasizes intervallic diversity available within the two alternatives for major seconds and thirds.215

Ex. 46. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 62

The following two bars outline a fourteen-tone equally-tempered sléndro scalar descent with the pitches C, AÍ, A², G², F¢, E³, D³, and C – equal to 1200.000, 1028.571, 857.143, 685.714, 514.286, 342.857, and 171.429 cents respectively. The melodic line is further supported via the harmonic progression of E³ and C, D³ and AÍ, C and A², or the 7-et grave or small major sixth ( [7 2]5 ); and C and G², or the 7-et grave or small fifth ( [7 2]4 ); followed singularly by F¢, or the 7-et acute or large fourth ( [7 2]3 ); E³, or the 7-et neutral third ( [);7 2]2 and D³, or the 7-et grave or small tone ( 7 2 ), to a resolution provided by the simultaneous sonority of C and E³, or the 7-et neutral third ( [7 2]2 ), which implies a fourteen-tone equally-tempered major triad.

Ex. 47. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 63-64

Additional simultaneous sonorities include the pitches C and F¢, and F¢ and AÍ, or the 7-et acute or large fourth ( [);7 2]3 D¹ and GÚ, or the 7-et grave or small fifth ( [);7 2]4 and D¹ and AÍ, or the 14-et

215 Daniélou, Tableau Comparatif des Intervalles Musicaux 109. The Equally-Tempered Archetype 143 augmented sixth ( [14 2]11 ). The following table depicts the tonal resources of fourteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 79. The fourteen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D¹ 14-et Pythagorean limma 1.050757 274.905 85.714 ß14 ÐÓ D³ 7-et grave or small tone 1.104090 288.858 171.429 ß29 ÐÔ DË / Eì 14-et five quarter-tones 1.160129 303.520 257.143 +57 ÐÕ E³ 7-et neutral third 1.219014 318.925 342.857 ß57 ÐÖ E£ 14-et acute or large major third 1.280887 335.113 428.571 +29 Ð× F¢ 7-et acute or large fourth 1.345900 352.122 514.286 +14 ÐØ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÐÙ G² 7-et grave or small fifth 1.485994 388.774 685.714 ß14 ÑÐ GÚ 14-et augmented fifth 1.561418 408.507 771.429 ß29 ÑÑ A² 7-et grave or small major sixth 1.640671 429.241 857.143 ß43 ÑÒ AÚ 14-et augmented sixth 1.723946 451.028 942.857 ß57 ÑÓ AÍ / Bí 7-et twenty-one quarter-tones 1.811447 473.921 1028.571 +29 ÑÔ B¢ 14-et Pythagorean major seventh 1.903390 497.976 1114.286 +14

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 48. 14-tone equal temperament

144 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 80. 14-tone equal temperament tuning matrix no. 1 (Program 15)

NOTE C D¹ D³ E³ E£ F¢ F¥ G² GÚ A² AÍ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß14 ß29 +43 +29 +14 +00 ß14 ß29 ß43 +29 +14

Table 81. 14-tone equal temperament tuning matrix no. 2 (Program 16)

NOTE – – – – – – DË – – – – – – – – – – AÚ – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 ß43 +00 +00 +00 +00 +00 +43 +00 +00

Fifteen-Tone Equal Temperament

Fifteen-tone equal temperament ( 15 2 ), with its twofold tripartite division of the octave, presents a subdivision of the 5-et supermajor second ( 5 2 ) into three equal tones 80.000 cents in size, which represent the 15-et grave or small just chromatic semitone (approximately 22/21).216 The temperament may alternatively be expressed as the juxtaposition of three sets of five-tone equally-tempered intervals, or five sets of three-tone equally-tempered intervals interlocked a 15-et grave or small just chromatic semitone ( 15 2 ) apart. Blackwood describes the temperament thus: “One of the most fascinating of the equal tunings, this contains triads that are sufficiently in tune to serve to serve as the final harmony in cadences. The major scale, however, is so strange that even the most common diatonic progressions are disturbing unless precautions are taken.”217 One new triad (essentially a fifteen-tone equally-tempered minor triad) is introduced to the work via this temperament, which consists of the pitches C, EÝ, and G¢, and presents a minor third 320.000 cents in size ( [15 2]4 , approximately 77/64, or 15-et just minor third) and a perfect fifth 720.000 cents in size ( [5 2]3 , or 5-et acute or large fifth).218 The temperament produces the just minor third (6/5) and just perfect fifth (3/2) with a falsity of +4.359 and +18.045 cents on each count. G¢4 (396.550Hz), or 5-et acute or large fifth ( [5 2]3 ), presents 8.224 beats between the third harmonic of C4 and the second

216 Daniélou, Tableau Comparatif des Intervalles Musicaux 22. 217 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 218 Daniélou, Tableau Comparatif des Intervalles Musicaux 82. The Equally-Tempered Archetype 145 harmonic of G¢4 (793.100Hz), and 16.447 beats between the sixth harmonic of C4 and the fourth harmonic of G¢4 (1586.201Hz); while EÝ4 (314.742Hz), or the 15-et just minor third ( [15 2]4 ), 3.957 beats between the sixth harmonic of C4 and the fifth harmonic of EÝ4 (1573.711Hz).

Ex. 49. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 66

Table 82. The beating characteristics of the fifteen-tone equally-tempered minor triad

C4 FREQUENCY EÝ4 FREQUENCY BEATS G¢4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 314.742 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 396.550 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 629.484 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 793.100 8.224 – – – – – – – – 3 944.226 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1189.651 – – – – – – – – – – – – 4 1258.968 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1573.711 3.957 4 1586.201 16.447 – – – – – – – – 6 1888.453 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1982.751 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The closest approximation of the major scale available in fifteen-tone equal temperament in both sléndro and pélog forms is considered as presenting the series: 0.000, 240.000, 400.000, 480.000, 720.000, 880.000, 1120.000, and 1200.000 cents, and therefore representing the just major tone (9/8), just major 146 The Equally-Tempered Archetype third (5/4), just perfect fourth (4/3), just perfect fifth (3/2), just major sixth (5/3), and just major seventh (15/8) with falsity of +36.090, +13.686, ß18.045, +18.045, ß4.359, and +31.731 cents on each count. Major, minor, and diminished triads produced via this temperament presents major thirds 400.000 cents in size, minor thirds with a range between 240.000 and 320.000 cents, perfect fifths with a range between 640.000 and 720.000 cents, and a diminished fifth 560.000 cents in size. The most problematic triad being D minor, or the pitches D£, F², and A, with a minor third and perfect fifth represented by the 5-et supermajor second ( 5 2 ) and 15-et thirteen quarter-tones ( [15 2]8 , 741/512, or 640.000 cents),219 and a deviation from just intervals of ß75.641 and ß61.955 cents. In harmonic terms, the two intervals of the D minor triad essentially approximate a supermajor second (59049/51300, or 243.545 cents) and the division of the disjunction between GÝ and G³, or thirteen quarter-tones (90/62, or 645.188 cents), and therefore contributing nothing with any possibility of being orally interpreted as any form of a major triad.

Ex. 50. The fifteen-tone equally-tempered major scale

The closest approximation of the minor scale on the other hand presents the series: 0.000, 240.000, 320.000, 480.000, 720.000, 800.000, 1120.000, and 1200.000 cents, and therefore representing the just major tone (9/8), just minor third (6/5), just perfect fourth (4/3), just perfect fifth (3/2), just minor sixth (8/5), and acute or large minor seventh (9/5) with a falsity of +36.090, +4.359, ß18.045, +18.045, ß13.686, and +2.404 cents on each count.

Ex.51. The fifteen-tone equally-tempered minor scale

The sléndro scalar scheme provides two options for the approximation of the major second, being D³ or D£, or the 15-et grave or small tone ( [15 2]2 , approximately 34/31, or 160.000 cents) and 5-et supermajor second ( 5 2 ); while pélog, two options for the tritone, being FË or FÍ, or 15-et eleven quarter-tones ( [15 2]7 , 1024/741, or 560.000 cents) and 15-et thirteen quarter-tones ( [15 2]8 ), as well as two options for the major seventh, being B³ or B£, or the 15-et neutral seventh ([15 2]13 , approximately 31/17, or 1040.000 cents) and 15-et just diatonic major seventh ( [15 2]14 , approximately 489/256, or

219 Daniélou, Tableau Comparatif des Intervalles Musicaux 137. The Equally-Tempered Archetype 147

1120.000 cents).220 Bar 68 is representative of intonational reinterpretations of a simultaneous sonority based on the two major seconds, which represent the just major tone (9/8) with a falsity of ß43.013 and +39.635 cents on each count. The latter interval is further represented by the simultaneous sonorities of D£ and F², and G¢ and AÚ in bar 66. The mean deviation of the two intervals from the just major tone (9/8) is significantly close to the interval of the great diesis (128/125), or “the defect of three major thirds from an octave.”221

Ex. 52. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 68

Bars 69 and 70 depicts the two microtonal inflections of the major seventh, or the 15-et neutral seventh ( [15 2]13 ) and 15-et just diatonic major ( [15 2]14 ).

Ex. 53. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 69-70

Bar 72 on the other hand, depicts the two microtonal inflections of the tritone, 15-et eleven quarter- tones ( [15 2]7 ) and 15-et thirteen quarter-tones ( [15 2]8 ); while the following bar outlines a fifteen-tone equally-tempered pélog scalar descent (omitting CÚ) with the pitches G¢, FÍ, FË, F², and EÝ – equal to 720.000, 640.000, 560.000, 480.000, and 320.000 cents respectively, and therefore a scalar progression with equal tones 80.000 cents in size, or the 15-et grave or small just chromatic semitone

220 Daniélou, Tableau Comparatif des Intervalles Musicaux 23-136. 221 The juxtaposition of three just major thirds (5/4), equal to 1158.941 cents is a figure that diminishes the octave (2/1) by 41.059 cents, and the interval termed as the ‘great diesis’ with a frequency ratio of 128/125. For a further discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 453. 148 The Equally-Tempered Archetype

( 15 2 ). Framed as simultaneous sonorities with repeated C's, the resolution is provided by the resounding of the fundamental in the downbeat of sixteen-tone equal temperament.

Ex. 54. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 72-73

The following table depicts the tonal resources of fifteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 83. The fifteen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CÚ 15-et grave or small just chromatic semitone 1.047294 273.999 80.000 ß20 ÐÓ D³ 15-et grave or small tone 1.096825 286.957 160.000 ß40 ÐÔ D£ 5-et supermajor second 1.148698 300.529 240.000 +40 ÐÕ EÝ 15-et just minor third 1.203025 314.742 320.000 +20 ÐÖ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 Ð× F² 5-et grave or small fourth 1.319508 345.217 480.000 ß20 ÐØ FË / Gì 15-et eleven quarter-tones 1.381913 361.544 560.000 +60 ÐÙ FÍ / Gí 15-et thirteen quarter-tones 1.447269 378.643 640.000 +40 ÑÐ G¢ 5-et acute or large fifth 1.515717 396.550 720.000 +20 ÑÑ A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÑÒ A 15-et just major sixth 1.662476 434.946 880.000 ß20 ÑÓ AÚ 5-et augmented sixth 1.741101 455.517 960.000 ß40 ÑÔ B³ 15-et neutral seventh 1.823445 477.060 1040.000 ß60 ÑÕ B£ 15-et just diatonic major seventh 1.909683 499.622 1120.000 +20

ÐÑ C octave 2.000000 523.251 1200.000 +00

The Equally-Tempered Archetype 149

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 55. 15-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 84. 15-tone equal temperament tuning matrix no. 1 (Program 17)

NOTE C CÚ D£ EÝ E¢ F² FË G¢ A¸ A A! B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß20 +40 +20 +00 ß20 ß40 +20 +00 ß20 ß40 +20

Table 85. 15-tone equal temperament tuning matrix no. 2 (Program 18)

NOTE – – – – D³ – – – – – – FÍ – – – – – – B³ – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 ß40 +00 +00 +00 +40 +00 +00 +00 +40 +00

150 The Equally-Tempered Archetype

Sixteen-Tone Equal Temperament

Sixteen-tone equal temperament ( 16 2 ) presents a series of sixteen equal tones 75.000 cents in size, which represent the 16-et grave or small just chromatic semitone (approximately 47/45),222 and is what may be interpreted as being a scale made up of sixteen equal three-quarter-tone steps. The temperament may alternatively be expressed as the juxtaposition of two sets of eight-tone equally- tempered intervals, four sets of four-tone equally-tempered intervals, or eight sets of two-tone equally- tempered intervals interlocked a 16-et grave or small just chromatic semitone ( 16 2 ) apart. Blackwood makes the following observations about the temperament: “Triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. But keys can still be established by successions of altered subdominant and dominant harmonies.”223 The work outlines a sixteen-tone equally-tempered minor triad, within an added-note chord incorporating CÚ, and therefore essentially what could be interpreted as being a form of a Cmin(add"9) tetrad if enharmonically reinterpreted. The triad consists of the pitches C, E¸, and G², and presents a minor third 300.000 cents in size ( 4 2 , or the equal minor third) and a perfect fifth 675.000 cents in size ( [16 2]9 , 16-et grave or small fifth, or approximately 189/128).224 The comparison with the just minor third (6/5) and just perfect fifth (3/2) reveal a falsity of +4.359 and +18.045 cents on each count. G²4 (386.375Hz), or 16-et grave or small fifth ( [16 2]9 ), presents 12.126 beats between the third harmonic of C4 and the second harmonic of G²4 (772.751Hz), and 24.251 beats between the sixth harmonic of C4 and the fourth harmonic of G¢4 (1545.502Hz); while E¸4, or the equal minor third ( 4 2 ), 14.118 beats between the sixth harmonic of C4 and the fifth harmonic of E¸4.

Ex. 56. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 78

222 Daniélou, Tableau Comparatif des Intervalles Musicaux 20. 223 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 224 Daniélou, Tableau Comparatif des Intervalles Musicaux 129. The Equally-Tempered Archetype 151

Table 86. The beating characteristics of the sixteen-tone equally-tempered minor triad

C4 FREQUENCY E¸4 FREQUENCY BEATS G²4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 311.127 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 386.375 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 622.254 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 772.751 12.126 – – – – – – – – 3 933.381 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1159.126 – – – – – – – – – – – – 4 1244.508 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1555.635 14.118 4 1545.502 24.251 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1866.762 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1931.877 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The pélog scalar scheme now provides two options for the approximation of the augmented unison, being CÚ or CÍ, or the 16-et grave or small just chromatic semitone ( 16 2 ) and equal three-quarter-tone ( 8 2 ); two for the fourth, being Fí or F¢, or nine equal quarter-tones ( [8 ]2 3 ) and the 16-et acute or large fourth [16 ]2 7 , approximately 256/189, or 525.000 cents); two for the fifth, being G² or G£, or the 16-et grave or small fifth ( 16 ]2[ 9 ) and fifteen equal quarter-tones ( 8 ]2[ 5 ); as well as two options for the major seventh, being B³ or B£, or twenty-one equal quarter-tones ( [8 ]2 7 ) and the 16-et acute or large major seventh ( [16 ]2 15 , approximately 90/47, or 1125.000 cents). Sléndro accommodates the fourth and fifth. The microtonal nuances of these pitches are exploited both melodically and harmonically throughout the section of the work demarcated as ‘16-tone equal temperament’. Simultaneous sonorities include the dyads F¢ and G², or the equal three-quarter-tone ( 8 2); F¢ and G£, or 16-et acute or large tone ( 16 ]2[ 3 , approximately 41/36, or 225.000 cents); CÚ and F¢, or the equal minor third ( 4 2), CÍ and F¢, or the 16-et grave or small major third ( [16 ]2 3 , approximately 77/62, or 375.000 cents); D¢ and G², and CÚ and F¥, or nine equal quarter-tones ( [);8 ]2 3 D¢ and G£, and E¸ and B¹, or the 16-et acute or large fourth ( 16 ]2[ 7 ); CÚ and GÍ, or the equal tritone ( 2 2 ); C and G², and F¢ and C, or the 16- et grave or small fifth ( [16 ]2 9 ); C and G£, or fifteen equal quarter-tones ( [);8 ]2 5 E¸ and B£, C and GÍ, B¹ and F¥, and G² and E¸, or 16-et seventeen quarter-tones ( [16 ]2 11 , approximately 124/77, or 825.000 cents); as well as C and B¹, or the 16-et grave or small minor seventh ( 16 ]2[ 13 , approximately 152 The Equally-Tempered Archetype

72/41, or 975.000 cents).225 The following table depicts the tonal resources of sixteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 87. The sixteen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CÚ 16-et grave or small just chromatic semitone 1.044274 273.209 75.000 ß25 ÐÓ CÍ / Dí equal three-quarter-tone 1.090508 285.305 150.000 +50 ÐÔ D¢ 16-et acute or large tone 1.138789 297.936 225.000 +25 ÐÕ E¸ equal or Pythagorean minor third, or trihemitone 1.189207 311.127 300.000 +00 ÐÖ E² 16-et grave or small major third 1.241858 324.902 375.000 ß25 Ð× EË / Fí nine equal quarter-tones 1.296840 339.286 450.000 ß50 ÐØ F¢ 16-et acute or large fourth 1.354256 354.308 525.000 +25 ÐÙ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÑÐ G² 16-et grave or small fifth 1.476826 386.375 675.000 ß25 ÑÑ G£ fifteen equal quarter-tones, or superfifth 1.542211 403.482 750.000 +50 ÑÒ GÍ / Aí 16-et seventeen quarter-tones 1.610490 421.345 825.000 +25 ÑÓ A¢ equal or Pythagorean major sixth 1.681793 440.000 900.000 +00 ÑÔ B¹ 16-et grave or small minor seventh 1.756252 459.480 975.000 ß25 ÑÕ B³ twenty-one equal quarter-tones, or neutral seventh 1.834008 479.823 1050.000 ß50 ÑÖ B£ 16-et acute or large major seventh 1.915207 501.067 1125.000 +25

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 57. 16-tone equal temperament

225 Daniélou, Tableau Comparatif des Intervalles Musicaux 21-128. The Equally-Tempered Archetype 153

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 88. 16-tone equal temperament tuning matrix no. 1 (Program 19)

NOTE C CÚ D¢ E¸ E² F¢ F¥ G² GÍ A¢ B¹ B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß25 +25 +00 ß25 +25 +00 ß25 +25 +00 ß25 +25

Table 89. 16-tone equal temperament tuning matrix no. 2 (Program 20)

NOTE – – CÍ – – – – – – Fí – – G£ – – – – – – B³

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +50 +00 +00 +00 ß50 +00 +50 +00 +00 +00 ß50

Seventeen-Tone Equal Temperament

Seventeen-tone equal temperament is generated by the factor 17 2 , and with its seventeen equal tones 70.588 cents in size, which represent the 17-et grave or small just chromatic semitone, is able to approximate the important interval of just intonation termed as the grave or small just chromatic semitone (25/24) with extreme accuracy – a value expressing a falsity of ß0.084 cents.226 This is the ratio in just intonation “applied to any diatonic note to produce the sharpened form of that note.”227 Blackwood describes the temperament thus: “17-note triads are very discordant due to the large major third, so the fundamental consonant harmony of the tuning is a minor triad with an added minor seventh. The scale is very good due to the relatively small minor second.”228 Blackwood is of course referring to the major third 423.529 cents in size (E£), which may be referred to as the 17-et acute or large major third ( [17 2]6 , or approximately 327/256), and deviates from the just major third (5/4) by +37.216 cents. The present scheme relegates this third to the position of auxiliary tone, due to the better approximation of

226 Daniélou, Tableau Comparatif des Intervalles Musicaux 20. 227 John Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” Perspectives of New Music 29.2 (Summer, 1991): 109. 228 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 154 The Equally-Tempered Archetype the just major third (5/4) provided by E³ ([17 2]5 , or approximately 38/31), the 17-et neutral third, at 352.941 cents.229 The alternative theorization produces a dissonant triad nevertheless, yet seventeen- tone equal temperament is the second temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognizability”.230 The seventeen-tone equally-tempered major triad consists of the pitches C, E³, and G, and presents a major third 352.941 cents in size ( [17 2]5 , or 17-et neutral third) and a perfect fifth 705.882 cents in size ( [17 2]10 , or 17-et just perfect fifth). The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß33.373 and +3.927 cents on each count. G4 (393.330Hz), or 17-et just perfect fifth ( [17 2]10 ), presents 1.783 beats between the third harmonic of C4 and the second harmonic of G4 (786.659Hz), and 3.565 beats between the sixth harmonic of C4 and the fourth harmonic of G4 (1573.318Hz); while E³4 (307.972Hz), or the 17-et neutral third ( [17 2]5 ), 76.241 beats between the fifth harmonic of C4 and the fourth harmonic of E³4 (1231.189Hz).

Ex. 58. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 87

229 Daniélou, Tableau Comparatif des Intervalles Musicaux 90-106. 230 Blackwood, The Structure of Recognizable Diatonic Tunings 197. The Equally-Tempered Archetype 155

Table 90. The beating characteristics of the seventeen-tone equally-tempered major triad

C4 FREQUENCY E³4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 307.972 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 393.330 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 615.943 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 786.659 1.783 – – – – – – – – 3 923.915 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1179.989 – – – – 5 1308.128 4 1231.189 76.241 – – – – – – – – – – – – – – – – – – – – 5 1539.858 – – – – – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1573.318 3.565 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1847.830 – – – – 5 1966.648 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Blackwood’s consonant minor triad is located in this scheme within the pitches C, DÚ, and G, which when enharmonically reinterpreted as C, EÝ, and G produces a minor third 282.353 cents in size, and therefore a just minor third (6/5) with a falsity of ß33.288 cents. The exercise illustrates that the principal minor triad is no more consonant than the principal major triad in seventeen-tone equal temperament. Seventeen-tone equal temperament provides two options for the approximation of the augmented unison, being CÚ or CÍ, or the 17-et grave or small just chromatic semitone ( 17 2 ) and 17-et three-quarter-tone ( [17 2]2 , approximately 243/224, or 141.176 cents); two for the major third, being E³ or E£, or the 17-et neutral third ( [17 2]5 ) and 17-et acute or large major third ( [17 2]6 ); two for the tritone, being FÚ or GÝ, or the 17-et grave or small augmented fourth ( [17 2]8 , approximately 709/512, or 564.706 cents) and 17-et acute or large diminished fifth ( [17 2]9 , approximately 739/512, or 635.294 cents); two for the major sixth, being A³ or A£, or the 17-et neutral sixth ( [17 2]12 , approximately 835/512, or 847.059 cents, and 17-et acute or large major sixth ( [17 2]13 , approximately 435/256, or 917.647 cents); as well as two options for the major seventh, being B² or B£, or the 17-et grave or small major seventh ( [17 2]15 , approximately 448/243, or 1058.824 cents) and 17-et grave or small major seventh ( [17 2]16 , approximately 48/25, or 1129.412 cents). Simultaneous sonorities include the dyads A£ and C, or the 17-et augmented second ( [17 2]4 , approximately 512/435, or 282.353 cents); G and C, and B¸ and DÚ, or the 17-et just perfect fourth 156 The Equally-Tempered Archetype

( [17 2]7 , approximately 681/512, or 494.118 cents); A³ and C, and C and E³, or the 17-et neutral third ( [17 2]5 ); C and E£, or the 17-et acute or large major third ( [17 2]6 ); C and FÚ, F and B², FÚ and B£, GÝ and C, G and CÚ, and GÚ and CÍ, or the 17-et grave or small augmented fourth ( [17 2]8 ); C and GÝ, CÚ and G, and CÍ and GÚ, or the 17-et acute or large diminished fifth ( [17 2]9 ); C and G, and DÚ and B¸, or 17-et just perfect fifth ( [);17 2]10 DÚ and B£, and C and A³, or the 17-et neutral sixth ([17 2]12 ); C and A£, or the 17-et acute or large major sixth ( [17 2]13 ); CÚ and B², and C and B¸, or the 17-et Pythagorean minor seventh ( [17 2]14 , approximately 3584/2025, or 988.235); as well as C and B², or the 17-et grave or small major seventh ( [).17 2]15 231 The following table depicts the tonal resources of seventeen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 91. The seventeen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CÚ 17-et grave or small just chromatic semitone 1.041616 272.513 70.588 ß29 ÐÓ CÍ / Dí 17-et three-quarter-tone 1.084964 283.854 141.176 +41 ÐÔ D 17-et just major tone 1.130116 295.667 211.765 +12 ÐÕ DÚ 17-et augmented second 1.177147 307.972 282.353 ß18 ÐÖ E³ 17-et neutral third 1.226135 320.788 352.941 ß47 Ð× E£ 17-et acute or large major third 1.277162 334.138 423.529 +24 ÐØ F 17-et just and Pythagorean perfect fourth 1.330312 348.044 494.118 ß06 ÐÙ FÚ 17-et grave or small augmented fourth 1.385674 362.528 564.706 ß35 ÑÐ GÝ 17-et acute or large diminished fifth 1.443341 377.615 635.294 +35 ÑÑ G 17-et just and Pythagorean perfect fifth 1.503407 393.330 705.882 +06 ÑÒ GÚ 17-et augmented fifth 1.565972 409.698 776.471 ß24 ÑÓ A³ 17-et neutral sixth 1.631142 426.748 847.059 ß53 ÑÔ A£ 17-et acute or large major sixth 1.699024 444.508 917.647 +18 ÑÕ B¸ 17-et Pythagorean minor seventh 1.769730 463.007 988.235 ß12 ÑÖ B² 17-et grave or small major seventh 1.843379 482.275 1058.824 ß41 Ñ× B£ 17-et acute or large major seventh 1.920093 502.346 1129.412 +29

ÐÑ C octave 2.000000 523.251 1200.000 +00

231 Daniélou, Tableau Comparatif des Intervalles Musicaux 21-139. The Equally-Tempered Archetype 157

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 59. 17-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 92. 17-tone equal temperament tuning matrix no. 1 (Program 21)

NOTE C CÍ D D! E³ F F! G G! A£ B¸ B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +41 +12 ß18 ß47 ß06 ß35 +06 ß24 +18 ß12 ß41

Table 93. 17-tone equal temperament tuning matrix no. 2 (Program 22)

NOTE – – CÚ – – – – E£ – – GÝ – – A³ – – – – B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß29 +00 +00 +24 +00 +35 +00 +47 +00 +00 +29

Eighteen-Tone Equal Temperament

Eighteen-tone equal temperament ( 18 2 ) presents a subdivision of the equal major third ( 3 2 ) into six equal tones 66.667 cents in size, or equal third-tones, which represent the 18-et grave or small just chromatic semitone (approximately 133/128).232 The temperament may alternatively be expressed as the juxtaposition of two sets of nine-tone equally-tempered intervals, three sets of six-tone equally-tempered

232 Daniélou, Tableau Comparatif des Intervalles Musicaux 18. 158 The Equally-Tempered Archetype intervals, six sets of three-tone equally-tempered intervals, or nine sets of two-tone equally-tempered intervals interlocked an 18-et grave or small just chromatic semitone ( 18 2 ) apart. Blackwood makes the following observations about the temperament: “The perfect fifths are so out of tune that even seventh chords are disturbingly discordant.”233 Major, minor, and diminished triads produced via the major scale derived from this temperament presents major thirds with a range between 333.333 and 400.000 cents; minor thirds between 266.667 and 333.333 cents; perfect fifths between 666.667 and 733.333 cents; and a diminished fifth 600.000 cents in size. In comparison with the just perfect fifth (3/2), fifths within the specified range present a deviation of between ß31.378 and +31.378 cents, which is roughly an equal sixth-tone ( 36 2 ), and approaching two syntonic commas (6561/6400, or 43.013) – clearly outside of the boundaries set by Blackwood in his theory of “perfect fifths within the range of

4 3 234 recognizability” and its formula: 7 adiminished triad (B², D, and F²) produces the just minor third (6/5) and just tritone (45/32) with a falsity of +17.692 and +9.776 cents.

Ex. 60. The eighteen-tone equally-tempered major scale

The work outlines an implied eighteen-tone equally-tempered minor triad, within an arpeggiated added- note chord incorporating B¸, and therefore essentially a Cmin7 tetrad. The triad consists of the pitches C, Eí and G£, and presents a minor third 333.333 cents in size ( [18 2]5 , 18-et seven quarter-tones, or approximately 40/33) and a perfect fifth 733.333 cents in size ( [18 2]11 , 18-et superfifth, or approximately 189/128).235 Eí, although equal to 18-et seven quarter-tones ( [18 2]5 ), functions harmonically as a minor third. The comparison with the just minor third (6/5) and just perfect fifth (3/2) reveal a falsity of +17.692 and +31.378 cents on each count. G£4 (399.616Hz), or the 18-et superfifth ( [18 2]11 ), presents 14.355 beats between the third harmonic of C4 and the second harmonic of G£4

233 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 234 Blackwood, The Structure of Recognizable Diatonic Tunings 197. 235 Daniélou, Tableau Comparatif des Intervalles Musicaux 18-129. The Equally-Tempered Archetype 159

(799.232Hz), and 28.711 beats between the sixth harmonic of C4 and the fourth harmonic of G£4 (1598.464Hz); while Eí4 (317.175Hz), or 18-et seven quarter-tones ( [18 2]5 ), 16.124 beats between the sixth harmonic of C4 and the fifth harmonic of Eí4 (1585.877Hz). Bar 88 also serves to highlight the highly dissonant nature of the two fifths available in eighteen-tone equal temperament.

Ex. 61. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 88

Table 94. The beating characteristics of the eighteen-tone equally-tempered minor triad

C4 FREQUENCY Eí4 FREQUENCY BEATS G£4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 317.175 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 399.616 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 634.351 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 799.232 14.355 – – – – – – – – 3 951.526 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1198.848 – – – – – – – – – – – – 4 1268.702 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1585.877 16.124 4 1598.464 28.711 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1903.053 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1998.080 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Eighteen-tone equal temperament provides two options for the approximation of the augmented unison/minor second, being CÚ or DÝ, or the 18-et grave or small just chromatic semitone ( 18 2 ) and 9-et 160 The Equally-Tempered Archetype great limma ( 9 2 ); two for the minor third, being Eì or Eí, or 9-et five quarter tones ( [9 2]2 ) and 18-et seven quarter-tones ( [18 2]5 ); two for the fourth, being F² or F£, or the 18-et grave or small fourth ( [18 2]7 , approximately 512/391, or 466.667 cents) and 9-et superfourth ( [9 2]4 ); two for the fifth, being G³ or G£, or the 9-et subfifth ( [9 2]5 ) and 18-et superfifth ( [18 2]11 ); two for the major sixth, being A² or AË, or the 18-et grave or small major sixth ( [18 2]13 , approximately 33/20, or 866.667 cents) and 9-et nineteen quarter-tones ( [9 2]7 , approximately 12/7, or 933.333 cents); as well as two options for the major seventh, being B² or B£, or the 9-et grave or small major seventh ( [9 2]8 , approximately 1024/553, or 1066.667 cents) and 18-et acute or large major seventh ( [18 2]17 , approximately 256/133, or 1133.333 cents).236 Bars 89 and 90 illustrate the melodic utilization of the two alternative fifths, fourths and minor thirds, each deviating approximately an equal sixth-tone ( 36 2 ) in either direction of its associated justly intoned interval.

Ex. 62. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 89-90

The utilization of simultaneous sonorities in this temperament are somewhat limited due to the focus on melodic development, although includes the dyads C and Eì, or the 9-et five quarter tones ( [9 2]2 ), C and A², or the 18-et grave or small major sixth ( [18 2]13 ), and C and AË, or 9-et nineteen quarter-tones ( [18 2]14 ). The following table depicts the tonal resources of eighteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

236 Daniélou, Tableau Comparatif des Intervalles Musicaux 31-116. The Equally-Tempered Archetype 161

Table 95. The eighteen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CÚ 18-et grave or small just chromatic semitone 1.039259 271.897 66.667 ß33 ÐÓ DÝ 9-et great limma, or large half-tone 1.080060 282.571 133.333 +33 ÐÔ D equal or just major tone 1.122462 293.665 200.000 +00 ÐÕ DË / Eì 9-et five quarter-tones 1.166529 305.194 266.667 +67 ÐÖ DÍ / Eí 18-et seven quarter-tones 1.212326 317.175 333.333 +33 Ð× E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÐØ F² 18-et grave or small fourth 1.309385 342.568 466.667 ß33 ÐÙ F£ 9-et superfourth 1.360790 356.017 533.333 +33 ÑÐ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÑÑ G³ 9-et subfifth 1.469734 384.520 666.667 ß33 ÑÒ G£ 18-et superfifth 1.527435 399.616 733.333 +33 ÑÓ A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÑÔ A² 18-et grave or small major sixth 1.649721 431.609 866.667 ß33 ÑÕ AË / Bì 9-et nineteen quarter-tones 1.714488 448.554 933.333 +33 ÑÖ B¸ equal or Pythagorean minor seventh 1.781797 466.164 1000.000 +00 Ñ× B² 9-et grave or small major seventh 1.851749 484.465 1066.667 ß33 ÑØ B£ 18-et acute or large major seventh 1.924448 503.485 1133.333 +33

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 63. 18-tone equal temperament

162 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 96. 18-tone equal temperament tuning matrix no. 1 (Program 23)

NOTE C DÝ D DÍ E¢ F² F¥ G³ A¸ A² B¸ B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +33 +00 +33 +00 ß33 +00 ß33 +00 ß33 +00 ß33

Table 97. 18-tone equal temperament tuning matrix no. 2 (Program 24)

NOTE – – CÚ – – DË – – F£ – – G£ – – AË – – B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß33 +00 ß33 +00 +33 +00 +33 +00 +33 +00 +33

Nineteen-Tone Equal Temperament

Nineteen-tone equal temperament is generated by the factor 19 2 , and presents nineteen equal tones 63.158 cents in size, which represent the 19-et just diatonic semitone (approximately 531/512).237 It is a temperament advocated by American musicologist, organist, and conductor Joseph Yasser (1893- 1981), and theorized in his book of 1932 entitled Theory of Evolving Tonality. A. R. McClure offers the following discussion:

“Yasser’s speculations on the evolution of music led him to conclude that the diatonic scale of seven tones plus five accidentals would, in time, give place to a supra-diatonic scale of twelve tones plus seven accidentals. Then, on certain premises of his own and from assumptions of physical theorists, he developed, by most ingenious logic, a fanciful scheme with fresh notation and terminology, designed to serve the needs of composers in the future – the detailed plan of a hypothetical new order.”238

The temperament was theoretically perceived by Joseph Yasser as the historical expansion of a diatonic system into a chromatic one (with the addition of five auxiliary tones), and a chromatic one into a ‘supra- diatonic scale’ (with the further addition of seven auxiliary tones). The nineteen-tone scale is accommodated by Yasser with a notation system incorporating a ten-line staff and two additional symbols for accidentals. “On the theoretical side there are many points of interest, for the nineteen-tone

237 Daniélou, Tableau Comparatif des Intervalles Musicaux 18. 238 A. R. McClure, “Studies in Keyboard Temperaments,” The Galpin Society Journal 1 (Mar., 1948): 32-34 The Equally-Tempered Archetype 163 system allowed one to actually distinguish augmented intervals and their inversions, which on the normal pianoforte have no separate existence,” points out McClure.239 Joseph Yasser envisaged systems of temperament beyond the ‘supra-diatonic scale’, with the theorization of the ‘ultra-diatonics’ and systems with tonal structures such as 19+12=31, and 31+19=50. The complete evolutionary chain is expressed as the following combinations of scale degrees: 2+3=5, 5+2=7, 7+5=12, 12+7=19, 19+12=31, and 31+19=50.240 With regards to the aesthetic qualities of nineteen-tone equal temperament, Blackwood makes the following observations: “Triads are smooth, but the scale sounds slightly out of tune because the leading tone seems low with respects to the tonic. Diatonic behaviour is virtually identical to that of 12-note tuning, but chromatic behaviour is very different. For example, a perfect fourth is divisible into two equal parts, while an augmented sixth and a diminished seventh sound identical.”241 The pitch allocated as the leading-tone that Blackwood refers to is B² at 1073.684 cents ( [19 2]17 , 19-et grave or small major seventh, or approximately 119/64),242 which represents the just diatonic major seventh (15/8) with a falsity of ß14.585 cents. Blackwood also makes note that “nineteen-note equal tuning contains diatonic scales in which a major second spans three chromatic degrees, while a minor second spans two.”243 The nineteen-tone division of the octave (along with five, seven, twelve, thirty-one, forty-one, fifty- three, three-hundred and six, and three-hundred and forty-seven) are represented in the list of nine seminal propositions by theorists with regards to the cyclic division of the octave, or the Pythagorean cycle of fifths. Although, the fact that the nineteenth 3/2 produces a tone 137.145 cents (3ñù/2óð, or cyclic great limma) above the eleventh 2/1 in the nineteen cycle, reveals somewhat of an inferiority when compared to the twelve-tone cycle, where the twelfth 3/2 produces a tone 23.460 cents (3ñò/2ñù, or Pythagorean comma) above the seventh 2/1. Just minor thirds and major sixths (6/5s and 5/3s) are represented well in this temperament, with a falsity of +0.148 and ß0.148 cents on each count, while just major thirds and minor sixths (5/4s and 8/5s), represented adequately, with a falsity +7.366 and ß7.366 cents on each count. This is a significant improvement to the falsities of 15.641 cents for 6/5 and 5/3, and 13.686 cents for 5/4 and 8/5 expounded by twelve-tone equal temperament.244 Nineteen-tone equal temperament is the third

239 Read, 20th-Century Microtonal Notation 98. 240 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 431-32. 241 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 242 Daniélou, Tableau Comparatif des Intervalles Musicaux 37. 243 Easley Blackwood, “Modes and Chord Progressions in Equal Tunings,” Perspectives of New Music 29.2 (Summer, 1991): 168. 244 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 400-32. 164 The Equally-Tempered Archetype temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognisability.”245 The nineteen-tone equally-tempered major scale generates consistent major triads, which produce the just major third (5/4) and just perfect fifth (3/2) with a falsity of +7.366 and ß7.218 cents, as in C major (C, E, and G), F major (F, A, and C), and G major (G, B², and D²). Equally consistent minor triads are available, which produce the just minor third (6/5) and just perfect fifth with a falsity of +0.148 and ß7.218 cents, as in D minor (D, F, and A), E minor (E, G, and B²), and A minor (A, C, and E); while the B diminished triad (B², D², and F), produces the just minor third (6/5) and just tritone (45/32) with a falsity of +0.148 and +41.355 cents. “In sum, all diatonic progressions of triads and seventh chords have the same behaviour and produce the same musical effect in twelve-note and nineteen-note tuning,” comments Blackwood, “save for slight differences only, the most noticeable being the peculiar tuning of the nineteen-note major scale.” In striking contrast, “many nineteen-note chromatic progressions will bring about alien melodic intervals.”246

Ex. 64. The nineteen-tone equally-tempered major scale

The work outlines a nineteen-tone equally-tempered minor triad in first inversion, which includes the pitches C, EÝ, and G, and presents a minor third 315.789 cents in size ( [19 ]2 5 , 19-et just minor third, or approximately 6/5) and a perfect fifth 694.737 cents in size ( [19 ]2 11 , 19-et just perfect fifth, or approximately 115/77).247 The comparison with the just minor third (6/5) and perfect fifth (3/2) reveal a falsity of +0.148 and ß7.218 cents on each count. G4 (390.806Hz), or the 19-et just perfect fifth ( [19 ]2 11 ), presents 3.266 beats between the third harmonic of C4 and the second harmonic of G4 (781.611Hz), and 6.531 beats between the sixth harmonic of C4 and the fourth harmonic of G4 (1563.222Hz); while EÝ4 (313.978Hz), or the 19-et just minor third ( [19 ]2 5 ), 0.134 beats between the sixth harmonic of C4 and the fifth harmonic of EÝ4 (1569.888Hz).

245 Blackwood, The Structure of Recognizable Diatonic Tunings 197. 246 Blackwood, “Modes and Chord Progressions in Equal Tunings,” Perspectives of New Music 169-72. 247 Daniélou, Tableau Comparatif des Intervalles Musicaux 82-125. The Equally-Tempered Archetype 165

Ex. 65. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 97

Table 98. The beating characteristics of the nineteen-tone equally-tempered minor triad

C4 FREQUENCY EÝ4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 313.978 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 390.806 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 627.955 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 781.611 3.266 – – – – – – – – 3 941.933 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1172.417 – – – – – – – – – – – – 4 1255.910 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1569.888 0.134 4 1563.222 6.531 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1883.865 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1954.028 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Nineteen-tone equal temperament provides two options for the approximation of the minor second, being D¸ or DÝ, or the 19-et just diatonic semitone ( 19 2 ) and 19-et great limma [19 2]2 , approximately 128/119, or 126.316 cents); two for the major second, being D² or DË, or the 19-et just minor tone ( [19 2]3 , approximately 512/459, or 189.474 cents) and 19-et five quarter-tones ( [19 2]4 , approximately 81/70, or 252.632 cents); two for the major third, being E or EË, or the 19-et just major third ( [19 2]6 , approximately 61/49, or 378.947 cents) and 19-et nine quarter-tones ( [19 2]7 , approximately 661/512, or 442.105 cents); two for the tritone, being FÚ or GÝ, or the 19-et grave or small augmented fourth ( [19 2]9 , approximately 711/512, or 568.421 cents) and 19-et acute or large diminished fifth ( [19 2]10 , 166 The Equally-Tempered Archetype approximately 1024/711, or 631.579 cents); two for the minor sixth, being A¸ or Aí, or the 19-et Pythagorean minor sixth ( [19 2]12 , approximately 793/512, or 757.895 cents) and 19-et seventeen quarter-tones ( [19 2]13 , approximately 98/61, or 821.053 cents); two for the major sixth, being A or AË, or the 19-et just major sixth ( [19 2]14 , approximately 5/3, or 884.211 cents) and 19-et nineteen quarter- tones ( [19 2]15 , approximately 140/81, or 947.368 cents); as well as two options for the major seventh, being B² or BË, or the 19-et grave or small major seventh ( [19 2]17 ) and 19-et twenty-three quarter-tones ( [19 2]18 , approximately 27/14, or 1136.842 cents).248 Simultaneous sonorities include the dyads A and C, or the 19-et just minor third ( [);19 2]5 DÚ and G, EÚ and A, Aí and D¸, or 19-et nine quarter-tones ( [19 2]6 ); C and F, D² and G, EÝ and Aí, E and A, F and BÝ, G and C, or the 19-et just perfect fourth ( [19 2]8 , approximately 154/115, or 505.263 cents); Aí and D², BÝ and E, AÚ and EÝ, C and FÚ, DÝ and G, or the 19-et grave or small augmented fourth ( [);19 2]9 A¸ and D², A and EÝ, C and GÝ, or the 19-et acute or large diminished fifth ( [19 2]10 ); C and G, or the 19- et just perfect fifth ( [19 2]11); and C and Aí, or 19-et seventeen quarter-tones ( [19 2]13 ). Bar 94 and 95 highlights intonational reinterpretations of simultaneous sonorities based on the two options for the major second (D² and G, and DÚ and G); two for the major third (E and A, and EÚ and A); two for the minor sixth (A¸ and D², and Aí and D²); two for the major sixth (A and EÝ, and AÚ and EÝ); and the two options for the tritone (C and FÚ, and C and GÝ).249

Ex. 66. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 94-95

The following table depicts the tonal resources of nineteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

248 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-139. 249 The juxtaposition of three just major thirds (5/4), equal to 1158.941 cents is a figure that diminishes the octave (2/1) by 41.059 cents, and the interval termed as the ‘great diesis’ with a frequency ratio of 128/125. For a further discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 453. The Equally-Tempered Archetype 167

Table 99. The nineteen-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ D¸ 19-et just diatonic semitone, or major half-tone 1.037155 271.346 63.158 ß37 ÐÓ DÝ 19-et great limma, or large half-tone 1.075691 281.428 126.316 +26 ÐÔ D² 19-et just minor tone 1.115658 291.885 189.474 ß11 ÐÕ DË / Eì 19-et five quarter-tones 1.157110 302.730 252.632 +53 ÐÖ EÝ 19-et just minor third 1.200103 313.978 315.789 +16 Ð× E 19-et just major third 1.244693 325.643 378.947 ß21 ÐØ EË / Fí 19-et nine quarter-tones 1.290939 337.743 442.105 +42 ÐÙ F 19-et just and Pythagorean perfect fourth 1.338904 350.292 505.263 +05 ÑÐ FÚ 19-et grave or small augmented fourth 1.388651 363.307 568.421 ß32 ÑÑ GÝ 19-et acute or large diminished fifth 1.440247 376.805 631.579 +32 ÑÒ G 19-et just and Pythagorean perfect fifth 1.493759 390.806 694.737 ß05 ÑÓ A¸ 19-et Pythagorean minor sixth 1.549260 405.326 757.895 ß42 ÑÔ GÍ / Aí 19-et seventeen quarter-tones 1.606822 420.386 821.053 ß79 ÑÕ A 19-et just major sixth 1.666524 436.005 884.211 ß16 ÑÖ AË / Bì 19-et nineteen quarter-tones 1.728444 452.205 947.368 +47 Ñ× BÝ 19-et acute or large minor seventh 1.792664 469.007 1010.526 +11 ÑØ B² 19-et grave or small major seventh 1.859271 486.433 1073.684 ß26 ÑÙ BË / Cí 19-et twenty-three quarter-tones 1.928352 504.506 1136.842 +37

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 67. 19-tone equal temperament

168 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 100. 19-tone equal temperament tuning matrix no. 1 (Program 25)

NOTE C DÝ D² EÝ E F F! G Aí A BÝ B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +26 ß11 +16 ß21 +05 ß32 ß05 +21 ß16 +11 ß26

Table 101. 19-tone equal temperament tuning matrix no. 2 (Program 26)

NOTE – – D¸ – – DË EË – – GÝ – – A¸ AË – – BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß37 +00 ß47 +42 +00 +32 +00 ß42 +47 +00 +37

Twenty-Tone Equal Temperament

Twenty-tone equal temperament ( 20 2 ) presents a subdivision of the 5-et supermajor second ( 5 2 ) into four equal tones 60.000 cents in size, which represent the 20-et Greek enharmonic quarter-tone (approximately 265/256).250 The temperament may alternatively be expressed as the juxtaposition of two sets of ten-tone equally-tempered intervals, four sets of five-tone equally-tempered intervals, five sets of four-tone equally-tempered intervals, or ten sets of two-tone equally-tempered intervals interlocked a 20-et Greek enharmonic quarter-tone ( 20 2 ) apart. Blackwood makes the following statement with regards to the temperament: “Triads are very bad, and the most consonant harmony this tuning offers is a minor triad with an added major sixth, along with its inversions.”251 The minor triad with an added major sixth that Blackwood refers to includes the pitches C, E¸, G¢, and A¢, and is made up of the equal major third ( 3 2 ), the 5-et acute or large fifth ( [5 ]2 3 ), and the equal major sixth ( [4 ]2 3 ). All these pitches are also available in four-tone, eight-tone, twelve-tone, sixteen-tone, and twenty-four-tone equal temperaments, but for G¢ ([5 ]2 3 , or 5-et acute or large fifth), which is only available in five-tone, ten- tone, fifteen-tone, and twenty-tone equal temperaments.

250 Daniélou, Tableau Comparatif des Intervalles Musicaux 16. 251 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. The Equally-Tempered Archetype 169

Twenty-tone equal temperament provides two options for the approximation of the minor second, being Dì or D¸, or the 20-et Greek enharmonic quarter-tone ( 20 2 ) and 10-et just diatonic semitone ( 10 2 ); two for the major second, being D² or D£, or the 20-et just minor tone ( [20 2]3 , approximately 81/73, or 180.000 cents) and 5-et supermajor second ( 5 2 ); two for the major third, being E² or E£, or the 10-et grave or small major third ( [10 2]3 ) and 20-et acute or large major third ( [20 2]7 , approximately 25088/19683, or 420.000 cents); two for the fifth, being G³ or G¢, or the 20-et subfifth ( [20 2]11, approximately 41/28, or 660.000 cents) and 5-et acute or large fifth ( [5 2]3 ); two for the major sixth, being A³ or A¢, or the 10-et neutral sixth ( [10 2]7 , or approximately 13/8) and equal major sixth ( [);4 2]3 two for the augmented sixth/minor seventh, being AÚ or BÝ, or the 5-et augmented sixth ( [5 2]4 ) and 20- et acute or large minor seventh ( [20 2]17 , approximately 146/81, or 1020.000 cents); as well as two options for the major seventh, being B or BË, or the 10-et just diatonic major seventh ( [,10 2]9 approximately 28/15, or 1080.000 cents) and 20-et twenty-three quarter-tones ( [20 2]19 , approximately 989/512, or 1140.000 cents).252 Simultaneous sonorities include the dyads C and E¸, or the equal minor third ( 4 2); E¸ and G³, or the 10-et grave or small major third ( [);10 2]3 E¸ and G¢, and C and E£, or the 20-et acute or large major third ( [20 2]7 ); as well as C and A³, or the 10-et neutral sixth ( [10 2]7 ). Bar 100 illustrates the utilization of intonational reinterpretations of simultaneous sonorities based on the two options for the fifth, G³ and G¢, or the 20-et subfifth ( [20 2]11 ) and 5-et acute or large fifth ( [).5 2]3 In this case, the juxtaposition with E¸, or the equal minor third ( 4 2 ) generates the 20-et acute or large major third ( [20 2]7 ) and 10-et grave or small major third ( [10 2]3 ), which in each case present the just major third (5/4) with a falsity of +33.686 and ß26.314 cents.

Ex. 68. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 100

The following table depicts the tonal resources of twenty-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

252 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-133. 170 The Equally-Tempered Archetype

Table 102. The twenty-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CË / Dì 20-et Greek enharmonic or septimal quarter-tone 1.035265 270.852 60.000 ß40 ÐÓ D¸ 10-et just diatonic semitone, or major half-tone 1.071773 280.403 120.000 +20 ÐÔ D² 20-et just minor tone 1.109569 290.292 180.000 ß20 ÐÕ D£ 5-et supermajor second 1.148698 300.529 240.000 +40 ÐÖ E¸ equal or Pythagorean minor third, or trihemitone 1.189207 311.127 300.000 +00 Ð× E² 10-et grave or small major third 1.231144 322.099 360.000 ß40 ÐØ E£ 20-et acute or large major third 1.274561 333.458 420.000 +20 ÐÙ F² 5-et grave or small fourth 1.319508 345.217 480.000 ß20 ÑÐ F£ 20-et superfourth 1.366040 357.391 540.000 +40 ÑÑ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÑÒ G³ 20-et subfifth 1.464086 383.042 660.000 ß40 ÑÓ G¢ 5-et acute or large fifth 1.515717 396.550 720.000 +20 ÑÔ GÚ 20-et augmented fifth 1.569168 410.535 780.000 ß20 ÑÕ A³ 10-et neutral sixth 1.624505 425.012 840.000 ß60 ÑÖ A¢ equal or Pythagorean major sixth 1.681793 440.000 900.000 +00 Ñ× AÚ 5-et augmented sixth 1.741101 455.517 960.000 ß40 ÑØ BÝ 20-et acute or large minor seventh 1.802501 471.580 1020.000 +20 ÑÙ B 10-et just diatonic major seventh 1.866066 488.211 1080.000 ß20 ÒÐ BË / Cí 20-et twenty-three quarter-tones 1.931873 505.427 1140.000 +40

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 69. 20-tone equal temperament

The Equally-Tempered Archetype 171

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 103. 20-tone equal temperament tuning matrix no. 1 (Program 27)

NOTE C D¸ D² E¸ E² F² F¥ G¢ G! A£ BÝ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +20 ß20 +00 ß40 ß20 +00 +20 ß20 +00 +20 ß20

Table 104. 20-tone equal temperament tuning matrix no. 2 (Program 28)

NOTE – – Dì D£ – – E£ F£ – – G³ A³ – – A! BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß40 +40 +00 +20 +40 +00 ß40 +40 +00 ß40 +40

Twenty-One-Tone Equal Temperament

Twenty-one-tone equal temperament ( 21 2 ), with its twofold tripartite division of the octave, presents a subdivision of the 7-et grave or small tone ( 7 2 ) into three equal tones 57.143 cents in size, which represent the 21-et Greek enharmonic quarter-tone (approximately 1323/1280).253 The temperament may alternatively be expressed as the juxtaposition of three sets of seven-tone equally-tempered intervals interlocked an equal 21-et Greek enharmonic quarter-tone ( 21 2 ) apart. According to Blackwood, “Major and minor triads and keys are relatively consonant here. Scales sound somewhat out of tune, however, due to the impossibility of dividing a major third into two equal parts.”254 The twenty-one-tone equally-tempered major scale generates inconsistent major triads, which produce the just major third (5/4) and just perfect fifth (3/2) with a falsity of +13.686 cents and ß16.241 cents, as in C major (C, E¢, and G²); ß43.457 and ß16.241 cents, as in F major (F¢, A², and C); and +13.686 and +40.902 cents, as in G major (G², B, and D¢). Equally inconsistent minor triads are available, deviating from the just minor third (6/5) and just perfect fifth (3/2) by ß29.927 and ß73.384 cents, as in D minor (D, F¢, and A²); ß29.927 and ß16.241 cents, as in E minor (E¢, G², and B); and +27.216 and +40.902 cents, as in A

253 Daniélou, Tableau Comparatif des Intervalles Musicaux 16. 254 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 172 The Equally-Tempered Archetype minor (A², C, and E¢); while the B diminished triad (B, D¢, and F¢), deviating from the just minor third (6/5) and just tritone (45/32) by +27.216 and +38.348 cents. The D minor triad produces a just perfect fifth (3/2) with a falsity of ß73.384 cents, which is an extremely dissonant interval 628.571 cents in size ( [21 2]11, 21-et acute or large diminished fifth, or approximately 23/16), and therefore certainly no form of recognizable perfect fifth.255

Ex. 70. The twenty-one-tone equally-tempered major scale

Twenty-one-tone equal temperament provides two options for the approximation of the minor second, being Dì or D¸, or the 21-et Greek enharmonic quarter-tone ( 21 2 ) and 21-et just diatonic semitone ( [21 2]2 , approximately 47/44, or 114.286 cents); two for the major second, being D³ or D¢, or the 7-et grave or small tone ( 7 2 ) and 21-et acute or large tone ( [21 2]4 , approximately 81/71, or 228.571 cents); two for the major third, being E³ or E¢, or the 7-et neutral third ( [7 2]2 ) and equal major third ( 3 2 ); two for the fourth, being F³ or F¢, or the 21-et subfourth ( [21 2]8 , approximately 125/96, or 457.143 cents) and 7-et acute or large fourth ( [7 2]3 ); two for the tritone, being FÚ or GÝ, or the 21-et grave or small augmented fourth ( [21 2]10 , approximately 32/23, or 571.429 cents) and 21-et acute or large diminished fifth ( [21 2]11); two for the fifth, being G² or G£, or the 7-et grave or small fifth ( [7 2]4 ) and 21-et superfifth ( [21 2]13 , approximately 43/28, or 742.857 cents); two for the major sixth, being A² or A¢, or the 7-et grave or small major sixth ( [7 2]5 ) and 21-et Pythagorean major sixth ( [,21 2]16 approximately 27783/16384, or 914.286 cents); two for the minor seventh, being B¹ or Bí, or the 21-et grave or small minor seventh ( [21 2]17 , approximately 142/81, or 971.429 cents) and 7-et twenty-one quarter-tones ( [7 2]6 ); as well as two options for the major seventh, being B or BË, or the 21-et just diatonic major seventh ( [21 2]19 , approximately 88/47, or 1085.714 cents) and 21-et twenty-three quarter-tones ( [21 2]20 , approximately 2560/1323, or 1142.857 cents).256 Bar 107 highlights a descending melodic passage, which incorporates all the primary, secondary, and auxiliary pitches available within the pélog scheme – the series C, (BË), B, Bí, (B¹), A¸, (G£), G², (GÝ), FÚ, F¢, (F³), E¸, D¸, (Dì), and C.

255 Daniélou, Tableau Comparatif des Intervalles Musicaux 141. 256 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-140. The Equally-Tempered Archetype 173

Ex. 71. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 107

Simultaneous sonorities utilized in this temperament are limited to C and G², or the 7-et grave or small fifth ( [),7 ]2 4 and F¢ and Bí, and F³ and B¹, or the 7-et acute or large fourth ( [7 ]2 3 ). The following table depicts the tonal resources of twenty-one-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 105. The twenty-one-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CË / Dì 21-et Greek enharmonic or septimal quarter-tone 1.033558 270.405 57.143 ß43 ÐÓ D¸ 21-et just diatonic semitone, or major half-tone 1.068242 279.479 114.286 +14 ÐÔ D³ 7-et grave or small tone 1.104090 288.858 171.429 ß29 ÐÕ D¢ 21-et acute or large tone 1.141140 298.551 228.571 +29 ÐÖ EÝ 21-et just minor third 1.179434 308.570 285.714 ß14 Ð× E³ 7-et neutral third 1.219014 318.925 342.857 +43 ÐØ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÐÙ F³ 21-et subfourth 1.302201 340.689 457.143 ß43 ÑÐ F¢ 7-et acute or large fourth 1.345900 352.122 514.286 +14 ÑÑ FÚ 21-et grave or small augmented fourth 1.391066 363.938 571.429 ß29 ÑÒ GÝ 21-et acute or large diminished fifth 1.437747 376.151 628.571 +29 ÑÓ G² 7-et grave or small fifth 1.485994 388.774 685.714 ß14 ÑÔ G£ 21-et superfifth 1.535861 401.820 742.857 +43 ÑÕ A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÑÖ A² 7-et grave or small major sixth 1.640671 429.241 857.143 ß43 Ñ× A¢ 21-et Pythagorean major sixth 1.695728 443.646 914.286 +14 ÑØ B¹ 21-et grave or small minor seventh 1.752633 458.534 971.429 ß29 ÑÙ AÍ / Bí 7-et twenty-one quarter-tones 1.811447 473.921 1028.571 ß71 ÒÐ B 21-et just diatonic major seventh 1.872235 489.825 1085.714 ß14 ÒÑ BË / Cí 21-et twenty-three quarter-tones 1.935064 506.262 1142.857 +43

ÐÑ C octave 2.000000 523.251 1200.000 +00 174 The Equally-Tempered Archetype

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 72. 21-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 106. 21-tone equal temperament tuning matrix no. 1 (Program 29)

NOTE C D¸ D¢ E¸ E¢ F¢ F! G² A¸ A² Bí B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +14 +29 ß14 +00 +14 ß29 ß14 +00 ß43 +29 ß14

Table 107. 21-tone equal temperament tuning matrix no. 2 (Program 30)

NOTE – – Dì D³ E³ – – F³ GÝ G£ – – A¢ B¹ BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß43 ß29 +43 +00 ß43 +29 +43 +00 +14 ß29 +43

Twenty-Two-Tone Equal Temperament

Twenty-two-tone equal temperament ( 22 2 ) presents a subdivision of the 11-et just diatonic semitone ( 11 2 ) into two equal tones 54.545 cents in size, which represent the 22-et Greek enharmonic quarter- tone (approximately 4096/3969).257 The temperament may alternatively be expressed as the juxtaposition of two sets of eleven-tone equally-tempered intervals, or eleven sets of two-tone equally-

257 Daniélou, Tableau Comparatif des Intervalles Musicaux 14. The Equally-Tempered Archetype 175 tempered intervals interlocked a 22-et Greek enharmonic quarter-tone ( 22 2 ) apart. Blackwood makes the following observations: “This tuning contains triads that are very smooth – in some respects, even a bit smoother than those of 12-note tuning. The smoothness of the triads is offset, however, by the out- of-tune scale which sharply restricts the tuning’s diatonic vocabulary.”258 Twenty-two-tone equal temperament is the fourth temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognisability.”259 The twenty-two-tone equally-tempered major scale generates consistent major triads, which produce the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß4.496 cents and +7.136 cents, as in C major (C, E, and G), F major (F, A², and C), and G major (G, B, and D¢). A very dissonant minor triad is available, which presents the just minor third (6/5) and perfect fifth (3/2) with a falsity of by ß42.914 and ß47.410 cents, as in D minor (D, F, and A²); while another two, display more consistency, as well as consonance, with a deviation of +11.631 and +7.136 cents, as in E minor (E, G, and B) and A minor (A², C, and E). The B diminished triad (B, D¢, and F) presents the just minor third (6/5) and just tritone (45/32) with a falsity of +11.631 and ß9.776 cents. The out-of-tune scale Blackwood is referring to is made up of tones and semitones deviating from just major tones (9/8s), just minor tones (10/9s), and just diatonic semitones (16/15s) by +14.272, ß18.767, and ß2.640 cents. Twelve-tone equal temperament yields ß3.910, +17.596, and ß11.731 cents on the same basis.

Ex. 73. The twenty-two-tone equally-tempered major scale

The work outlines the triad C, DÚ, and G, which when enharmonically reinterpreted as C, E¹, and G may be stated as being a twenty-two-tone equally-tempered minor triad with the minor third and perfect fifth 272.727 and 709.091 cents in size respectively (the interval of the augmented second converted into a subminor third). The comparison with the just minor third (6/5) and just perfect fifth (3/2) reveal a falsity of ß42.914 and +7.136 cents on each count. G4 (394.059Hz), or the 22-et just perfect fifth ( [,22 ]2 13 or approximately 122/81), presents 3.242 beats between the third harmonic of C4 and the second harmonic of G4 (788.119Hz), and 6.484 beats between the sixth harmonic of C4 and the fourth harmonic of G4 (1576.237Hz); while E¹4 (306.264Hz), or 22-et subminor third ( [22 ]2 5 , or approximately 2560/2187), 38.433 beats between the sixth harmonic of C4 and the fifth harmonic of E¹4 (1531.320Hz).260

258 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 259 Blackwood, The Structure of Recognizable Diatonic Tunings 197. 260 Daniélou, Tableau Comparatif des Intervalles Musicaux 72-123. 176 The Equally-Tempered Archetype

Ex. 74. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 109

Table 108. The beating characteristics of the nineteen-tone equally-tempered minor triad

C4 FREQUENCY E¹4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 306.264 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 394.059 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 612.528 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 788.119 3.242 – – – – – – – – 3 918.792 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1182.178 – – – – – – – – – – – – 4 1225.056 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1531.320 38.433 4 1576.237 6.484 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1837.585 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1970.296 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Twenty-two-tone equal temperament provides two options for the approximation of the major second, being D³ or D¢, or the 22-et grave or small tone ( [22 2]3 , approximately 256/233, or 163.636 cents) and 11-et acute or large tone ( [11 2]2 ); two for the augmented second, being DÚ or DÍ, or the 22-et augmented second ( [22 2]5 ) and 11-et seven quarter-tones ( [11 2]3 ); two for the major third, being E or EË, or the 22-et just major third ( [22 2]7 , approximately 96/77, or 381.818 cents) and 11-et nine quarter- tones ( [11 2]4 ); two for the fourth, being F or F£, or the 22-et just perfect fourth ( [22 2]9 , approximately 81/61, or 490.909 cents) and 11-et superfourth ( [11 2]5 , approximately 37/27, or 545.455 cents); two for the fifth, being G³ or G, or the 11-et subfifth ( [11 2]6 ) and 22-et just perfect fifth ( [22 2]13 ); two for the minor sixth, being Aì or AÝ, or 11-et fifteen quarter-tones ( [11 2]7 , approximately 199/128, or 763.636 cents) and the 22-et just minor sixth ( [22 2]15 , approximately 77/48, or 818.182 cents); two for The Equally-Tempered Archetype 177 the major sixth, being A² or A£, or the 11-et grave or small major sixth ( [11 2]8 , approximately 48/29, or 872.727 cents) and 22-et acute or large major sixth ( [22 2]17 , approximately 2187/1280, or 927.273 cents); two for the minor seventh, being B¹ or Bí, or the 11-et grave or small minor seventh ( [11 2]9 , approximately 432/245, or 981.818 cents) and 22-et twenty-one quarter-tones ( [22 2]19 , approximately 233/128, or 1036.364 cents); as well as two options for the major seventh, being B or BË, or the 11-et just diatonic major seventh ( [11 2]10 , approximately 77/41, or 1090.909 cents) and 22-et twenty-three quarter-tones [22 2]21, approximately 3969/2048, or 1145.455 cents).261 A microtonal cluster in bar 108 made up of the pitches D¸, Eí, F', F¥, G, AÝ, Bí, and B' highlights four significant twenty-two-tone equally-tempered intervals, measuring 54.545, 109.091, 163.636, and 218.182 cents.

Ex. 75. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 108

The only simultaneous sonority utilized in this temperament is Bí and F, or the 11-et subfifth ( [11 2]6 ). The following table depicts the tonal resources of twenty-two-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

261 Daniélou, Tableau Comparatif des Intervalles Musicaux 31-134. 178 The Equally-Tempered Archetype

Table 109. The twenty-two-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CË / Dì 22-et Greek enharmonic or septimal quarter-tone 1.032008 270.000 54.545 +55 ÐÓ D¸ 11-et just diatonic semitone, or major half-tone 1.065041 278.642 109.091 +09 ÐÔ D³ 22-et grave or small tone 1.099131 287.561 163.636 ß36 ÐÕ D¢ 11-et acute or large tone 1.134313 296.765 218.182 +18 ÐÖ DÚ 22-et augmented second 1.170620 306.264 272.727 ß27 Ð× DÍ / Eí 11-et seven quarter-tones 1.208089 316.067 327.273 +27 ÐØ E 22-et just major third 1.246758 326.184 381.818 ß18 ÐÙ EË / Fí 11-et nine quarter-tones 1.286665 336.624 436.364 +36 ÑÐ F 22-et just and Pythagorean perfect fourth 1.327849 347.399 490.909 ß09 ÑÑ F£ 11-et superfourth 1.370351 358.519 545.455 +45 ÑÒ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÑÓ G³ 11-et subfifth 1.459480 381.837 654.545 ß45 ÑÔ G 22-et just and Pythagorean perfect fifth 1.506196 394.059 709.091 +09 ÑÕ GË / Aì 11-et fifteen quarter-tones 1.554406 406.672 763.636 ß36 ÑÖ AÝ 22-et just minor sixth 1.604160 419.689 818.182 +18 Ñ× A² 11-et grave or small major sixth 1.655507 433.123 872.727 ß27 ÑØ A£ 22-et acute or large major sixth 1.708496 446.986 927.273 +27 ÑÙ B¹ 11-et grave or small minor seventh 1.763183 461.294 981.818 ß18 ÒÐ AÍ / Bí 22-et twenty-one quarter-tones 1.819619 476.059 1036.364 ß64 ÒÑ B 11-et just diatonic major seventh 1.877862 491.297 1090.909 ß09 ÒÒ BË / Cí 22-et twenty-three quarter-tones 1.937969 507.022 1145.455 +45

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 76. 22-tone equal temperament

The Equally-Tempered Archetype 179

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 110. 22-tone equal temperament tuning matrix no. 1 (Program 31)

NOTE C D¸ D¢ DÍ E F F¥ G AÝ A² Bí B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +09 +18 +27 ß18 ß09 +00 +09 +18 ß27 +36 ß09

Table 111. 22-tone equal temperament tuning matrix no. 2 (Program 32)

NOTE – – CË D³ D! EË F£ – – G³ Aì A£ B¹ BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß45 ß36 ß27 +36 +45 +00 ß45 ß36 +27 ß18 +45

Twenty-Three-Tone Equal Temperament

Twenty-three-tone equal temperament is generated by the 23 2 , and presents twenty-three equal tones 52.174 cents in size, which represent the 23-et Greek enharmonic quarter-tone (approximately 34/33), and approximate the equal quarter-tone ( 24 2 ) with a falsity of +2.174 cents.262 According to Blackwood, twenty-three-tone equal temperament presents the following characteristics: “A particular challenge, 23-note tuning contains no diatonic configurations and no chromatic structures in common with any of the other tunings of this study. However, it does contain an intriguing arrangement of the two distinct pentatonic modes of Java and Bali, known as sléndro and pélog – modes that cannot be realistically approximated in 12-note tuning.”263 The twenty-three-tone equally-tempered major scale generates inconsistent major triads, which produce the just major third (5/4) and just perfect fifth (3/2)

262 Daniélou, Tableau Comparatif des Intervalles Musicaux 14. 263 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 180 The Equally-Tempered Archetype with a falsity of ß21.096 cents and ß23.694 cents, as in C major (C, E², and G²) and F major (F¢, A, and C); and with +31.078 cents and +28.480 cents, as in G major (G, B, and D). Somewhat more consonant minor triads are available, which produce the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß2.598 and ß23.694 cents, as in D minor (D, F¢, and A) and A minor (A, C, and E²); and with ß2.598 and +28.480 cents, as in E minor (E, G², and B); while the B diminished triad (B, D, and F¢) produces the just minor third (6/5) and just tritone (45/32) with a falsity of ß2.598 and +35.863 cents.

Ex. 77. The twenty-three-tone equally-tempered major scale

The sléndro scale is represented in twenty-three-tone equal temperament with the pitches C, D, F¢, G², and B¸, and equal to 0.000, 208.696, 521.739, 678.261, and 991.304 cents; while the pélog scale, with the pitches C, D¸, EÝ, F¢, G², A¸, and B, and equal to 0.000, 104.348, 313.043, 521.739, 678.261, 782.609, and 1095.652 cents. A comparison with the tuning of the gamelan gedhé analyzed by Perlman (presenting sléndro measurements equal to 0.000, 238.000, 475.000, 727.000, and 950.000 cents; and pélog measurements equal to 0.000, 116.000, 281.000, 550.000, 669.000, 769.000, and 861.000 cents) reveals a falsity of between 9.261 and 32.043 cents on the first count (not accounting for the +134.652 cent falsity of B), and a falsity between 29.304 and 48.739 cents on the second count.264 A further comparison of the two scales with five-tone and seven-tone equal temperaments produces an even greater level of error. The work outlines a twenty-three-tone equally-tempered minor triad, which includes the pitches C, EÝ, and G², and produces a minor third 313.043 cents in size ( [23 2]6 , 23-et just minor third, or approximately 1024/855) and a perfect fifth 678.261 cents in size ([23 2]13 , 23-et grave or small fifth, or approximately 262144/177147).265 The comparison with the just minor third (6/5) and perfect fifth (3/2) reveal a falsity of ß2.598 and ß23.694 cents on each count. G²4 (387.104Hz), or the 23-et grave or small fifth ( [23 2]13 ), presents 10.669 beats between the third harmonic of C4 and the second harmonic of G²4 (774.208Hz), and 21.338 beats between the sixth harmonic of C4 and the fourth harmonic of G²4 (1548.416Hz); while EÝ4 (313.480Hz), or the 23-et just minor third ( [23 2]6 ), 2.354 beats between the sixth harmonic of C4 and the fifth harmonic of EÝ4 (1567.400Hz).

264 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41. 265 Daniélou, Tableau Comparatif des Intervalles Musicaux 82-129. The Equally-Tempered Archetype 181

Ex. 78. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 112

Table 112. The beating characteristics of the nineteen-tone equally-tempered minor triad

C4 FREQUENCY EÝ4 FREQUENCY BEATS G²4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 313.480 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 387.104 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 626.960 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 774.208 10.669 – – – – – – – – 3 940.440 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1161.312 – – – – – – – – – – – – 4 1253.920 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1567.400 2.354 4 1548.416 21.338 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1880.880 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1935.520 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Twenty-three-tone equal temperament provides two options for the approximation of the major second, being D³ or D, or the 23-et grave or small tone ( [23 2]3 , approximately 81/74, or 156.522 cents) and 23- et just major tone ( [23 2]4 , approximately 44/39, or 208.696 cents); two for the augmented second/minor third, being DË or EÝ, or 23-et five quarter-tones ( [23 2]5 , approximately 93/80, or 260.870 cents) and the 23-et just minor third ( [23 2]6 ); two for the major third, being E² or E¢, or the 23-et grave or small major third ( [23 2]7 , approximately 2401/1944, or 365.217 cents) and 23-et Pythagorean major third ( [23 2]8 , approximately 14/11, or 417.391 cents); two for the fourth, being F² or F¢, or the 23-et grave or small fourth ( [23 2]9 , approximately 101/77, or 469.565 cents) and 23-et acute or large fourth ( [23 2]10 , approximately 173/128, or 521.739 cents); two for the tritone, being FÚ or GÝ, or the 23-et grave or small augmented fourth ( [23 2]11 , approximately 1024/735, or 573.913 cents) and 23-et acute 182 The Equally-Tempered Archetype or large diminished fifth ( [23 2]12 , approximately 735/512, or 626.087 cents); two for the fifth, being G² or G¢, or the 23-et grave or small fifth ([23 2]13 ) and 23-et acute or large fifth ( [23 2]14 , approximately 154/101, or 730.435 cents); two for the major sixth, being A³ or A, or the 23-et neutral sixth ([23 2]16 , approximately 3888/2401, or 834.783 cents) and 23-et just major sixth ( [23 2]17 , approximately 855/512, or 886.957 cents); as well as two options for the major seventh, being B³ or B, or the 23-et neutral seventh ( [23 2]20 , approximately 148/81, or 1043.478 cents) and 23-et just diatonic major seventh ( [23 2]21, approximately 145/77, or 1095.652 cents). The two approximations for the augmented second/minor third and fifth is also highlighted in bar 112 with the twenty-three-tone equally- tempered minor triad variant of C, DË, and G¢, which presents a triad with a just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß2.598 and ß23.694 cents.266 Simultaneous sonorities include the dyads F¢ and A¸, or 23-et five quarter-tones ( [);23 2]5 G¢ and B³, and B¸ and D¸, or the 23-et just minor third ( [23 2]6 ); GÝ and B¸, or the 23-et grave or small major third ( [23 2]7 ); A¸ and C, and B³ and DË, or the 23-et Pythagorean major third ( [23 2]8 ); C and F², and DË and G¢, or the 23-et grave or small fourth ( [);23 2]9 G² and C, and D¸ and GÝ, or the or 23-et acute or large fourth ( [23 2]10 ). The following table depicts the tonal resources of twenty-three-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

266 Daniélou, Tableau Comparatif des Intervalles Musicaux 29-141. The Equally-Tempered Archetype 183

Table 113. The twenty-three-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CË / Dì 23-et Greek enharmonic or septimal quarter-tone 1.030596 269.630 52.174 +52 ÐÓ D¸ 23-et just diatonic semitone, or major half-tone 1.062127 277.880 104.348 +04 ÐÔ D³ 23-et grave or small tone 1.094624 286.382 156.522 ß43 ÐÕ D 23-et just major tone 1.128114 295.143 208.696 +09 ÐÖ DË / Eì 23-et five quarter-tones 1.162629 304.174 260.870 ß39 Ð× EÝ 23-et just minor third 1.198201 313.480 313.043 +13 ÐØ E² 23-et grave or small major third 1.234860 323.071 365.217 ß35 ÐÙ E¢ 23-et Pythagorean major third, or ditone 1.272642 332.956 417.391 +17 ÑÐ F² 23-et grave or small fourth 1.311579 343.143 469.565 ß30 ÑÑ F¢ 23-et acute or large fourth 1.351707 353.641 521.739 +22 ÑÒ FÚ 23-et grave or small augmented fourth 1.393063 364.461 573.913 ß26 ÑÓ GÝ 23-et acute or large diminished fifth 1.435685 375.612 626.087 +26 ÑÔ G² 23-et grave or small fifth 1.479610 387.104 678.261 ß22 ÑÕ G¢ 23-et acute or large fifth 1.524880 398.948 730.435 +30 ÑÖ A¸ 23-et Pythagorean minor sixth 1.571534 411.154 782.609 ß17 Ñ× A³ 23-et neutral sixth 1.619616 423.733 834.783 ß65 ÑØ A 23-et just major sixth 1.669169 436.697 886.957 ß13 ÑÙ AË / Bì 23-et nineteen quarter-tones 1.720239 450.058 939.130 +39 ÒÐ B¸ 23-et Pythagorean minor seventh 1.772870 463.828 991.304 ß09 ÒÑ B³ 23-et neutral seventh 1.827112 478.019 1043.478 ß56 ÒÒ B 23-et just diatonic major seventh 1.883014 492.645 1095.652 ß04 ÒÓ BË / Cí 23-et twenty-three quarter-tones 1.940626 507.717 1147.826 ß52

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

Ex. 79. 23-tone equal temperament

184 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 114. 23-tone equal temperament tuning matrix no. 1 (Program 33)

NOTE C D¸ D EÝ E² F¢ F! G² A¸ A B¸ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +04 +09 +13 ß35 +22 ß26 ß22 ß17 ß13 ß09 ß04

Table 115. 23-tone equal temperament tuning matrix no. 2 (Program 34)

NOTE – – CË D³ DÍ E¢ F² GÝ G¢ A³ AË B³ Cí

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß48 ß43 ß39 +17 ß30 +26 +30 +35 +39 +44 +48

Twenty-Four-Tone Equal Temperament

Twenty-four-tone equal temperament ( 24 2 ), with its division of the octave into twenty-four equally- tempered quarter-tones 50.000 cents in size, marks the concluding tuning modulation of the work. The temperament may alternatively be expressed as the juxtaposition of two sets of twelve-tone equally- tempered intervals, three sets of eight-tone equally-tempered intervals, four sets of six-tone equally- tempered intervals, six sets of four-tone equally-tempered intervals, eight sets of three-tone equally- tempered intervals, or twelve sets of two-tone equally-tempered intervals interlocked an equal quarter- tone ( 24 2 ) apart. Blackwood makes the following observations: “This familiar ‘quarter-tone’ tuning is actually one of the most difficult to deal with in a practical situation. Without exception, the notes ‘in the cracks’ make extreme discords with the other notes, and there are only a very few satisfactory harmonies The Equally-Tempered Archetype 185 that combine the two.”267 Twenty-four-tone equal temperament is the fifth temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognisability.”268 Twenty-four-tone equal temperament provides two options for the approximation of the minor second, being Dì or D¹ , or the equal quarter-tone ( 24 2 ) and equal semitone ( 12 2 ); two for the major second, being D or D£, or the equal major tone ( 6 2 ) and five equal quarter-tones ( [,24 ]2 5 approximately 52/45, or 250.000 cents); two for the major third, being E³ or E, or seven equal quarter- tones ( 24 ]2[ 7 , approximately 60/49, or 350.000 cents) and the equal major third ( 3 2 ); two for the fourth, being Fí or F, or nine equal quarter-tones ( [8 ]2 3 ) and the equal perfect fourth ( [12 ]2 5 ); two for the tritone, being FË or F¥, or eleven equal quarter-tones ( 24 ]2[ 11, approximately 703/512, or 550.000 cents) and the equal tritone ( 2 2 ); two for the fifth, being G or G£, or the equal perfect fifth ( 12 ]2[ 7 ) and fifteen equal quarter-tones ( [8 ]2 5 ); two for the minor sixth, being A² or A¢, or seventeen equal quarter-tones ( 24 ]2[ 17 , approximately 49/30, or 850.00 cents) and the equal major sixth ( 4 ]2[ 3 ); two for the augmented sixth/minor seventh, being AÚ or B¸, or nineteen equal quarter-tones ( 24 ]2[ 19 , approximately 45/26, or 950.000 cents) and the equal minor seventh ( [6 ]2 5 ); as well as two options for the major seventh, being B³ or B¢, or twenty-one equal quarter-tones ( 8 ]2[ 7 ) and the equal major seventh ( 12 ]2[ 11 ).269 A five-figure rhythmically perpetuating descending melodic passage in bar 115 highlights all the primary, secondary, and auxiliary pitches available within the pélog scheme.

Ex. 80. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 115

Simultaneous sonorities in this temperament are limited to the dyads G and C, or the equal perfect fourth ( 12 ]2[ 5 ), and its inversion of C and G, or the equal perfect fifth ( 12 ]2[ 7 ). American composer Charles Edward Ives’s (1874-1954) experiments with twenty-four-tone equal temperament conducted during the 1925-62 period revealed that the division of the equal perfect fifth ( [12 ]2 7 ) into seven equal quarter- tones, or neutral thirds ( 24 ]2[ 7 ) to produce a twenty-four-tone equally-tempered neutral triad, resulted in a “chord that is heard as an ‘out of tune’ major or minor triad, not as an entity in itself,” but by the addition of a fourth tone (“in a quarter-tone relationship to the root or fifth”) the simultaneous sonority

267 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag. 268 Blackwood, The Structure of Recognizable Diatonic Tunings 197. 269 Daniélou, Tableau Comparatif des Intervalles Musicaux 66-134. 186 The Equally-Tempered Archetype

“seemed to establish an identity of its own.”270 The following table depicts the tonal resources of twenty- four-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 116. The twenty-four-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1.000000 261.626 0.000 +00 ÐÒ CË / Dì equal, Greek enharmonic or septimal quarter-tone 1.029302 269.292 50.000 +50 ÐÓ D¹ equal semitone, or Pythagorean limma 1.059463 277.183 100.000 +00 ÐÔ CÍ / Dí equal three-quarter-tone 1.090508 285.305 150.000 ß50 ÐÕ D equal or just major tone 1.122462 293.665 200.000 +00 ÐÖ D£ five equal quarter-tones, or supermajor second 1.155353 302.270 250.000 +50 Ð× E¸ equal or Pythagorean minor third, or trihemitone 1.189207 311.127 300.000 +00 ÐØ E³ seven equal quarter-tones, or neutral third 1.224054 320.244 350.000 ß50 ÐÙ E¢ equal or Pythagorean major third, or ditone 1.259921 329.628 400.000 +00 ÑÐ EË / Fí nine equal quarter-tones 1.296840 339.286 450.000 ß50 ÑÑ F equal, or just and Pythagorean perfect fourth 1.334840 349.228 500.000 +00 ÑÒ FË / Gì eleven equal quarter-tones 1.373954 359.461 550.000 ß50 ÑÓ F¥ equal or acute or large tritone, or augmented fourth 1.414214 369.994 600.000 +00 ÑÔ FÍ / Gí thirteen equal quarter-tones 1.455653 380.836 650.000 ß50 ÑÕ G equal, or just and Pythagorean perfect fifth 1.498307 391.995 700.000 +00 ÑÖ G£ fifteen equal quarter-tones, or superfifth 1.542211 403.482 750.000 +50 Ñ× A¸ equal or Pythagorean minor sixth 1.587401 415.305 800.000 +00 ÑØ A² seventeen equal quarter-tones 1.633915 427.474 850.000 ß50 ÑÙ A¢ equal or Pythagorean major sixth 1.681793 440.000 900.000 +00 ÒÐ AÚ nineteen equal quarter-tones, or augmented sixth 1.731073 452.893 950.000 ß50 ÒÑ B¸ equal or Pythagorean minor seventh 1.781797 466.164 1000.000 +00 ÒÒ B³ twenty-one equal quarter-tones, or neutral seventh 1.834008 479.823 1050.000 ß50 ÒÓ B¢ equal or Pythagorean major seventh 1.887749 493.883 1100.000 +00 ÒÔ BË / Cí twenty-three equal quarter-tones 1.943064 508.355 1150.000 ß50

ÐÑ C octave 2.000000 523.251 1200.000 +00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to represent sléndro and pélog scales.

270 Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 5. The Equally-Tempered Archetype 187

Ex. 81. 24-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically within the program memory of the Akai S3000XL.

Table 117. 24-tone equal temperament tuning matrix no. 1 (Program 35)

NOTE C D¹ D E¸ E¢ F F¥ G A¸ A¢ B¸ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Table 118. 24-tone equal temperament tuning matrix no. 2 (Program 36)

NOTE CË Dí D£ E³ Fí FÍ Gí G£ A² A! B³ Cí

KEY C C! D D! E F F! G G! A A! B

CENTS +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50

188 The Equally-Tempered Archetype

Blackwood’s Dictum

Blackwood offers the following summary of some the characteristics of equal tuning principles, and ‘the equally-tempered archetype’:

“Mostly, I’ve explored the equal tunings from twelve to twenty-four notes per octave. The choice of which tuning to use depends largely on the desired style. Certain tunings are more versatile or prettier than others. Twelve, seventeen, nineteen, twenty-two, and twenty-four contain recognizable diatonic scales. If random dissonance is what you want, then the worst of all tunings is twelve-note equal, because it contains a greater concentration of relatively consonant intervals and harmonies in a smaller number of notes. The most effective temperament for random dissonance is eleven notes. There aren’t two notes in that tuning that make any kind of a consonance. Certain others tend toward modal arrangements that coexist in twelve-note equal. For example, if the number of notes is divisible by four, you always have families of octatonic scales. If the number of notes is divisible by six, there are always families of whole- tone scales that can arise as altered chords, as they normally do in twelve-note tuning. If the number of notes is divisible by three, there is a symmetric mode that alternates minor thirds with minor seconds, creating a chromatic world all its own.”271

271 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 177. 3. The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer Quartet

Just Intonation

“Extended just intonation is a tuning system based on the ‘pure’ intervals of the overtone series: intervals generated therefore from whole number ratios. This is in contrast to temperaments, both equal and unequal, which use compromised intervals, often based on irrational numbers,” explains John Fonville. American composer and theorist Benjamin Burwell Johnston (1926-)272 personally elaborates on the matter within a statement contained in the performance notes of the score to his String Quartet No. 9, headed On the Performance Practice of Extended Just Intonation:

“Just intonation is simply the easiest way to tune musical intervals by ear. It results in greatly heightened purity and clarity of sound for two reasons: it eliminates acoustic beats to the maximum possible, and second, it exploits resonances by utilizing harmonically simple combinations of pitches. The term extended refers to the use of higher overtones than the first six partials.”273

In order to further understand the basic concept of just intonation, one must refer to the natural acoustic phenomenon known as the ‘harmonic series’. The overtones of a specific pitch are generally referred to as the ‘harmonic series’, and the musical scale derived from this series is constructed around ‘pure’ or ‘just’ intervals. This system of just intonation is strikingly dissimilar to the twelve-tone equally-tempered division of the octave, which is based on the division of the octave into twelve equal intervals; the frequency ratio of each semitone therefore mathematically representing the twelfth root of two.274 Leta E. Miller and Fredric Lieberman describe just intonation thus: “Pure intervals arise when the frequencies of the individual tones reflect the precise mathematical proportions that occur in the series: 3/2 for the fifth, 4/3 for the fourth, etcetera.” Intervals manifested naturally within the harmonic series are particularly favourable in just intonation, and certainly ones with “superparticular vibration ratios,” where the

272 “Johnston’s credentials are impressive. He studied with Harry Partch, Darius Milhaud, John Cage, and Burrill Phillips, and although some of these teachers influenced Johnston’s early music, he has remained an individual and followed his own creative path. Yet he has written in many different styles: jazz resulting from his Naval service, neoclassicism, and serialism (which he has adapted to serve a functional purpose in just intonation). Much of his work has intuitively predated important trends in contemporary music. He used combinatoriality in the late 1950’s, quotations in the 1960’s, and returned to tonality in the early 1970’s. He even tried indeterminacy and electronic music but has been less successful in these areas.” For a further discussion, see Heidi Von Gunden, The Music of Ben Johnston (Metuchen, N.J.:The Scarecrow Press, 1986) vii. 273 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 106-07. 274 Olson, Music, Physics and Engineering 46-47. 190 The Harmonic Consideration

“numerator exceeds the denominator by one” such as the just perfect fifth (3/2), just perfect fourth (4/3), just major third (5/4) and just minor third (6/5).275

The Harmonic and Subharmonic Series

The concept of the vibrating string and its connection to music – via the sounding of an open string to produce the fundamental (or the 1/1 ratio), with the string stopped at the midpoint producing the octave (or the 2/1 ratio), and at the two-third point, the perfect fifth (or the 3/2 ratio) – is first presented in the West by Pythagoras of Samos (570-504 B.C.), although it is not until Galileo Galilei (1564-1642) and Marin Mersenne (1588-1648) that the connection between the actual time or period of the cycle or vibration, and string length, tension and density is made. John Wallis (1616-1703) and Joseph Sauveur (1653-1716) are acknowledged for contributing further to the study with their discovery that proportional vibrations (in the ratios of 1/1, 1/2, 1/3, 1/4, and so on) are also simultaneously produced – this fact elucidating the notion of the vibrating string as an amalgam of the fundamental and octave proposed by Aristotle (382-322 B.C.), and consequently stimulating scientific study that in time reveals more and more upper partials. “Thus the note produced in the ear by a vibrating string was shown to be composed of a series of pure tones (partials), sounding simultaneously, and corresponding to component or partial vibrations of the main vibration whose periods (or string lengths) formed an harmonic series,” explains Lloyd and Boyle.276 The following table illustrates the harmonic characteristics of the first eight partials of the harmonic series.277

Table 119. The beating characteristics of the first eight partials of the harmonic series

PARTIAL NOTE INTERVAL RATIO RATIO FREQUENCY CENTS NUMBER (FRACTION) (DECIMAL) (HERTZ) 1 C4 fundamental 1/1 1.000000 261.626 0.000 2 C5 octave 2/1 2.000000 523.251 1200.000 3 G5 twelfth 6/2 3.000000 784.877 1901.955 4 C6 fifteenth, or double octave 4/1 4.000000 1046.502 2400.000 5 E6 tierce (octave tenth) 20/4 5.000000 1308.128 2786.314 6 G6 octave twelfth 12/2 6.000000 1569.753 3101.955 7 B;6 harmonic seventh 28/4 7.000000 1831.379 3368.826 8 C7 triple octave 8/1 8.000000 2093.005 3600.000

275 Leta E. Miller, and Fredric Lieberman, Lou Harrison: Composing a World (New York: Oxford U. Press, 1998) 107. 276 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 2-4. 277 R. H. M. Bosanquet, An Elementary Treatise on Musical Intervals and Temperament, ed. Rudolf Rasch (Utrecht, The Netherlands: Diapason Press, 1987) 16. The Harmonic Consideration 191

When just intonation practitioners speak of eliminating “acoustic beats to the maximum possible” they are of course referring to the beats that occur when unmatched pure tones sounding simultaneously move in and out-of-phase, which generate shifts in amplitude, and hence difference tones that are manifested as beats per second.278 The chart for the beating characteristics of the mistuned and properly tuned unison illustrates the fact that the partials of a properly tuned unison correspond precisely with those generated by the fundamental, and result in an amalgam “absolutely smooth and free from any disturbance.” In striking contrast, the consequence of a mistuned unison (1.000Hz sharp) is a continuous series of mistuned pairs of partials.279

Table 120. The beating characteristics of the mistuned and properly tuned unison

FUNDAMENTAL MISTUNED UNISON PROPERLY TUNED UNISON C4 FREQUENCY C4 FREQUENCY BEATS C4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 1 262.626 1.000 1 261.626 0.000 2 523.251 2 525.251 2.000 2 523.251 0.000 3 784.877 3 787.877 3.000 3 784.877 0.000 4 1046.502 4 1050.502 4.000 4 1046.502 0.000 5 1308.128 5 1313.128 5.000 5 1308.128 0.000 6 1569.753 6 1575.753 6.000 6 1569.753 0.000 7 1831.379 7 1838.379 7.000 7 1831.379 0.000 8 2093.005 8 1050.502 8.000 8 2093.005 0.000

The case for the octave is identical but for the fact that every second partial of a properly tuned octave corresponds precisely with those generated by the fundamental, while a mistuned octave (1.000Hz sharp) exponentially beats at rate of an additional 1.000Hz every second partial.280

278 Roderick D. Gordon, The World of Musical Sound (Dubuque, IA: Kendall/Hunt, 1979) 41. 279 “Beats, which have now to be considered, are essentially distinguished from combinational tones as follows: In combinational tones the composition of vibrations in the elastic vibrating bodies which are either within or without the ear, undergoes certain disturbances, although the ear resolves the motion which is finally conducted to it, into a series of simple tones, according to the usual law. In beats, on the contrary, the objective motions of the elastic bodies follow the simple law; but the composition of the sensations is disturbed. As long as several simple tones of a sufficiently different pitch enter the ear together, the sensation due to each remains undisturbed in the ear, probably because entirely different bundles of nerve fibres are affected. But tones of the same, or of nearly the same pitch, which therefore affect the same nerve fibres, do not produce a sensation which is the sum of the two they would have separately excited, but new and peculiar phenomena arises which we term ‘interference’, when caused by two perfectly equal simple tones, and ‘beats’ when due to nearly equal simple tones.” For a further discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 159-60. 280 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 20. 192 The Harmonic Consideration

Table 121. The beating characteristics of the mistuned and properly tuned octave

FUNDAMENTAL MISTUNED OCTAVE PROPERLY TUNED OCTAVE C4 FREQUENCY C5 FREQUENCY BEATS C5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – 2 523.251 1 524.251 1.000 1 523.251 0.000 3 784.877 – – – – – – – – – – – – – – – – – – – – – – – – 4 1046.502 2 1048.502 2.000 2 1046.502 0.000 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 3 1572.753 3.000 3 1569.753 0.000 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – 8 2093.005 4 2097.005 4.000 4 2093.005 0.000

According to Helmholtz, ‘dissonance’ is explained as a beating condition between the corresponding partials of a simultaneous sonority, and “there are certain determinate ratios between pitch numbers, for which this rule suffers an exception, and either no beats at all are formed, or at least only such as have so little intensity that they produce no unpleasant disturbance of the united sound. These exceptional cases are called consonances.” James Tenney explains: “Helmholtz equates the dissonance of a simultaneous aggregate with ‘roughness’ of the sensation caused by beats between adjacent partials (and to a lesser extent, between ‘combinational tones’) in the combined spectrum of the tones forming the aggregate.”281 Helmholtz categorizes consonances as “the most perfect consonances,” or “absolute,” which include the octave (2/1), twelfth (6/2), and double octave (4/1). Next come “perfect consonances” such as the fifth (3/2) and fourth (4/3), which are followed by “medial consonances” such as major sixth (5/3) and major third (5/4); with the minor third (6/5) and minor sixth (8/5) categorized as “imperfect consonances.” The “essence of dissonance” is ultimately summarized by Helmholtz as follows:

“It is apparent to the simplest natural observation that the essence of dissonance consists merely in very rapid beats. The nerves of hearing feel these rapid beats as ‘rough’ and unpleasant, because every intermittent excitement of any nervous apparatus affects us more powerfully than one that lasts unaltered. The individual pulses of tone in a dissonant combination give us certainly the same impression of separate pulses as slow beats, although we are unable to recognize them separately and count them; hence they form a ‘tangled’ mass of tone, which cannot be analyzed into its constituents. The cause of the unpleasantness of dissonance we attribute to this ‘roughness’ and ‘entanglement’. The meaning of this distinction may be thus briefly stated: ‘Consonance is a continuous, dissonance an intermittent sensation of tone.’ Two consonant tones flow quietly side by side in an undisturbed stream; dissonant tones cut one another up into separate pulses of tone. This description of the distinction at which we have arrived

281 James Tenney, A History of ‘Consonance’ and ‘Dissonance’ (New York: Excelsior Music Publishing Co., 1988) 87-88. The Harmonic Consideration 193

agrees precisely with Euclid’s old definition, ‘Consonance is the blending of a higher with a lower tone. Dissonance is incapacity to mix, when two tones cannot blend, but appear rough to the ear.’”282

Tenney offers the following observations with regards to the general acceptance of Helmholtz’s beat theory by music theorists: “The fact that the consonance or dissonance predicted by the beat theory for a given dyad would vary with the absolute frequencies of its tones, rather than simply the intervals between them, has been pointed out by many other writers – and generally used as an argument against the validity of Helmholtz’s theory.”283 It is interesting to note that Bosanquet makes a distinction “between ‘beating dissonances’ and ‘unsatisfied combinations’,” classifying the harmonic seventh (7/4), along with the just perfect fourth (4/3) dyads in the latter group;284 while Johnston, “between different kinds of dissonance: the dissonance of complex ratios and the dissonance of higher prime numbers,” utilizing the numerical similarity of the grave or small just chromatic semitone (25/24) and tridecimal third tone (26/25, or 67.900 cents) as examples of the relationship between the dissonant curve and prime limit.285 An important factor in just intonation rationale is the further notion of the complement or mirror image of the harmonics series, which is referred to as the ‘subharmonic series’, and represents the foundation of Partch’s theory of ‘otonalities’ (pitches derived from the ascending series) and ‘utonalities’ (pitches derived from the descending series). “Unlike the harmonic series, the subharmonic series is not represented in the partials of any known sounding bodies,” notes Doty. “Theorists in earlier centuries anxiously sought sounds in nature with subharmonic partials, but none were ever discovered.”286 Henry Cowell presents the following discussion on the musical significance of research conducted on the subharmonic series:

“A very interesting approach to the theoretical explanation of minor is a consideration of the theory of undertones. Until recently undertones were a theory only. Their existence was contested by scientists on the ground that a string or vibrating body could not vibrate at a length greater than its complete length, which gives the fundamental tone. Hence, it was contended, no deeper tones in such a series would be possible of formation on the string. Now, however, Professor Nicolas A. Garbusov, of the Moscow State Institute for Musicology, has built an instrument on which at least the first nine undertones are easily heard without the aid of resonators. The principle is not that the original sounding body produces the undertones, but that it is difficult to avoid them in resonation.”287

282 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 194-226. 283 Tenney, A History of ‘Consonance’ and ‘Dissonance’ 90. 284 R. H. M. Bosanquet, “Temperament; Or, the Division of the Octave (Part II),” Proceedings of the Musical Association, 2nd Sess. (1874-75): 127. 285 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 202. 286 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 28-29. 287 Henry Cowell, New Musical Resources (Cambridge: Cambridge U. Press, 1996) 21-22. 194 The Harmonic Consideration

Ex. 82. The harmonics and subharmonics of the fundamental C – first partial, through the 16th partial

The following comparative table of intonation depicts the twelve basic intervals of just intonation, Pythagorean intonation, meantone temperament, and equal temperament, indicating interval, ratio, and cents for each system of tuning.

Table 122. Comparative table of intonation

COMPARATIVE JUST PYTHAGOREAN MEANTONE EQUAL TABLE INTONATION INTONATION TEMPERAMENT TEMPERAMENT INTERVAL RATIO CENTS RATIO CENTS RATIO CENTS RATIO CENTS unison 1/1 0.000 1/1 0.000 1.000000 0.000 1.000000 0.000 diatonic semitone 16/15 111.731 2187/2048 113.685 1.044907 76.049 1.059463 100.000 major tone 9/8 203.910 9/8 203.910 1.118034 193.157 1.122462 200.000 minor third 6/5 315.641 32/27 294.135 1.196279 310.265 1.189207 300.000 major third 5/4 386.314 81/64 407.820 1.250000 386.314 1.259921 400.000 perfect fourth 4/3 498.045 4/3 498.045 1.337481 503.422 1.334840 500.000 tritone 45/32 590.224 729/512 611.730 1.397542 579.471 1.414214 600.000 perfect fifth 3/2 701.955 3/2 701.955 1.495349 696.578 1.498307 700.000 minor sixth 8/5 813.686 128/81 792.180 1.600000 813.686 1.587401 800.000 major sixth 5/3 884.359 27/16 905.865 1.671851 889.735 1.681793 900.000 minor seventh 9/5 1017.596 16/9 996.090 1.788854 1006.843 1.781797 1000.000 major seventh 15/8 1088.269 243/128 1109.775 1.869186 1082.892 1.887749 1100.000 octave 2/1 1200.000 2/1 1200.000 2.000000 1200.000 2.000000 1200.000 Table 123. The harmonic series – first partial, through the 128th partial

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison (1st harmonic) 1/1 1.000000 261.626 0.000 +00 ÐÒ C. tridecimal comma (65th harmonic) 65/64 1.015625 265.713 26.841 +27

ÐÓ C, undecimal comma (33rd and 66th harmonic) 33/32 (66/64) 1.031250 269.801 53.273 ß47 ÐÔ 67th harmonic 67/64 1.046875 273.889 79.070 ß21

ÐÕ CP septendecimal chromatic semitone (17th, 34th and 68th harmonic) 17/16 (34/32, 68/64) 1.062500 277.977 104.955 +05 ÐÖ CX( trivigesimal chromatic semitone (69th harmonic) 69/64 1.078125 282.065 130.229 +30

Ð× D+ septimal neutral second (35th and 70th harmonic) 35/32 (70/64) 1.093750 286.153 155.140 ß45 ÐØ 71st harmonic 71/64 1.109375 290.241 179.697 ß20

ÐÙ D just major tone (9th, 18th, 36th and 72nd harmonic) 9/8 (18/16, 36/32, 72/64) 1.125000 294.329 203.910 +04 ÑÐ 73rd harmonic 73/64 1.140625 298.417 227.789 +28

ÑÑ 37th and 74th harmonic 37/32 (74/64) 1.156250 302.505 251.344 ß49 ÑÒ D! augmented second (75th harmonic) 75/64 1.171875 306.592 274.582 ß25

ÑÓ EW nonadecimal subminor, or overtone minor third (19th, 38th and 76th harmonic) 19/16 (38/32, 76/64) 1.187500 310.680 297.513 ß02

ÑÔ E© undecimal neutral third (77th harmonic) 77/64 1.203125 314.768 320.144 +20 T eHroi osdrto 195 Consideration Harmonic he

ÑÕ EO tridecimal grave or small neutral third (39th and 78th harmonic) 39/32 (78/64) 1.218750 318.856 342.483 +42 ÑÖ 79th harmonic 79/64 1.234375 322.944 364.537 ß35

Ñ× E just major third (5th, 10th, 20th, 40th and 80th harmonic) 5/4 (10/8, 20/16, 40/32, 80/64) 1.250000 327.032 386.314 ß14 ÑØ E( Pythagorean major third, or ditone (81st harmonic) 81/64 1.265625 331.120 407.820 +08

ÑÙ 41st and 82nd harmonic 41/32 (82/64) 1.281250 335.208 429.062 +29 ÒÐ 83rd harmonic 83/64 1.296875 339.296 450.047 ß50

ÒÑ F+( septimal subfourth (21st, 42nd and 84th harmonic) 21/16 (42/32, 84/64) 1.312500 343.384 470.781 ß29 9 h amncConsideration Harmonic The 196 DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÒÒ EP septendecimal superaugmented third (85th harmonic) 85/64 1.328125 347.471 491.269 ß09

ÒÓ 43rd and 86th harmonic 43/32 (86/64) 1.343750 351.559 511.518 +12 ÒÔ F6( nonavigesimal superfourth (87th harmonic) 87/64 1.359375 355.647 531.532 +32

ÒÕ F, undecimal superfourth (11th, 22nd, 44th and 88th harmonic) 11/8 (22/16, 44/32, 88/64) 1.375000 359.735 551.318 ß49 ÒÖ 89th harmonic 89/64 1.390625 363.823 570.880 ß29

Ò× F!( just tritone, or augmented fourth (45th and 90th harmonic) 45/32 (90/64) 1.406250 367.911 590.224 ß10 ÒØ GÞ tridecimal diminished fifth (91st harmonic) 91/64 1.421875 371.999 609.354 +09

ÒÙ FX( trivigesimal superaugmented fourth (23rd, 46th and 92nd harmonic) 23/16 (46/32, 92/64) 1.437500 376.087 628.274 +28 ÓÐ F`( untrigesimal superaugmented fourth (93rd harmonic) 93/64 1.453125 380.175 646.991 +47

ÓÑ 47th and 94th harmonic 47/32 (94/64) 1.468750 384.263 665.507 ß34 ÓÒ G2 nonadecimal subfifth (95th harmonic) 95/64 1.484375 388.350 683.827 ß16

ÓÓ G just and Pythagorean perfect fifth (3rd, 6th, 12th, 24th, 48th and 96th harmonic) 3/2 (6/4, 12/8, 24/16, 48/32, 96/64) 1.500000 392.438 701.955 +02 ÓÔ 97th harmonic 97/64 1.515625 396.526 719.895 +20

ÓÕ AŒ( septimal diminished sixth (49th and 98th harmonic) 49/32 (98/64) 1.531250 400.614 737.652 +38 ÓÖ G, undecimal superfifth (99th harmonic) 99/64 1.546875 404.702 755.228 ß45

Ó× GÚ augmented fifth (25th, 50th and 100th harmonic) 25/16 (50/32, 100/64) 1.562500 408.790 772.627 ß27 ÓØ 101st harmonic 101/64 1.578125 412.878 789.854 ß10

ÓÙ GP septendecimal superaugmented fifth (51st and 102nd harmonic) 51/32 (102/64) 1.593750 416.966 806.910 +07 ÔÐ 103rd harmonic 103/64 1.609375 421.054 823.801 +24

ÔÑ AO tridecimal grave or small neutral, or overtone sixth (13th, 26th, 52nd and 104th harmonic) 13/8 (26/16, 52/32, 104/64) 1.625000 425.142 840.528 +41 ÔÒ A+( septimal neutral sixth (105th harmonic) 105/64 1.640625 429.229 857.095 ß43

ÔÓ 53rd and 106th harmonic 53/32 (106/64) 1.656250 433.317 873.505 ß26 ÔÔ 107th harmonic 107/64 1.671875 437.405 889.760 ß10 DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ)

ÔÕ A( Pythagorean major sixth (27th, 54th and 108th harmonic) 27/16 (54/32, 108/64) 1.687500 441.493 905.865 +06 ÔÖ 109th harmonic 109/64 1.703125 445.581 921.821 +22

Ô× A, undecimal supermajor sixth (55th and 110th harmonic) 55/32 (110/64) 1.718750 449.669 937.632 +38 ÔØ 111th harmonic 111/64 1.734375 453.757 953.299 ß47

ÔÙ B; septimal subminor seventh (7th, 14th, 28th, 56th and 112th harmonic) 7/4 (14/8, 28/16, 56/32, 112/64) 1.750000 457.845 968.826 ß31 ÕÐ 113th harmonic 113/64 1.765625 461.933 984.215 ß16

ÕÑ BW nonadecimal subminor seventh (57th and 114th harmonic) 57/32 (114/64) 1.781250 466.021 999.468 ß01 ÕÒ AX( trivigesimal superaugmented sixth (115th harmonic) 115/64 1.796875 470.108 1014.588 +15

ÕÓ B_ nonavigesimal grave or small neutral seventh (29th, 58th and 116th harmonic) 29/16 (58/32, 116/64) 1.812500 474.196 1029.577 +30 ÕÔ BO tridecimal neutral seventh (117th harmonic) 117/64 1.828125 478.284 1044.438 +44

ÕÕ 59th and 118th harmonic 59/32 (118/64) 1.843750 482.372 1059.172 ß41 ÕÖ Bß septendecimal neutral seventh (119th harmonic) 119/64 1.859375 486.460 1073.781 ß26

Õ× B just diatonic major seventh (15th, 30th, 60th and 120th harmonic) 15/8 (30/16, 60/32, 120/64) 1.875000 490.548 1088.269 ß12 ÕØ BÏ) grave or small neutral seventh (121st harmonic) 121/64 1.890625 494.636 1102.636 +03

ÕÙ 61st and 122nd harmonic 61/32 (122/64) 1.906250 498.724 1116.885 +17 ÖÐ 123rd harmonic 123/64 +31 1.921875 502.812 1131.017 T eHroi osdrto 197 Consideration Harmonic he ÖÑ B8 untrigesimal supermajor seventh (31st, 62nd and 124th harmonic) 31/16 (62/32, 124/64) 1.937500 506.900 1145.036 +45 ÖÒ B! augmented seventh (125th harmonic) 125/64 1.953125 510.987 1158.941 ß41

ÖÓ C+( septimal subdiminished octave (63rd and 126th harmonic) 63/32 (126/64) 1.968750 515.075 1172.736 ß27 ÖÔ 127th harmonic 127/64 1.984375 519.163 1186.422 ß14

ÐÑ C octave (2nd harmonic) 2/1 2.000000 523.251 1200.000 +00

(4th, 8th, 16th, 32nd, 64th and 128th harmonic) (4/2, 8/4, 16/8, 32/16, 64/32, 128/64) 198 The Harmonic Consideration

It is important to note that odd-numbered partials in the harmonic and subharmonic series represent unique entities, while even-numbered partials, merely their respective octave duplications, and therefore in harmonic analysis the former needs only be considered. The term designated to these unique entities of the harmonic and subharmonic series is ‘identities’.288

The Monochord

A monochord is a simple instrument that consists of a metal string extended over two bridges, and is a useful tool for illustrating the nature of ratios (a bamboo pipe, or air column deliver similar results). If for example, striking the open string produced 100 cycles when set in vibration (cycles referring to the number of vibrations per second which a tone makes), striking half of that length (the string stopped via a third bridge placed in the centre) would produce 200 cycles, or a 200 to 100 ratio (2/1); sounding the perfect octave of the original pitch. The logic then follows that striking a third of the full length would produce 300 cycles, or a 300 to 200 ratio (3/2) of the half length, and hence now sounding the perfect fifth.289 Cecil Adkins provides the following historical perspective:

“The monochord in its early form, and in the form utilized throughout the Middle Ages was a table or plank (AC) upon which were erected two fixed bridges (EB and FD). The string was stretched across the bridges (EF) and securely fastened at the ends (AC). A movable bridge (K) was then placed underneath the string, dividing it into two sections (EK and KF). The marks indicating the placement of the movable bridge were then inscribed on the table underneath the string, between the two end bridges (B and D). The resonating box, generally considered an integral part of the instrument, is not mentioned in the treatises of the Middle Ages, but is depicted in miniatures after the twelfth century. It was probably a late medieval addition directed at increasing the portability of the instrument as much as enhancing its tone.”

Fig. 6. Medieval figure of the monochord

The traditional function of the monochord is the facilitation of aural representations of intervals or scales, established via mathematical calculation, and generally articulated via the principles of proportions, string lengths, or cents. The term ‘monochord division’ in this context refers to what Adkins prefers to call the “manual division,” and therefore implicates the actual physical application of the mathematical formula

288 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 30. 289 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 79-81. The Harmonic Consideration 199 required to produce a specific interval. The Pythagorean principle of “monochord division by proportions” takes into account the “arithmetic mean and the harmonic mean” in a system of four principal intervals directly derived from the “smallest whole integers” that is capable of expressing the intervallic “relationships of these two means.” In this case, the principle is represented by the numbers 6, 8, 9, and 12, and by the resulting ratios of 12/6 (the diapason, or octave), 9/6 and 12/8 (the diapente, or fifth), 8/6 and 12/9 (the diatessaron, or fourth), and 9/8 (the tone, or major second). In different terms, these four intervals may be further theorized as the “multiplex and superparticular proportions” of 2/1 (dupla), 3/2 (sesquialtera), 4/3 (sesquitertia), and 9/8 (sesquioctava).290 The employment of string lengths and cents as a measuring device in the Pythagorean chromatic context represent 2/1 (octave) by the figure 314928, or 1200.000 cents; 3/2 (just perfect fifth) by 419904, or 701.955 cents; 4/3 (just perfect fourth) by 472392, or 498.045 cents; and 9/8 (just major tone) by 559872, or 203.910 cents. The interval of the unison is represented by the string length of 629856, or 0.000 cents.291

Table 124. The Pythagorean monochord

DEGREE NOTE LENGTHS RATIO CENTS FALSITY NUMBER (DECIMAL) (CENTS) ÐÑ C4 629856 1.000000 0.000 +0.000 ÐÒ C!4 589824 1.067871 113.685 +1.954 ÐÓ D4 559872 1.125000 203.910 +0.000 ÐÔ D!4 531441 1.185185 294.135 ß21.506 ÐÕ E4 497664 1.265625 407.820 +21.506 ÐÖ F4 472392 1.333333 498.045 +0.000 Ð× F!4 442368 1.423828 611.730 +21.506 ÐØ G4 419904 1.500000 701.955 +0.000 ÐÙ G!4 393216 1.601807 815.640 +1.954 ÑÐ A4 373248 1.687500 905.865 +21.506 ÑÑ A!4 354294 1.777778 996.090 ß21.506 ÑÒ B4 321776 1.898438 1109.775 +21.506

ÐÑ C5 314928 2.000000 1200.000 +0.000

290 Cecil Adkins, “The Technique of the Monochord,” Acta Musicologica 39 (Jan.-Jun., 1967): 34-37. 291 Barbour, Tuning and Temperament: A Historical Survey 90. 200 The Harmonic Consideration

Combinational Tones

The discovery of combinational tones in 1745 may be attributed to German organist Georg Andreas Sorge (1703-1778), and was followed by the later classification of Italian violinist Giuseppe Tartini (1692- 1770). According to Helmholtz, “These tones are heard whenever two musical tones of different pitches are sounded together, loudly and continuously.” There are two classes of combinational tones, with the first being differential tones (whereby the resultant tone equals f2 minus f1), and the second, summation tones (whereby the resultant tone equals f2 plus f1), which represent Helmholtz’s own contribution to the theory. Primary and upper partial tones are both capable of producing the two classes of combinational tones. Differential tones are more prominent when the interval between the two generating tones is inferior to an octave, as this condition produces a resultant tone below the primary tones. The theory may be further expanded to differentiate between first and second-order differential tones, with the latter being phenomena of an infinite nature, and the result of the secondary differential tones produced by the combination of the primary differential tones and the actual generating tones. Summation tones are less prominent than combinational tones; and in view of the fact that the resultant frequency is equal to the sum of the two primary tones, the pitch of these tones will always exceed their generators. It should be further noted that upper partial tones are also capable of producing combinational tones.292 The following example illustrates the differential tones produced by the octave (2/1), just perfect fifth (3/2), just perfect fourth (4/3), just major third (5/4), just minor sixth (8/5), just minor third (6/5), and just major sixth (5/3).

Ex. 83. Differential tones produced by 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, and 5/3

Summation tones for the same generating tones on the other hand present the following intervals: just perfect fifth (3/2), just major third (5/4), septimal subminor third (7/6), just major tone (9/8), tridecimal subdiminished fourth (13/10), undecimal acute or large neutral second (11/10), and just perfect fourth (4/3).

292 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 152-56. The Harmonic Consideration 201

Ex. 84. Summation tones produced by 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, and 5/3

Periodicity Pitch

“When two or more pure tones separated by intervals greater than the critical band are sounded simultaneously, the auditory mechanism is sensitive to the period or frequency of the resulting composite waveform. This gives rise to a phenomenon variously known as the periodicity pitch, virtual pitch, subjective pitch, residue tone, or the missing fundamental,” explains Doty. The simultaneous sounding of

m Ï two dissimilar pitched musical tones may be mathematically explained as f2= n f1, with m and n representative of the relevant integers (numerator and denominator) that denote the frequency ratio of

 1 Ï the interval. The formula for calculating the periodicity pitch of a dyad is 0= n 1, with n, the denominator, utilized as the dividing factor, and applied to the fundamental, or lower tone. The process may be illustrated via the calculations of the just perfect fifth (3/2) and just perfect fourth (4/3) in relation

1 Ï 4 3 1 Ï to the unison (1/1), which presents 2 261.626Hz (or C )=130.8136Hz (or C ); and 3 261.626Hz (or C4)=87.209Hz (or F2). Doty alludes to the facts that the “smaller the value of n, the shorter the period and the higher the frequency of the resulting pattern,” and that “musical consonance is associated with high periodicity pitch.” Dyads generated via simple integer-ratio intervals with integers inferior to eight or nine produce “unambiguous periodicity pitch,” while superior integers generally produce an “ambiguous” approximation of a simpler ratio.293

Ex. 85. Periodicity pitches produced by 2/1, 3/2, 4/3, 5/3, 5/4, 7/4, 6/5, 7/5, 8/5, and 7/6

293 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 17-18. 202 The Harmonic Consideration

Sethares has the following to say about the “periodicity theory of pitch perception,” and its theoretical significance in relation to JND, or Just Noticeable Difference:

“An alternative hypothesis (to JND, or Just Noticeable Difference), called ‘periodicity’ theory of pitch perception suggests that information is extracted directly from the time behaviour of the sound. For instance, the time interval over which a signal repeats may be used to determine its frequency. In fact, there is now (and has been for the past hundred years or so) considerable controversy between advocates of the place and periodicity theories, and it is probably safe to say that there is not enough evidence to decide between them. Indeed, J. R. Pierce (Periodicity and Pitch Perception, 1889-93 [1991]) suggests that both mechanisms may operate in tandem, and a growing body of recent neurophysiological research (such as P. Cariani and his co-workers [Temporal Coding of Periodicity Pitch in the Auditory System: An Overview [1999], and A Temporal Model for Pitch Multiplicity and Tonal Consonance [2004]) reinforce this.”294

Prime Numbers, Primary Intervals, and Prime Limits

Another aspect of just intonation principles is the theory of prime limits, which is based on the concept of prime numbers, or integers with only factors of one and themselves. This is an infinite series, but musical significance is accorded to primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. The primary intervals of these eleven primes are 2/1, 3/2, 5/4, 7/4, 11/8, 13/8, 17/16, 19/16, 23/16, 29/16, and 31/16 (produced by the formula: p/2n=p [prime], with “2n being the greatest power of 2 less than p”), while its subharmonic complements may be represented by 2/1, 4/3, 8/5, 8/7, 16/11, 16/13, 32/17, 32/19, 32/32, 32/29, and 32/31 (produced by the formula: 2n/p=p [prime], with “2n being the smallest power of 2 less than p”). An example of this concept is ‘five limit just intonation’, which should be taken to mean an intonation system based only on the primes 2, 3, and 5, with all intervals directly derived from 2/1, 3/2, and 5/4 (the octave, just perfect fifth, and just major third), or 2/1, 4/3, and 8/5 (octave, just perfect fourth, and just minor sixth) from a subharmonic perspective.295

The Just Diatonic Scale

The construction of a just diatonic scale begins with the fundamental or unison (the frequency ratio 1/1), and the establishment of a just major triad in the ratio of 1/1:5/4:3/2.296 The harmonic relationship of this root position chord, which has the 1, 5, and 3 identities of the harmonic series, is also expressible as 4:5:6, or in the key of C major as the pitches C, E, and G. In first inversion, the triad is represented by the

294 Sethares, Tuning, Timbre, Spectrum, Scale 44. 295 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 28-30. 296 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 108. The Harmonic Consideration 203 ratio 5:6:8, and in second, by 3:4:5. This triad is beatless, made up of the first, fifth, and third partials of the harmonic series, and contains the intervals of the unison (1/1), just major third (5/4), and just perfect fifth (3/2). Analysis of this simultaneous sonority further reveals a 6/5 ratio (or just minor third) between the third and the fifth. G4 (392.438Hz), or the just perfect fifth (3/2), presents 0.000 beats between the third harmonic of C4 (1/1) and the second harmonic of G4 (784.877Hz), and 0.000 beats between the sixth harmonic of C4 and the fourth harmonic of G4 (1569.753Hz); while E4 (327.032Hz), or the just major third (5/4) then consistently also presents 0.000 beats between the fifth harmonic of C4 and the fourth harmonic of E4 (1308.128Hz).

Table 125. The beating characteristics of the just major triad

C4 FREQUENCY E4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 327.032 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 392.438 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 654.064 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 784.877 0.000 – – – – – – – – 3 981.096 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1177.315 – – – – 5 1308.128 4 1308.128 0.000 – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1569.753 0.000 – – – – – – – – 5 1635.160 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1962.192 – – – – 5 1962.192 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The next stage in the generation of a just diatonic scale requires simple mathematics, or the multiplication of ratios, to deal with the addition of ratios. If one were to take the C major triad as an example, which has a 6/5 ratio between 5/4 and 3/2, one can ascertain that adding 6/5 to 5/4 will produce the equation: 6/5 plus 5/4=(6Ï5):(5Ï4)=30/20=3/2. In this particular case, 3/2 represents the reduction of the unnecessary larger 30/20 ratio. Subtraction of ratios is alternatively resolved via the multiplication of ratios with the subtraction in question inverted. In other words, the subtraction of 6/5 from 3/2 is resolved thus: 3/2 minus 6/5=(3Ï5):(2Ï6)=15/12=5/4. In cases where addition results in a figure that exceeds the octave (2/1), or where the “numerator is greater than twice the denominator,” subtraction of an appropriate number of 2/1s should be undertaken. For example, the equation of 3/2 plus 204 The Harmonic Consideration

3/2=(3Ï3):(2Ï2)=9/4 should be followed by 9/4 minus 2/1=(9Ï1):(4Ï2)=9/8. On the other hand, in cases where subtraction results in a figure that exceeds the unison (1/1), or where the result is a “ratio with a denominator greater than its numerator”, addition of an appropriate number of 2/1s should be undertaken. For example, 9/8 minus 3/2=(9Ï2):(8Ï3)=18/24=3/4 should be followed by 3/4 plus 2/1=(3Ï2):(4Ï1)=6/4=3/2.297 The final stage in the establishment of the just diatonic scale involves the further generation of just major triads on the fifth degree, 3/2 (or G) and its inversion, 4/3 (or F), which produce the intervals 3/2, 15/8, and 9/8 (or the pitches G, B, and D), and 4/3, 5/3, and 1/1 (or the pitches F, A, and C). The equations involved in obtaining 4:5:6 harmonic relationships (or appropriate 5/4s and 3/2s) for 1/1, 3/2, and 4/3 are:

i. C (unison) generated by the ratio 1/1 ii. E (just major third) by the equation 1/1 plus 5/4=(1Ï5):(1Ï4)=5/4 iii. G (just and Pythagorean perfect fifth) by 1/1 plus 3/2=(1Ï3):(1Ï2)=3/2

iv. G (just and Pythagorean perfect fifth) by the ratio 3/2 v. B (just diatonic major seventh) by the equation 3/2 plus 5/4=(3Ï5):(2Ï4)=15/8 vi D (just major tone) by 3/2 plus 3/2=(3Ï3):(2Ï2)=9/4; and 9/4 minus 2/1=(9Ï1):(4Ï2)=9/8

vii. F (just and Pythagorean perfect fourth) by the ratio 4/3 viii. A (just major sixth) by the equation 4/3 plus 5/4=(4Ï5):(3Ï4)=20/12=5/3 ix. C (unison) by 4/3 plus 3/2=(4Ï3):(3Ï2)=12/6=2/1; and 2/1 minus 2/1=(2Ï1):(1Ï2)=2/2=1/1

Ex. 86. Just major triads on 1/1, 3/2, and 4/3

The resulting scale consists of two dissimilar tetrachords separated by a just major tone, or the ratio 9/8, which in relation to the fundamental, or unison (1/1), presents the intervals of the just major tone (9/8), just major third (5/4), just perfect fourth (4/3), just perfect fifth (3/2), just major sixth (5/3), just diatonic major seventh (15/8), and octave (2/1).

297 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 22-26. The Harmonic Consideration 205

Ex. 87. The harmonic characteristics of the just diatonic scale

The Just Chromatic Scale

The next stage involves the construction of a “secondary group of triads” with 4:5:6 harmonic relationships to D, E, A, and B, or 9/8, 5/4, 5/3, and 15/8, which are produced via the following equations:

i. D (just major tone) generated by the ratio 9/8 ii. F!( (just tritone, or augmented fourth) by the equation 9/8 plus 5/4=(9Ï5):(8Ï4)=45/32 iii. A( (Pythagorean major sixth) by 9/8 plus 3/2=(9Ï3):(8Ï2)=27/16

iv. E (just major third) by the ratio 5/4 v. G! (augmented fifth) by the equation 5/4 plus 5/4=(5Ï5):(4Ï4)=25/16 vi. B (just diatonic major seventh) by 5/4 plus 3/2=(5Ï3):(4Ï2)=15/8

vii. A (just major sixth) by the ratio 5/3 viii. C! (grave or small just chromatic semitone) by the equation 5/3 plus 5/4=(5Ï5):(3Ï4)=25/12; and 25/12 minus 2/1=(25Ï1):(12Ï2)=25:24 ix. E (just major third) by 5/3 plus 3/2=(5Ï3):(3Ï2)=15/6; and 15/6 minus 2/1=(15Ï1):(6Ï2)=15/12=5/4

x. B (just diatonic major seventh) by the ratio 15/8 xi. D! (augmented second) by 15/8 plus 5/4=(15Ï5):(8Ï4)=75/32; and 75/32 minus 2/1=(75Ï1):(32Ï2)=75/64 xii. F!( (just tritone, or augmented fourth) by 15/8 plus 3/2=(15Ï3):(8Ï2)=45/16; and 45/16 minus 2/1=(45Ï1):(16Ï2)=45/32

206 The Harmonic Consideration

Ex. 88. Just major triads on 9/8, 5/4, 5/3, and 15/8

Of particular note is the fact that the A major triad now presents the grave or small just chromatic semitone (25/24), or what may be assumed to be the ‘sharp’ (!) ratio, and hence the ratio “applied to a diatonic note to produce the sharpened form of that note.” Following from that, in order to generate the correct ratios for the 4:5:6 triads built on D and B will require the raising of the 25/24 ratio by an additional interval of a syntonic comma (81/80). Theorists generally refer to this interval (the sum of 25/24 and 81/80) as the large limma (135/128, or 92.179 cents), which may be mathematically expressed as: 25/24 plus 81/80=(25Ï81):(24Ï80)=2025/1920=135/128. 135/128, and its reciprocal, 128/135, may also be theoretically referred to as the sharp, or greater limma, and flat, or hypolimma respectively.298 The generation of flats then utilizes Partch’s otonal and utonal theoretical basis to incorporate pitches derived from the complement or mirror image of the harmonics series. The retrograde of a C major triad – the initiating factor – is therefore utilized to produce an F minor triad, or what Fonville describes as an inversion explained as a “subharmonic relationship or subharmonic series descending from 1/1,” or in different terms; the intervallic complements of 1/1, 5/4, and 3/2, equal to 1/1, 8/5, and 4/3. The equations involved in obtaining 5:6:10 relationships (or appropriate 8/5s and 4/3s) for 1/1, 3/2, and 4/3 are:

i. C (unison) generated by the ratio 1/1 ii. A" (just minor sixth) by the equation 1/1 minus 5/4=(1Ï4):(1Ï5)=4/5; and 4/5 plus 2/1=(4Ï2):(5Ï1)=8/5 iii. F (just and Pythagorean perfect fourth) by 1/1 minus 3/2=(1Ï2):(1Ï3)=2/3; and 2/3 plus 2/1=(2Ï2):(3Ï1)=4/3

iv. G (just and Pythagorean perfect fifth) by the ratio 3/2 v. E" (just minor third) by the equation 3/2 minus 5/4=(3Ï4):(2Ï5)=12/10=6/5 vi C (unison) by 3/2 minus 3/2=(3Ï2):(2Ï3)=6/6=1/1

vii. F (just and Pythagorean perfect fourth) by the ratio 4/3

298 Alexander J. Ellis, “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments with Fixed Tones,” Proceedings of the Royal Society of London 13 (1863-64): 95. The Harmonic Consideration 207

viii. D") (just diatonic semitone, or major half-tone) by the equation 4/3 minus 5/4=(4Ï4):(3Ï5)=16/15 ix. B") (Pythagorean minor seventh) by 4/3 minus 3/2=(4Ï2):(3Ï3)=8/9; and 8/9 plus 2/1=(8Ï2):(9Ï1)=16/9

Ex. 89. Just minor triads on 1/1, 3/2, and 4/3

The chromatic sonorities in the F and G minor triads now present the subtraction of a grave or small just chromatic semitone (25/24) from a diatonic note, and hence, the introduction of the flat (") symbol. The symbol could for example be applied to D, or 9/8 (or any other diatonic interval for that matter) to produce the flattened form of that note or ratio simply by the subtraction of 25/24 from 9/8, or 9/8 minus 25/24=(9Ï24):(8Ï25)=216/200=27/25. Theorists generally refer to this interval as the great limma, acute or large half-tone, which measures 133.238 cents. The correct ratios for the 5:6:10 triad built on F require the lowering of the 9/8 and 9/5 ratios by an additional syntonic comma (81/80). For example, D"), or the just diatonic semitone (16/15) is the result of the subtraction of 25/24 and 81/80 (equal to 135/128) from 9/8, or 9/8 minus 135/128=(9Ï128):(8Ï135)=1152/1080=16/15. The otonal process produces the just minor scale (aeolian mode, natural minor, or descending melodic minor scale). The scale consists of two dissimilar tetrachords separated by a just major tone, or the ratio 9/8, which in relation to the fundamental, or unison (1/1) presents the intervals of just major tone (9/8), just minor third (6/5), just perfect fourth (4/3), just perfect fifth (3/2), just minor sixth (8/5), acute or large minor seventh (9/5), and octave (2/1).

Ex. 90. The just minor scale

208 The Harmonic Consideration

The process is then continued to include 5:6:10 relationships (or appropriate 8/5s and 4/3s) for 9/8, 5/4, 5/3, and 15/8, which are produced via the following equations:

i. D (just major tone) generated by the ratio 9/8 ii. B" (acute or large minor seventh) by the equation 9/8 minus 5/4=(9Ï4):(8Ï5)=36/40=9/5 iii. G (just and Pythagorean perfect fifth) by 9/8 minus 3/2=(9Ï2):(8Ï3)=18/24=3/4; and 3/4 plus 2/1=(3Ï2):(4Ï1)=6/4=3/2

iv. E (just major third) by the ratio 5/4 v. C (unison) by the equation 5/4 minus 5/4=(5Ï4):(4Ï5)=20/20=1/1 vi. A (just major sixth) by 5/4 minus 3/2=(5Ï2):(4Ï3)=10/12; and 10/12 plus 2/1=(10Ï2):(12Ï1)=20/12=5/3

vii. A (just major sixth) by the ratio 5/3 viii. F (just and Pythagorean perfect fourth) by the equation 5/3 minus 5/4=(5Ï4):(3Ï5)=20/15=4/3 ix. D) (just minor tone) by 5/3 minus 3/2=(5Ï2):(3Ï3)=10/9

x. B (just diatonic major seventh) by the ratio 15/8 xi. G (just and Pythagorean perfect fifth) by 15/8 minus 5/4=(15Ï4):(8Ï5)=60/40=3/2 xii. E (just major third) by 15/8 minus 3/2=(15Ï2):(8Ï3)=30/24=5/4

Ex. 91. Just minor triads on 9/8, 5/4, 5/3, and 15/8

Repeating the process of triadic construction to include all major and minor triads, as well as dominant triads to all the basic just scales degrees (1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, and 15/8) results in a twenty-five-note just enharmonic scale.299

299 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 108-11. The Harmonic Consideration 209

Ex. 92. The just enharmonic scale

Ben Johnston’s Fifty-Three-Tone Just Intonation Scale

Johnston’s elaboration of the twenty-five-note just enharmonic scale – premised on a system of infinite intervallic regeneration – firstly produces a twelve-note just scale, via the combination of the just diatonic and just minor scales, and the further subdivision of the two just major tones of the structure; the process resulting in the addition of D" and G" between C and D, and F and G respectively. The scale is then developed into a nineteen-note just enharmonic scale, via the establishment of an enharmonic equivalent of the previous twelve-note model, and therefore inclusive of C!, D!, F!, G!, and A!, as well as E! and F", and B! and C"; and finally, expanded into a 2, 3, 5-limit just intonation fifty-three-tone enharmonic scale that confers each of the nineteen pitches with a function as root, fifth, major third, and minor third – a process limited to the overlapping of the “twelve chromatic regions of the octave.” Each whole-tone, minor whole-tone, and diatonic semitone in Johnston’s just intonation fifty-three- tone enharmonic scale features the identical pattern of adjacent intervals, which include either the syntonic comma (81/80), diaskhisma (2048/2025),300 or grave or small diesis (3125/3072, or 29.614 cents). The grave or small diesis may be characterized as the difference between the grave or small just chromatic semitone (25/24) and the great diesis (128/125).301

300 “The diaskhisma is a small interval composed of two major thirds down and four perfect fifths down = 532 ßß 2411 , also expressed as the 2,3,5-monzo [11ß4,ß2>. Its ratio is 2048/2025 [=0.20 semitones =~19.55256881 or ~195/9 cents]. The term was used by Alexander Ellis in his English translation of Helmholtz, On the Sensations of Tone (1875). It had been referred to earlier by Rameau (Traité de l’harmonie, 1722) as the ‘diminished comma’. The standard epimoric approximation to the diaskhisma is the ratio 89:88 (=~19.56217479 cents), whose 2,3,5,11,89-monzo is ß3, 0, 0, ß1, 1>. Tuning treatises before c.1970 sometimes defined the diaskhisma as the 89:88 ratio without emphasis on the fact of its being an approximation, particularly in the German literature of c.1850-1950. A good example is Helmholtz in On the Sensations of Tone.” For a further discussion, see Monzo, “Encyclopedia of Microtonal Music Theory,” n. pag. 301 Ben Johnston, “Scalar Order as a Compositional Resource,” Perspectives of New Music 2.2 (Summer, 1964): 69- 73. 1 h amncConsideration Harmonic The 210 Table 126. Ben Johnston’s fifty-three-tone just intonation scale

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ C( syntonic comma 81/80 1.012500 264.896 21.506 +22 ÐÓ D$) diminished second, or great diesis 128/125 1.024000 267.905 41.059 +41 ÐÔ C! grave or small just chromatic semitone, or minor half-tone 25/24 1.041667 272.527 70.672 ß29 ÐÕ C!( Pythagorean acute or large limma 135/128 1.054688 275.933 92.179 ß08 ÐÖ D") just diatonic semitone, or major half-tone 16/15 1.066667 279.067 111.731 +12 Ð× D" great limma, acute or large half-tone 27/25 1.080000 282.556 133.238 +33 ÐØ C#( acute or large double augmented octave 1125/1024 1.098633 287.430 162.851 ß37 ÐÙ D) just minor tone 10/9 1.111111 290.695 182.404 ß18 ÑÐ D just major tone (9th harmonic) 9/8 1.125000 294.329 203.910 +04 ÑÑ D( acute or large tone 729/640 1.139063 298.008 225.416 +25 ÑÒ E$ diminished third 144/125 1.152000 301.393 244.969 +45 ÑÓ D! augmented second (75th harmonic) 75/64 1.171875 306.592 274.582 ß25 ÑÔ E") Pythagorean minor third, or trihemitone 32/27 1.185185 310.075 294.135 ß06 ÑÕ E" just minor third 6/5 1.200000 313.951 315.641 +16 ÑÖ D# double augmented second 625/512 1.220703 319.367 345.255 +45 Ñ× E) grave or small major third 100/81 1.234568 322.995 364.807 ß35 ÑØ E just major third (5th harmonic) 5/4 1.250000 327.032 386.314 ß14 ÑÙ E( Pythagorean major third, or ditone (81st harmonic) 81/64 1.265625 331.120 407.820 +08 ÒÐ F" diminished fourth 32/25 1.280000 334.881 427.373 +27 ÒÑ E! augmented third 125/96 1.302083 340.658 456.986 ß43 ÒÒ F) grave or small fourth 320/243 1.316872 344.527 476.539 ß23 ÒÓ F just and Pythagorean perfect fourth 4/3 1.333333 348.834 498.045 ß02 ÒÔ F( acute or large fourth 27/20 1.350000 353.195 519.551 +20 ÒÕ G$) grave or small double diminished fifth 512/375 1.365333 357.206 539.104 +39 ÒÖ F! grave or small augmented fourth 25/18 1.388889 363.369 568.717 ß31 DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) Ò× F!( just tritone, or augmented fourth (45th harmonic) 45/32 1.406250 367.911 590.224 ß10 ÒØ G") diminished fifth 64/45 1.422222 372.090 609.776 +10 ÒÙ G" acute or large diminished fifth 36/25 1.440000 376.741 631.283 +31 ÓÐ F#( acute or large double augmented fourth 375/256 1.464844 383.241 660.896 ß39 ÓÑ G) grave or small fifth 40/27 1.481481 387.593 680.449 ß20 ÓÒ G just and Pythagorean perfect fifth (3rd harmonic) 3/2 1.500000 392.438 701.955 +02 ÓÓ G( acute or large fifth 243/160 1.518750 397.344 723.014 +23 ÓÔ A$ diminished sixth 192/125 1.536000 401.857 743.014 +43 ÓÕ G! augmented fifth (25th harmonic) 25/16 1.562500 408.790 772.627 ß27 ÓÖ A") Pythagorean minor sixth 128/81 1.580247 413.433 792.180 ß08 Ó× A" just minor sixth 8/5 1.600000 418.601 813.686 +14 ÓØ G# double augmented fifth 625/384 1.627604 425.823 843.300 +43 ÓÙ A) grave or small major sixth 400/243 1.646091 430.659 862.852 ß37 ÔÐ A just major sixth 5/3 1.666667 436.043 884.359 ß16 ÔÑ A( Pythagorean major sixth (27th harmonic) 27/16 1.687500 441.493 905.865 +06 ÔÒ B$) diminished seventh 128/75 1.706667 446.508 925.418 +25 ÔÓ A! augmented sixth 125/72 1.736111 454.211 955.031 ß45 ÔÔ A!( acute or large augmented sixth 225/128 1.757813 459.889 976.537 ß23 ÔÕ B") Pythagorean minor seventh 16/9 1.777778 465.112 996.090 ß04 ÔÖ B" acute or large minor seventh 9/5 1.800000 470.926 1017.596 +18 T Ô× A#( acute or large double augmented sixth 1875/1024 1.831055 479.051 1047.210 +47 211 Consideration Harmonic he ÔØ B) grave or small major seventh 50/27 1.851852 484.492 1066.762 ß33 ÔÙ B just diatonic major seventh (15th harmonic) 15/8 1.875000 490.548 1088.269 ß12 ÕÐ B( Pythagorean major seventh 243/128 1.898438 496.680 1109.775 +10 ÕÑ C" diminished octave 48/25 1.920000 502.321 1129.328 +29 ÕÒ B! meantone augmented seventh (125th harmonic) 125/64 1.953125 510.987 1158.941 ß41 ÕÓ C) grave or small octave 160/81 1.975309 516.791 1178.494 ß22

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 212 The Harmonic Consideration

Harry Partch’s Forty-Three-Tone Just Intonation Scale

Partch’s forty-three-tone just intonation scale, with its inclusion of eleven-limit intervals, represents a harmonic expansion of Johnston’s 2, 3, 5-limit just intonation fifty-three-tone enharmonic scale. The aesthetic rationale behind discontinuing beyond the seven-limit is illustrated by Partch within these colourful remarks:

“The reasons why Monophony proceeds to the limit of 11 are basic and quite specific, as will be seen, but the reason for resting at the limit of 11 is a purely personal and arbitrary one. When a hungry man has a large table of aromatic and unusual viands spread before him he is unlikely to go tramping along the seashore and in the woods for still another exotic fare. And however sceptical he is of the many warnings regarding the unwholesomeness of his fare – like the ‘poison’ of the ‘love-apple’ tomato of a comparatively few generations ago – he has no desire to provoke further alarums.”

According to Partch, the harmonic argument behind the abovementioned intonation system is based on the premise that “the expansion of identities 1-3-5 through 7-9-11 provides a new and highly intriguing triad, and immediately makes possible a wide variety in quality.” The identities 7-9-11 represent the intervals 7/4, 9/8, and 11/8, and the ratio 7:9:11, which form an unusual variety of major triad consisting of the septimal supermajor third (9/7, or 435.084 cents) and the undecimal augmented fifth (11/7, or 782.492 cents). Twenty unique triads, fifteen unique tetrads, as well as six unique pentads are now made possible via the combination of the six identities of each individual otonality and utonality, and this is excluding the further augmentation of harmonic resources effectuated via the inversion and extension of simultaneous sonorities.

Ex. 93. Eleven-limit simultaneous sonorities i. The twenty eleven-limit triads

ii. The fifteen eleven-limit tetrads

The Harmonic Consideration 213 iii. The six eleven-limit pentads

With regards to the historical argument, Partch gives reference to Alexandrian astronomer, mathematician, and geographer of the second century, Claudius Ptolemy (c.87-150).302

“In Ptolemy’s scales there is enough evidence to warrant the conclusion that his procedure was generally governed by the principle of appropriating the smallest-number ratios permissible to the purpose of the scale in question. In this light it is quite natural that he should have used all the ratios of the 11-limit as a body.”

Partch’s forty-three-tone scale firstly adopts pairs of complementary five-limit intervals made available by the first five partials of the harmonic series – the octave (2/1), unison (1/1), just perfect fifth (3/2), just perfect fourth (4/3), just major third (5/4), and just minor third (6/5) – to then incorporate seven-limit and eleven-limit intervallic ratios to construct a scale of twenty-nine degrees. To resolve some of the scalar discontinuities of this unequal scale, the system is then expanded to include secondary ratios, which result in a scale consisting of forty-three-tones. An example of one of these secondary ratios is 33/32 (undecimal comma, or 33rd harmonic), which is derived by calculating the 3/2 of 11/8 via the equation: 11/8 plus 3/2=(11Ï3):(8Ï2)=33/16; followed by 33/16 minus 2/1=(33Ï1):(16Ï2)=33/32. The final set of intervals include four types of tritones: the undecimal superfourth (11/8), septimal subdiminished fifth (7/5), and their complements; septimal tritone, or superaugmented fourth (10/7), and undecimal subfifth (16/11); six types of thirds: the septimal subminor third (7/6), Pythagorean minor third (32/27), just minor third (6/5), undecimal acute or large neutral third (11/9), just major third (5/4), septimal supermajor third (9/7), and their complements; the septimal subminor sixth (14/9), just minor sixth (8/5), undecimal grave or small neutral sixth (18/11), just major sixth (5/3), Pythagorean major sixth (27/16), and septimal supermajor sixth (12/7); seven types of seconds: the septimal diatonic semitone (21/20), just diatonic semitone (16/15), undecimal grave or small neutral second (12/11), undecimal acute or large neutral second (11/10), just minor tone (10/9), just major tone (9/8), septimal supermajor second (8/7), and their complements; the septimal subminor seventh (7/4), Pythagorean minor seventh (16/9), acute or large minor seventh (9/5), undecimal grave or small neutral seventh (20/11), undecimal acute or large neutral seventh (11/6), just diatonic major seventh (15/8), and septimal supermajor seventh (40/21).303

302 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 123-35. 303 Bob Gilmore, “On Harry Partch’s Seventeen Lyrics by Li Po,” Perspectives of New Music 30.2 (Summer, 1992): 26-27. 1 h amncConsideration Harmonic The 214 Table 127. Harry Partch’s forty-three-tone just intonation scale

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ C( syntonic comma 81/80 1.012500 264.896 21.506 +22 ÐÓ C, undecimal comma (33rd harmonic) 33/32 1.031250 269.801 53.273 ß47 ÐÔ D; septimal chromatic semitone 21/20 1.050000 274.707 84.467 ß16 ÐÕ D") just diatonic semitone, or major half-tone 16/15 1.066667 279.067 111.731 +12 ÐÖ D- undecimal grave or small neutral second 12/11 1.090909 285.410 150.637 ß49 Ð× DE) undecimal acute or large neutral second 11/10 1.100000 287.788 165.004 ß35 ÐØ D) just minor tone 10/9 1.111111 290.695 182.404 ß18 ÐÙ D just major tone (9th harmonic) 9/8 1.125000 294.329 203.910 +04 ÑÐ D*) septimal supermajor second 8/7 1.142857 299.001 231.174 +31 ÑÑ E; septimal subminor third 7/6 1.166667 305.230u 266.871 ß33 ÑÒ E") Pythagorean minor third, or trihemitone 32/27 1.185185 310.075 294.135 ß06 ÑÓ E" just minor third 6/5 1.200000 313.951 315.641 +16 ÑÔ EE) undecimal acute or large neutral third 11/9 1.222222 319.765 347.408 +47 ÑÕ E just major third (5th harmonic) 5/4 1.250000 327.032 386.314 ß14 ÑÖ FK( undecimal diminished fourth 14/11 1.272727 332.978 417.508 +18 Ñ× E* septimal supermajor third 9/7 1.285714 336.376 435.084 +35 ÑØ F+( septimal subfourth (21st harmonic) 21/16 1.312500 343.384 470.781 ß29 ÑÙ F just and Pythagorean perfect fourth 4/3 1.333333 348.834 498.045 ß02 ÒÐ F( acute or large fourth 27/20 1.350000 353.195 519.551 +20 ÒÑ F, undecimal superfourth (11th harmonic) 11/8 1.375000 359.735 551.318 ß49 DEGREE NOTE INTERVAL RATIO RATIO FREQUENCY CENTS TUNING NUMBER (FRACTION) (DECIMAL) (HERTZ) ÒÒ G; septimal subdiminished fifth 7/5 1.400000 366.276 582.512 ß17 ÒÓ F: septimal tritone, or superaugmented fourth 10/7 1.428571 373.751 617.488 +17 ÒÔ G- undecimal subfifth 16/11 1.454545 380.546 648.682 +49 ÒÕ G) grave or small fifth 40/27 1.481481 387.593 680.449 ß20 ÒÖ G just and Pythagorean perfect fifth (3rd harmonic) 3/2 1.500000 392.438 701.955 +02 Ò× G*) septimal superfifth 32/21 1.523810 398.668 729.219 +29 ÒØ A; septimal subminor sixth 14/9 1.555556 406.973 764.916 ß35 ÒÙ GJ) undecimal augmented fifth 11/7 1.571429 411.126 782.492 ß18 ÓÐ A" just minor sixth 8/5 1.600000 418.601 813.686 +14 ÓÑ A-( undecimal grave or small neutral sixth 18/11 1.636364 428.115 852.592 ß47 ÓÒ A just major sixth 5/3 1.666667 436.043 884.359 ß16 ÓÓ A( Pythagorean major sixth (27th harmonic) 27/16 1.687500 441.493 905.865 +06 ÓÔ A* septimal supermajor sixth 12/7 1.714286 448.501 933.129 +33 ÓÕ B; septimal subminor seventh (7th harmonic) 7/4 1.750000 457.845 968.826 ß31 ÓÖ B") Pythagorean minor seventh 16/9 1.777778 465.112 996.090 ß04 Ó× B" acute or large minor seventh 9/5 1.800000 470.926 1017.596 +18 T

ÓØ B- undecimal grave or small neutral seventh 20/11 1.818182 475.683 1034.996 +35 215 Consideration Harmonic he ÓÙ BE) undecimal acute or large neutral seventh 11/6 1.833333 479.647 1049.363 +49 ÔÐ B just diatonic major seventh (15th harmonic) 15/8 1.875000 490.548 1088.269 ß12 ÔÑ B*) septimal supermajor seventh 40/21 1.904762 498.334 1115.533 +16 ÔÒ C- undecimal subdiminished octave 64/33 1.939394 507.395 1146.727 +47 ÔÓ C) grave or small octave 160/81 1.975309 516.791 1178.494 ß22

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 216 The Harmonic Consideration

Adriaan Daniël Fokker’s Thirty-One-Tone Equally-Tempered Division of the Octave

According to Wolf, Dutch physicist, music theorist, and composer Adriaan Daniël Fokker (1987-1972)304 “appears to be the first theorist to represent a 7-limit tuning system graphically with three implied axes or dimensions of tonal space, the horizontal axis is assigned to fifths (3/2s), the vertical axis to major thirds (5/4s), and an oblique axis, implying a third dimension, to tones generated by the 7/4 relationship.” Fokker, inspired by the musical writings of Dutch mathematician, astronomer, and physicist Christian Huygens (1629-95), became an ardent advocate of the tonal resources of seven-limit just intonation, and their “practical realization” within the domain of thirty-one-tone equal temperament.305 Huygens, via the recognition of the significance of the interval of the great diesis (128/125), or one-fifth of a whole-tone

5 9 ( 8 ), which is calculated via the amount the octave (2/1) exceeds three consecutive just major thirds (2/1÷125/64[5/4Ï5/4Ï5/4]=128/125, or 1200ß1158.941=41.059 cents), developed a theoretical basis for the “octave partitioned into thirty-one steps.” Fokker’s extensive research into thirty-one-tone equal temperament appears in his 1966 publication of New Music with 31 Notes.306 Notably, the thirty-one-tone equally-tempered division of the octave ( 31 2 ) produces the just major third (5/4) and septimal subminor seventh (7/4), with a falsity of +0.783 and ß1.084 cents on each count, which is a “faithful rendering” of the former, while a “nearly exact reproduction” of the latter. The untrigesimal, or ‘tricesimoprimal’ equal temperament is therefore capable of approximating the septimal dominant seventh tetrad (4:5:6:7) with the intervallic measurements of 387.097, 696.774, and 967.742 cents, which are extremely close to the ‘true’ seven-limit equivalents (386.314, 701.955, and 968.826 cents).307 “Thirty-one equal has an honourable history,” notes Rapoport, and “partly because it has excellent approximations to the intervals represented by harmonics five and seven, and is the closed regular expansion of quarter-comma meantone temperament.”308

304 “Fokker was born to Dutch parents in Indonesia. He received his doctorate in physics at Leiden in 1913, pursued advanced work with Rutherford and Einstein (with whom he seems to have collaborated in research and publication), and became a distinguished theoretical physicist and professor, with important publications on relativity theory, atomic physics, acoustics, electricity, and magnetism.” For a further discussion, see Douglas Leedy, “Selected Musical Compositions (1948-1972),” Notes 46.1 (Sep., 1989): 224. 305 Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 8. 306 “New Music with 31 Notes appeared originally in German in 1966. It is a distillation of the ideas which have punctuated his theoretical writings and make a fine introduction to the works of this seminal figure. The book is in two parts, the first narrating the history of Fokker’s involvement in 31-tone temperament and the early fruits of this interest. The longer second part is theoretical and deals with the aspects of temperament which have especially preoccupied Fokker.” For a further discussion, see Joel Mandelbaum, “Toward the Expansion of Our Concepts of Intonation,” Perspectives of New Music 13.1 (Autumn-Winter, 1974): 220. 307 A. D. Fokker, “Equal Temperament and the Thirty-One-Keyed Organ,” The Scientific Monthly 81.4 (Oct., 1955): 162-63. 308 Paul Rapoport, “Towards the Infinite Expansion of Tonal Resources,” Tempo 144 (Mar., 1983): 9. Table 128. Adriaan Daniël Fokker’s thirty-one-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE NOTE INTERVAL FACTOR RATIO FREQUENCY CENTS TUNING NUMBER (DECIMAL) (HERTZ) ÐÑ C unison 1/1 1.000000 261.626 0.000 +00 ÐÒ Cà / D$ 31-et superoctave, or diminished second 31 2 1.022611 267.541 38.710 +39 ÐÓ C! / Dé 31-et augmented octave, or subminor second (31 2) 2 1.045734 273.591 77.419 ß23 ÐÔ Cè / D" 31-et superaugmented octave, or minor second (31 2)3 1.069380 279.777 116.129 +16 ÐÕ C# / Dá 31-et double augmented octave, or neutral second (31 2) 4 1.093560 286.103 154.839 ß45 ÐÖ D 31-et major tone (31 2) 5 1.118287 292.572 193.548 ß06 Ð× Dà / E$ 31-et supermajor second, or diminished third (31 2) 6 1.143573 299.188 232.258 +32 ÐØ D! / Eé 31-et augmented second, or subminor third (31 2)7 1.169431 305.953 270.968 ß29 ÐÙ Dè / E" 31-et superaugmented second, or minor third (31 2) 8 1.195873 312.871 309.677 +10 ÑÐ D# / Eá 31-et double augmented second, or neutral third (31 2) 9 1.222914 319.945 348.387 ß52 ÑÑ E 31-et major third (31 2)10 1.250566 327.180 387.097 ß13 ÑÒ Eà / F" 31-et supermajor third, or diminished fourth (31 2)11 1.278843 334.578 425.806 +26 ÑÓ E! / Fá 31-et augmented third, or subfourth (31 2)12 1.307759 342.143 464.516 ß35 ÑÔ F 31-et perfect fourth (31 2)13 1.337329 349.880 503.226 +03 ÑÕ Fà / G$ 31-et superfourth, or diminished fifth (31 2)14 1.367568 357.791 541.935 +42 ÑÖ F! / Gé 31-et augmented fourth, or subdiminished fifth (31 2)15 1.398491 365.881 580.645 ß19 Ñ× Fè / G" 31-et superaugmented fourth, or diminished fifth (31 2)16 1.430113 374.154 619.355 +19 ÑØ F# / Gá 31-et double augmented fourth, or subfifth (31 2)17 1.462450 382.614 658.065 ß42 ÑÙ G 31-et perfect fifth (31 2)18 1.495518 391.266 696.774 ß03 ÒÐ Gà / A$ 31-et superfifth, or diminished sixth (31 2)19 1.529334 400.113 735.484 +35 ÒÑ G! / Aé 31-et augmented fifth, or subminor sixth (31 2) 20 1.563914 409.160 774.194 ß26 è " 31-et superaugmented fifth, or minor sixth 31 21 1.599276 418.412 812.903 ÒÒ G / A ( 2) +13 T ÒÓ G# / Aá 31-et double augmented fifth, or neutral sixth (31 2)22 1.635438 427.872 851.613 ß48 217 Consideration Harmonic he ÒÔ A 31-et major sixth (31 2) 23 1.672418 437.547 890.323 ß10 ÒÕ Aà / B$ 31-et supermajor sixth, or diminished seventh (31 2)24 1.710234 447.441 929.032 +29 ÒÖ A! / Bé 31-et augmented sixth, or subminor seventh (31 2)25 1.748905 457.558 967.742 ß32 Ò× Aè / B" 31-et superaugmented sixth, or minor seventh (31 2) 26 1.788450 467.904 1006.452 +06 ÒØ A# / Bá 31-et double augmented sixth, or neutral seventh (31 2)27 1.828889 478.484 1045.161 ß55 ÒÙ B 31-et major seventh (31 2)28 1.870243 489.303 1083.871 ß16 ÓÐ Bà / C" 31-et supermajor seventh, or diminished octave (31 2) 29 1.912532 500.367 1122.581 +23 ÓÑ B! / Cá 31-et augmented seventh, suboctave (31 2)30 1.955777 511.681 1161.290 ß39

ÐÑ C octave 2/1 2.000000 523.251 1200.000 +00 218 The Harmonic Consideration

The notation of fifth tones, or the equal untrigesimal equal diesis ( 31 2 , or 38.710 cents) of thirty-one equal temperament, was devised by Fokker, together with Dutch composer Henk Badings (1907-87),309 and is represented with the following nine symbols: semi-sharp (à), sharp (!), sesqui-sharp (è), double sharp (#), semi-flat (á), flat ("), sesqui-flat (é), double flat ($), and natural ('). The selection of symbols also represents the recommendations made by the International Musicological Society, following their 1967 meeting in Ljubljana, Slovenia, where it was agreed upon that the “best microtonal symbology” should incorporate the “five standard accidental signs,” along with the semi-sharp and sesqui-sharp symbols, as well as the semi-flat and sesqui-flat symbols introduced by Italian composer and violinist Guiseppe Tartini (1692-1770) in 1754.310

La Homa Kanto for Harmonically Tuned Synthesizer Quartet

La Homa Kanto, or ‘The Human Song’ in Esperanto,311 is a dedication to the late American composer Lou Harrison (1917-2003),312 and its pitch material has been derived directly from Harrison’s five-tone scales, presented in Lou Harrison’s Music Primer: Various Items About Music to 1970. Included are the first five in the series, with the first (the diatonic or major pentatonic scale) acknowledged by Harrison as the “prime pentatonic,” and “practically the Human Song.” According to Harrison, “These first five are the most widespread, the core, the principal modes of Human Music. They also constitute the bone-work, the firmest compositional basis for seven-tone music.”313

Ex. 94. Major Pentatonic Scale “The Human Song”

309 Richard Orton, “The 31-Note Organ,” The Musical Times 107.1478 (Apr., 1966): 342. 310 Read, 20th-Century Microtonal Notation 19-20. 311 Esperanto is “an artificial language invented as a means of international communication.” The origin of the name derived from “Dr. Esperanto, a pen name of the inventor.” For a further discussion, see Catherine Soanes, ed., Oxford Dictionary of Current English, 3rd ed. (Lodon: Oxford U. Press, 2001) 304. 312 “Lou Harrison, who celebrated his eightieth birthday in 1997, has often been cited as one of America’s most original and influential composers. In addition to his prolific musical output, Harrison is also a skilled painter, calligrapher, essayist, critic, poet, and instrument-builder. During his long and varied career, he has explored dance, Asian music, tuning systems, and universal languages, and has actively championed political causes ranging from pacifism to gay rights.” For a further discussion, see Miller, and Lieberman, Lou Harrison: Composing a World n.pag. 313 Lou Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 (New York: C. F. Peters, 1971) 27- 29. The Harmonic Consideration 219

In his music primer, Harrison also states that the “fullest musical bounty of any tradition is the mode,” and expresses the following sentiments about five-tone modes:

“Time’s endless flowering of tone and tune here streams across to each; embodied matrix song; hearts gift of ancestry and dreams. Behind the singer’s song stands quietly (or dancing) the strange remembered body of the mode. Or is it fashioned piecewise (that body) from the rhymes and purlings of the tune? Sure, scales are not. These sit, emotive, on systematic right. Perhaps modes meld from melodies themselves.”314

Modes one, two, and five may be considered to be anhemitonic pentatonic forms (scales incorporating half steps, or semitones), while three and four, hemitonic pentatonic forms (scales not incorporating half steps, or semitones).315 Harrison’s five pentatonic scales include the diatonic or major pentatonic, minor pentatonic (or fifth mode major pentatonic), Japanese hirajoshi, fifth mode Indonesian pélog, and Indonesian sléndro (or second mode major pentatonic). The consideration of their complements presents the third mode major pentatonic, fourth mode major pentatonic, third mode Indonesian pélog, fourth mode Japanese hirajoshi, and Indonesian sléndro (or second mode major pentatonic).316

Ex. 95. Lou Harrison’s five pentatonic scales i. Lou Harrison pentatonic #2

ii. Lou Harrison pentatonic #2

314 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 27. 315 J. H. Kwabena Nketia, The Music of Africa (New York: W. W. Norton & Co., 1974) 118. 316 Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice 50-51. 220 The Harmonic Consideration

ii. Lou Harrison pentatonic #3

iv. Lou Harrison pentatonic #4

v. Lou Harrison pentatonic #5

The Harmonic Consideration 221

The Harpsichord Sample

The ‘1967 William Dowd French Double Harpsichord’ samples (recorded in a recital room space by sound designer and programmer Peter Grech at the School of Music, Victorian College of the Arts, Melbourne, Australia, utilizing an Audio-Technica AT4050/CM5 condenser microphone [with switchable cardioid, omnidirectional, or figure-8 operation, and a frequency response from 20Hz-20kHz] and an Akai S3000XL Midi Stereo Digital Sampler) include two sets of twenty (five-octave span) samples (encompassing the upper and lower manuals of the harpsichord), as well as one corresponding set of keyboard release clicks. All these sounds multi-sampled on an Akai S3000XL Midi Stereo Digital Sampler – tuned firstly to standard A=440Hz twelve-tone equal temperament, and then modified within nineteen patches collectively adhering to the tuning matrixes required to represent the ten unique systems of just intonation utilized in the composition. In view of the superior aesthetic qualities of the upper manual, and for the sake of harmonic clarity, the lower manual and keyboard release clicks are omitted from the final 63-key ‘1967 William Dowd French Double Harpsichord’ sample. The technical requirements for the realization of the live performance of the work is an Akai S3000XL Midi Stereo Digital Sampler, together with four 61-key Midi Keyboards.

Fig. 7. Alex Pertout, 1967 William Dowd French Double Harpsichord.317

317 1967 William Dowd French Double Harpsichord, personal photograph of Alex Pertout, 3 Feb. 2007. 222 The Harmonic Consideration

Ben Johnston’s System of Notation

The system of notation utilized in the score is based on Johnston’s ‘extended just intonation’. It contains twenty-three unique symbols, which may be combined in any manner – five identical to conventional sharp (Ú), flat (Û), double sharp (#), double flat ($), and natural (Ö) accidentals (raising, lowering or neutralizing a tone by 25/24 and 625/576, or 70.672 and 141.345 cents). Additional symbols include (() and ()), raising or lowering a tone by 81/80 (one syntonic comma), or 21.506 cents, as well as the following utonal and otonal sets of symbols for each partial chroma up to the thirty-first harmonic: (+) and (*), lowering or raising a tone by 36/35 (one septimal comma), or 48.770 cents; (,) and (-), raising or lowering a tone by 33/32 (one undecimal comma), or 53.273 cents; (.) and (/), raising or lowering a tone by 65/64 (one tridecimal comma), or 26.841 cents; (0) and (1), raising or lowering a tone by 51/50 (one septendecimal comma), or 34.283 cents; (2) and (3), lowering or raising a tone by 96/95 (one nonadecimal comma), or 18.128 cents; (4) and (5), raising or lowering a tone by 46/45 (one trivigesimal comma), or 38.051 cents; (6) and (7), raising or lowering a tone by 145/144 (one nonavigesimal comma), or 11.981 cents; and finally, (8) and (9), raising or lowering a tone by 31/30 (one untrigesimal comma), or 56.767 cents.318 Johnston “infinitely expandable” system of notation for extended just intonation categorizes commas above the seventh harmonic under the nomenclature of ‘chromas’. “These accidentals are used in combination for the more complex ratios,” notes Bob Gilmore, with reference to Johnston’s not uncommon use of “three such symbols applied to one note.”319

Table 129. Ben Johnston’s notational symbols for just intonation commas and chromas through thirty-one

RAISE LOWER INTERVAL RATIO CENTS AMOUNT BY EXCEEDS (FRACTION) WHICH… Ú Û chromatic semitone 25/24 70.672 5/4 6/5 ( ) syntonic comma 81/80 21.506 9/8 10/9 * + septimal comma 36/35 48.770 9/5 7/4 , - undecimal comma 33/32 53.273 11/8 4/3 . / tridecimal comma 65/64 26.841 13/8 8/5 0 1 septendecimal comma 51/50 34.283 17/16 25/24 3 2 nonadecimal comma 96/95 18.128 6/5 19/16 4 5 trivigesimal comma 46/45 38.051 23/16 45/32 6 7 nonavigesimal comma 145/144 11.981 29/16 9/5 8 9 untrigesimal comma 31/30 56.767 31/16 15/8

318 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 109-15. 319 Bob Gilmore, “Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben Johnston, and James Tenney,” Perspectives of New Music 33.1/2 (Winter-Summer, 1995): 480. The Harmonic Consideration 223

Compositional Strategy

Harrison offers the following commentary about the microtonal compositional process, and the utilization of just intonation principles:

“After only a brief study of intervals it becomes clear that there are two ways of composing with them: 1) arranging them into a fixed mode, or gamut, and then composing within that structure. This is Strict Style, and is the vastly predominant world method. However, another way is possible – 2) to freely assemble, or compose with whatever intervals one feels that he needs as he goes along. This is Free Style, and I used this method first in my Simfony in Free Style.”320

The compositional strategy for the work involves the creation of two complimentary sets of melodic material, consisting of what is essentially a four-bar melodic sentence based on Harrison’s first pentatonic (or major pentatonic), with an additional four-bar variant based on its complement (or third mode major pentatonic). In some aspects, this second sentence could be considered an almost comparable inversion, although it does not represent an ‘authentic’ inversion in the serial sense of the word, but rather an inversion of contour utilizing the inverted pitch material of the original major pentatonic. The third bar of both versions represents the only literal inversion of intervals. A further observation reveals the respective major and minor tonalities of the two four-bar melodic sentences.

Ex. 96. Two four-bar melodic sentences i. Four-bar melodic sentence

ii. Complimentary sentence

The work also features ten distinct tuning modulations: three-limit, five-limit, seven-limit, eleven-limit, thirteen-limit, seventeen-limit, nineteen-limit, twenty-three-limit, twenty-nine-limit, and thirty-one-limit just intonation systems, based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth, twenty-third, twenty-ninth, and thirty-first partials of the harmonic series – each system adding its own microtonal nuances to the recurring melodic material, which is further transformed via the introduction of alternative scalar material, as well as via harmonic development pertinent to each individual just intonation

320 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 6. 224 The Harmonic Consideration system. The structural framework of the composition is directly related to the primary motive of the four- bar melodic sentence. In five-limit terms, the primary motive denotes the following intervals: 5/4, 3/2, and 9/8, and 5/4, and 3/2; and is expressed in the score as section one (rehearsal letters A and B – incorporating Lou Harrison Pentatonic #1, and both three-limit and five-limit just intonation), which is designated a 5/4 metrical structure; section 2 (rehearsal letters C and D – incorporating Lou Harrison Pentatonic #2, and both seven-limit and eleven-limit just intonation), a 3/2 metrical structure; section 3 (rehearsal letters E and F – incorporating Lou Harrison Pentatonic #3, and both thirteen-limit and seventeen-limit just intonation), a 9/8 metrical structure; section 4 (rehearsal letters G and H – incorporating Lou Harrison Pentatonic #4, and both nineteen-limit and twenty-three-limit just intonation), a 5/4 metrical structure; and section 5 (rehearsal letters I and J – incorporating Lou Harrison Pentatonic #5, and both twenty-nine-limit and thirty-one-limit just intonation), a 12/8 metrical structure (equal to 3/2). A sequential series of metrical structures is therefore utilized in the work to reflect the primary motive and its frequency ratios of 5/4, 3/2, 9/8, 5/4, and 3/2.

Composing With Melodicles

According to Harrison, composing with melodicles, or neumes “in some form is the oldest known method of musical composition, probably deriving from Mesopotamia and Egypt. One makes a mosaic, so to speak.” Acknowledging Henry Cowell as the source for this technique, Harrison proposes that beginning with a selection of melodicles, the composer may: combine melodicles – essentially motivic material – to form phrases; diatonically transpose inside, or chromatically transpose outside of a specific mode; invert, retrograde, or retrograde-invert melodicles; while maintaining pitch integrity, alter rhythmic design; as well as individually or collectively displace octaves.321 The abovementioned is no doubt representative of a series of elementary suggestions later subjected to considerable development in the compositional process. The technique is adopted in La Homa Kanto; theorized utilizing the first bar of the work’s principal melodic sentence (a two-note motive), and remodelled within the following three categories of motivic manipulation: melodic transformation of motive, rhythmic transformation of motive, and harmonic transformation of motive.

Ex. 97. Original, complement, retrograde, complement retrograde of motive

321 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 1. The Harmonic Consideration 225

The following series of examples represent a selection of the first category of motivic manipulation, or the melodic transformation of motive.

Ex. 98. Melodic transformation of motive i. Motivic diatonic expansion (start)

ii. Motivic diatonic expansion (middle)

iii. Motivic diatonic expansion (end)

iv. Motivic diatonic expansion (upbeat)

v. Motivic chromatic expansion (start)

vi. Motivic chromatic expansion (middle)

vii. Motivic chromatic expansion (end)

226 The Harmonic Consideration viii. Motivic chromatic expansion (upbeat)

ix. Motivic exclusion

x. Intervallic inclusion

xi. Motivic diatonic ornamentation

xii. Motivic chromatic ornamentation

xiii. Intervallic augmentation

xiv. Intervallic diminution

xv. Intonational reinterpretation

The Harmonic Consideration 227 xvi. Intervallic saturation

xvii. Intervallic redirection

xviii. Intervallic reordering

xix. Diatonic transposition

xx. Chromatic transposition

xxi. Contour transposition

xxii. Conjunct octave displacement

xxiii. Disjunct octave displacement

228 The Harmonic Consideration

The following series of examples represent a selection of the second category of motivic manipulation, or the rhythmic transformation of motive.

Ex. 99. Rhythmic transformation of motive i. Pitch recapitulation

ii. Motivic recapitulation

iii. Motivic ostinato

iv. Rhythmic recapitulation

v. Rhythmic augmentation

vi. Rhythmic diminution

vii. Irregular rhythmic augmentation

The Harmonic Consideration 229 viii. Irregular rhythmic diminution

ix. Conjunct rhythmic displacement

x. Disjunct rhythmic displacement

xi. Rhythmic serialism

xii. Polyrhythmic representation

xiii. Time signature change

The following series of examples represent a selection of the third category of motivic manipulation, or the harmonic transformation of motive.

Ex. 100. Harmonic transformation of motive i. Diatonic harmonic extension

230 The Harmonic Consideration ii. Chromatic harmonic extension

iii. Two-part canon

iv. Two-part rhythmically displaced canon

v. Two-part inverted canon

vi. Two-part complementary canon

vii. Three-part canon

viii. Double canon

The Harmonic Consideration 231 ix. Augmentation canon

x. Diminution canon

Three-Limit Just Intonation

The primary interval of three-limit just intonation is the third harmonic, or just perfect fifth, which is represented by the frequency ratio 3/2, and in relation to C (1/1), notated simply as G. The complement, 4/3, or just perfect fourth, is notated simply as F. The notational symbols introduced in three-limit just intonation ([(] and [)], raising or lowering a tone by 81/80 [one syntonic comma], or 21.506 cents) are in reference to the amount a series of four fifths minus a major third exceeds the unison (3/2Ï3/2Ï3/2Ï3/2÷5/4=81/80).322 The essential three-limit pitch material is arrived at via a one-dimensional process, and the generation of an ascending series of fifths from the fundamental C (1/1) to produce the six ‘otonal’ pitches: G, D, A(, E(, B(, and F!(( (3/2, 9/8, 27/16, 81/64, 243/128, and 729/512); while a descending series to produce the other five ‘utonal’ pitches: F, B"), E"), A"), D")), and G")) (4/3, 16/9, 32/27, 128/81, 256/243, and 1024/729). The pitch material is then sequentially arranged as:

C D")) D E") E( F F!(( G")) G A") A( B") B 1 256 9 32 81 4 729 1024 3 128 27 16 40

1 243 8 27 64 3 512 729 2 81 16 9 21

The series of intervals presents the Pythagorean limma (256/243), just major tone (9/8), Pythagorean minor third (32/27), Pythagorean major third (81/64), just perfect fourth (4/3), Pythagorean tritone, or augmented fourth (729/512), Pythagorean diminished fifth (1024/729), just perfect fifth (3/2), Pythagorean minor sixth (128/81), Pythagorean major sixth (27/16), Pythagorean minor seventh (16/9), and Pythagorean major seventh (243/128). The following table depicts the essential tonal resources of three-limit just intonation, indicating degree, notation, interval, ratio, and cents.

322 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 452-53. 232 The Harmonic Consideration

Table 130. Essential three-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ D")) Pythagorean limma 256/243 1.053498 90.225 ÐÓ D just major tone (9th harmonic) 9/8 1.125000 203.910 ÐÔ E") Pythagorean minor third, or trihemitone 32/27 1.185185 294.135 ÐÕ E( Pythagorean major third, or ditone (81st harmonic) 81/64 1.265625 407.820 ÐÖ F just and Pythagorean perfect fourth 4/3 1.333333 498.045 Ð× F!(( Pythagorean tritone, or augmented fourth 729/512 1.423828 611.730 ÐØ G just and Pythagorean perfect fifth (3rd harmonic) 3/2 1.500000 701.955 ÐÙ A") Pythagorean minor sixth 128/81 1.580247 792.180 ÑÐ A( Pythagorean major sixth (27th harmonic) 27/16 1.687500 905.865 ÑÑ B") Pythagorean minor seventh 16/9 1.777778 996.090 ÑÒ B( Pythagorean major seventh 243/128 1.898438 1109.775

ÐÑ C octave 2/1 2.000000 1200.000

The adaptation of Harrison’s pentatonic #1 to three-limit intonation presents a collection of pitches that include: C, D, E(, G, and A( (1/1, 9/8, 81/64, 3/2, and 27/16); as well as the complements: C, B"), A"), F, and E") (2/1, 16/9, 128/81, 4/3, and 32/27). It must be noted that auxiliary pitches will be added to the original pentatonic forms in order to beget a heptatonic scalar source.

Ex. 101. Lou Harrison pentatonic #1 (three-limit intonation)

The Harmonic Consideration 233

The following tuning matrix contains the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 131. 3-limit tuning matrix no. 1 ‘key of C’ (Program 01)

NOTE C D")) D E") E( F – – G A") A( B") B(

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß10 +04 ß06 +08 ß02 +00 +02 ß08 +06 ß04 +10

The first four bars of the composition serve to introduce the four-bar melodic sentence in three-limit intonation, and what follows is essentially a two-part inverted canon highlighting the complement or inversion of the principal melodic material. The end of the second system presents a D"("5/omit 3) simultaneous sonority within a 7-1 suspension in the bass (the pitches D")), D")), and G, or the ratio 243:512:729), while the following harmony delineates D"("5)/F (the pitches F, D")) and G, or the ratio 324:512:729). In the latter example, D")) and G represents the interval of a Pythagorean tritone (729/512), and F and D")), a Pythagorean minor sixth (128/81, or 792.180 cents). The third system now introduces the four-bar variant based on the complement pentatonic scale. A hint of the C fundamental highlights two characteristic intervals of three-limit just intonation: B"), or Pythagorean minor seventh (16/9, or 996.090 cents), and A"), or Pythagorean minor sixth (128/81). Another two-part inverted canon in the fourth system produces the succession: E") and B"), A") and D, and A") and E(, which present the just perfect fifth (3/2), Pythagorean diminished fifth (1024/729, or 588.270 cents), and Pythagorean diminished fourth (8192/6561, or 384.360 cents). Another interesting simultaneous sonority presented in the fourth system is one proposed by pitches E") and G, which pronounce the Pythagorean major third, or ditone (81/64, or 407.820 cents). The nomenclature (ditone) is in direct reference to the resulting interval derived from the summation process of two just major tones, or 9/8×9/8=81/64. It is important to note that whilst the three-limit intonation system renders consonant just perfect fourths and fifths, as well as the just major tone (9/8), its one-dimensional process of juxtaposed ascending fifths (series of 3/2s) and descending fourths (series of 4/3s) presents thirds and sixths as complex and dissonant intervals. Kyle Gann offers the following discussion with regards to the 81/64 interval:

“Before the advent of meantone tuning, the French academy at Notre Dame (13th and 14th centuries) followed a medieval tradition since Boethius (4th century) in decreeing that only a series of perfect fifths could make up a scale; their ratio was 3/2, and 3, after all, was the perfect number, connoting the Trinity among other things. Thus the Pythagorean scale is a just intonation scale on a series of perfect fifths, all the ratio numbers powers of either 3 or 2. This was an appropriate scale for a music in which perfect fifths 234 The Harmonic Consideration

and fourths were the overwhelmingly dominant simultaneous sonority, and in which the pitches CÚ, FÚ, and GÚ hardly appeared if at all. Though used, the thirds were theoretical dissonances, and therefore avoided at final cadences.”323

The root position three-limit major triad is represented by the ratio 64:81:96, identities 1-81-3, intervals 1/1, 81/64, and 3/2, and the pitches C, E(, and G; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of +21.506 and +0.000 cents on each count, which is a deviation of a syntonic comma (81/80) for the major third. G4, or the just perfect fifth (3/2), presents 0.000 beats between the third harmonic of C4 (1/1) and the second harmonic of G4, and 0.000 beats between the sixth harmonic of C4 and the fourth harmonic of G4; while E(4 (331.120Hz), or the Pythagorean major third (81/64), presents 16.352 beats between the fifth harmonic of C4 and the fourth harmonic of E(4 (1324.479Hz).

Table 132. The beating characteristics of the three-limit major triad

C4 FREQUENCY E(4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 331.120 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 392.438 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 662.240 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 784.877 0.000 – – – – – – – – 3 993.360 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1177.315 – – – – 5 1308.128 4 1324.479 16.352 – – – – – – – – – – – – 6 1569.753 – – – – – – – – – – – – 4 1569.753 0.000 – – – – – – – – 5 1655.599 – – – – – – – – – – – – – – – – 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1896.719 – – – – 5 1962.192 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

Doty makes the following observation with regards to the intervallic properties of Pythagorean intonation and its application during the European Middle Ages:

“Although three-limit scales are poorly suited to chordal harmony, they work well melodically and are eminently suitable for polyphony based on fourths and fifths, such as the motets and organa of the

323 Kyle Gann, “An Introduction to Historical Tunings,” Kyle Gann’s Home Page, 1997, 15 December 2005, . The Harmonic Consideration 235

European Ars Antiqua period (c. 1110-1300 C.E.). Pythagorean scales were the sole theoretical basis for intonation throughout the European Middle Ages. The compositional practices of this period, which treat thirds and sixths as dissonances, are understandable in light of the properties of Pythagorean tuning.”324

The root position three-limit minor triad is represented by the ratio 54:64:81, identities 27-1-81, intervals 27/16, 1/1, and 81/64, and the pitches A(, C, and E(; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß21.506 and +0.000 cents on each count, which is a deviation of a syntonic comma (81/80) for the major third. E(5 (662.240Hz), or the Pythagorean major third (81/64), presents 0.000 beats between the third harmonic of A(4 (1324.479Hz), or the Pythagorean major sixth (27/16), and the second harmonic of E(5 (1324.479Hz), and 0.000 beats between the sixth harmonic of A(4 (2648.959Hz) and the fourth harmonic of D5 (2648.959Hz); while C5 (523.251Hz), or the octave (2/1), 32.703 beats between the sixth harmonic of A(4 and the fifth harmonic of C5 (2616.256Hz).

Table 133. The beating characteristics of the three-limit minor triad

A(4 FREQUENCY C5 FREQUENCY BEATS E(5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 441.493 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 662.240 – – – – 2 882.986 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1046.502 – – – – – – – – – – – – – – – – 3 1324.479 – – – – – – – – – – – – 2 1324.479 0.000 – – – – – – – – 3 1569.753 – – – – – – – – – – – – – – – – 4 1765.973 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1986.719 – – – – – – – – – – – – 4 2093.005 – – – – – – – – – – – – – – – – 5 2207.466 – – – – – – – – – – – – – – – – – – – – – – – – 6 2648.959 5 2616.256 32.703 4 2648.959 0.000 7 3090.452 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 3139.507 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 3311.199 – – – – 8 3531.945 – – – – – – – – – – – – – – – – – – – – – – – –

The concluding sonorities of the fourth system include a G("5)/B triad resolving to a C tonal centre within two successive 2-3 and 7-1 suspensions in the bass. This harmony presents the pitches B(, G, and

324 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 37-38. 236 The Harmonic Consideration

D")), and within the ratio 59049:93312:131072, characterizes the Pythagorean minor sixth (128/81), and Pythagorean diminished fifth (1024/729).

Five-Limit Just Intonation

The primary interval of five-limit just intonation is the fifth harmonic, or just major third, which is represented by the frequency ratio 5/4, and in relation to C (1/1), notated simply as E. The complement, 8/5, or just minor sixth, is notated simply as A". The essential five-limit pitch material is arrived at via a now two-dimensional process, which requires a two-dimensional 2, 3, 5-limit pitch lattice. Pitch lattices allow for the graphic representation of pitch relationships, with one dimension in this case represented by horizontal columns defining just major thirds (the sequential addition or subtraction of 5/4s to the right or left of any particular pitch), while the other dimension; vertical rows defining just perfect fifths (the sequential addition or subtraction of 3/2s to the top or bottom of any particular pitch). In different terms, north and south vertical links (0.00° and 180.00°) within the lattice produce 3/2 and 4/3 relationships; east and west horizontal links (90.00° and 270.00°), 5/4 and 8/5; northeast and southwest diagonal links (45.00° and 225.00°), 15/8 and 16/15; while northwest and southeast diagonal links (315.00° and 135.00°), 6/5 and 5/3.325 A five-limit scale is constructed via the establishment of just major triads (4:5:6) on the fundamental C (1/1), G (3/2), and D (9/8), which generate the six otonal pitches: D, E, F, G, A, and B (9/8, 5/4, 4/3, 3/2, 5/3, and 15/8); as well as the establishment of complement minor triads (5:6:10), which generate the six utonal pitches: B"), A", G, F, E", and D") (16/9, 8/5, 3/2, 4/3, 6/5, and 16/15).

Fig. 8. Five-limit otonal pitch generation

G (3/2) D (9/8) C (1/1)

C (1/1) E (5/4) G (3/2) B (15/8) F (4/3) A (5/3)

Fig. 9. Five-limit utonal pitch generation

A" (8/5) C (1/1) D") (16/15) F (3/2) E" (6/5) G (3/2)

F (4/3) B") (16/9) C (1/1)

325 Von Gunden, The Music of Ben Johnston 60. The Harmonic Consideration 237

F!(, or the just tritone, or augmented fourth (45/32) is adopted to represent the interval of an augmented fourth, along with the complement, G"), or the diminished fifth (64/45, or 609.776). The pitch material is then sequentially arranged as:

C D") D E" E F F!( G") G A" A B") B 1 16 9 6 5 4 45 64 3 8 5 16 15

1 15 8 5 4 3 32 45 2 5 3 9 8

The series of intervals presents the just diatonic semitone (16/15), just major tone (9/8), just minor third (6/5), just major third (5/4), just perfect fourth (4/3), just tritone, or augmented fourth (45/32), diminished fifth (64/45), just perfect fifth (3/2), just minor sixth (8/5), just major sixth (5/3), Pythagorean minor seventh (16/9), and just diatonic major seventh (15/8). The prime number five is a prerequisite for consonant thirds and sixths, and together with their respective complements, 5/4, 6/5, 5/3, and 8/5 represent the “principle consonances of the five-limit.” The dissonant Pythagorean major third (81/64) is now replaced by the consonant just major third with a superparticular vibrating ratio of 5/4. “This interval is the serene consonance we expect a major third to be,” notes Doty.326 The following table depicts the essential tonal resources of five-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 134. Essential five-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ D") just diatonic semitone, or major half-tone 16/15 1.066667 111.731 ÐÓ D just major tone (9th harmonic) 9/8 1.125000 203.910 ÐÔ E" just minor third 6/5 1.200000 315.641 ÐÕ E just major third (5th harmonic) 5/4 1.250000 386.314 ÐÖ F just and Pythagorean perfect fourth 4/3 1.333333 498.045 Ð× F!( just tritone, or augmented fourth (45th harmonic) 45/32 1.406250 590.224 ÐØ G just and Pythagorean perfect fifth (3rd harmonic) 3/2 1.500000 701.955 ÐÙ A" just minor sixth 8/5 1.600000 813.686 ÑÐ A just major sixth 5/3 1.666667 884.359 ÑÑ B") Pythagorean minor seventh 16/9 1.777778 996.090 ÑÒ B just diatonic major seventh (15th harmonic) 15/8 1.875000 1088.269

ÐÑ C octave 2/1 2.000000 1200.000

326 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 38-39. 3 h amncConsideration Harmonic The 238 Fig. 10. 2, 3, 5-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 239

Five-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #1 within the pitches C, D, E, G, and A (1/1, 9/8, 5/4, 3/2, and 5/3); and the complements: C, B"), A", F, and E" (2/1, 16/9, 8/5, 4/3, and 6/5).

Ex. 102. Lou Harrison pentatonic #1 (five-limit intonation)

The following two tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 135. 5-limit tuning matrix no. 1 ‘key of C’ (Program 02)

NOTE C D") D E" E F – – G A" A B") B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +12 +04 +16 ß14 ß02 +00 +02 +14 ß16 ß04 ß12

Table 136. 5-limit tuning matrix no. 2 ‘key of C’ (Program 03)

NOTE C D") D E" E F – – G A" A B" B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +12 +04 +16 ß14 ß02 +00 +02 +14 ß16 +18 ß12

As well as expressing a diatonic harmonic extension of the four-bar melodic sentence, the first simultaneous sonority of section B represents the 1, 3, and 5 identities of the harmonic series that form the consonant or ‘pure’ major triad of just intonation (4:5:6). “The five-limit major triad conforms in all aspects to our definition of a consonant chord in just intonation,” notes Doty. “It is free from disturbing interference beats (assuming a harmonic timbre), and is accompanied by first-order difference tones that 240 The Harmonic Consideration reinforce the identity of the series from which the chord derives.” The first-order difference tones that Doty refers to include C2 and C3 for the root position five-limit major triad (4:5:6), which in combination with the fundamental tones essentially form the pitch series C2, C3, C4, E4, and G4, or the first, second, fourth, fifth, and sixth partials of the harmonic series. The inversions of the triad on the other hand include C2, , C3, and G3 for the first inversion five-limit major triad (5:6:8); and C3 and C4 for the second inversion five-limit major triad (3:4:5). This chord is obtained via the two-dimensional five-limit process that now incorporates not only 3/2 vertical derivatives of the C fundamental, but also horizontal 5/4 relationships that produce the just major third (5/4), as well as the just perfect fifth (3/2) of three-limit just intonation; not to mention the complements – the 8/5 relationships that produce the just major sixth (5/3), as well as the just perfect fourth (4/3) of three-limit just intonation. “In root position, the fifth harmonic of the root (1 identity) coincides with the fourth harmonic of the major third (5 identity), and the sixth harmonic of the major third coincides with the fifth harmonic of the perfect fifth (3 identity),” explains Doty. The following example illustrates the consonant nature of the five-limit major triad. Differential tones are represented in the bass clef by filled noteheads, while periodicity pitch, by triangular noteheads.

Ex. 103. Differential tones and periodicity pitches produced by the five-limit major triad

The second most important chord of five-limit intonation is the five-limit minor triad (10:12:15), which according to Doty is more consonant than the three-limit Pythagorean minor triad, yet nevertheless less consonant than the five-limit major triad due to the fact of possessing identities 3, 5, and 15, but no 1 identity. Analysis of this simultaneous sonority further reveals a 6/5 ratio (or just minor third) between the fifth and the seventh. B4 (490.548Hz), or the just diatonic major seventh (15/8), presents 0.000 beats between the third harmonic of E4 (981.096Hz), or the just major third (5/4), and the second harmonic of B4 (981.096Hz), and 0.000 beats between the sixth harmonic of E4 (1962.192Hz) and the fourth harmonic of B4 (1962.192Hz); while G4, or the just perfect fifth (3/2), then consistently also presents 0.000 beats between the sixth harmonic of E4 and the fifth harmonic of G4 (1962.192Hz).327

327 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 46-47. The Harmonic Consideration 241

Table 137. The beating characteristics of the just minor triad

E4 FREQUENCY G4 FREQUENCY BEATS B4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 327.032 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 392.438 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 490.548 – – – – 2 654.064 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 784.877 – – – – – – – – – – – – – – – – 3 981.096 – – – – – – – – – – – – 2 981.096 0.000 – – – – – – – – 3 1177.315 – – – – – – – – – – – – – – – – 4 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1471.644 – – – – – – – – – – – – 4 1569.753 – – – – – – – – – – – – – – – – 5 1635.160 – – – – – – – – – – – – – – – – – – – – – – – – 6 1962.192 5 1962.192 0.000 4 1962.192 0.000 7 2289.224 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2354.630 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 2452.740 – – – – 8 2616.256 – – – – – – – – – – – – – – – – – – – – – – – –

The beating characteristics of the just minor triad display equal consonance in comparison to the just major triad, but upon analysis of the inherent differential tones and periodicity pitch it becomes evident that in the case of the three inversions (10:12:15, 12:15:20, and 15:20:24), one or more tones alien to the fundamental triad are introduced. These tones include A" and B", or 8/5 and 9/5, which have no relevance to the conventional musical application of the minor triad. Although, according to Partch’s theories, these two triads may be stated as being equally consonant, for they form the theoretical basis of ‘otonalities’ and ‘utonalities’ – the major and minor tonalities of the harmonics and subharmonic series. Partch makes the following observations:

“In utonality (‘minor’) the conception is somewhat different (to otonality [‘major’]), since the series of identities descends in pitch from its unity, though the practical results are exactly the same; the unity is here the ‘fifth of the chord’. The long controversy as to the correct location of the ‘root’ of the ‘minor’ triad is rhetoric, so far as creative music goes, since the composer needs no greater authority than his fancy to put the ‘root’ wherever he wants to put it.” 328

328 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 110-12. 242 The Harmonic Consideration

In opposition to the subharmonic argument for the minor triad, Doty states that, “the Partchian view, however, ignores the implications of difference tones and periodicity pitches. These phenomena will always reinforce a harmonic interpretation of a chord.”329

Ex. 104. Differential tones and periodicity pitches produced by the five-limit minor triad

Other simultaneous sonorities of interest in five-limit just intonation include the root position major seventh tetrad (8:10:12:15) with identities 1, 5, 3, and 15; minor seventh tetrad (10:12:15:18) with identities 5, 3, 15, and 9, but no 1 identity; major ninth pentad (8:10:12:15:18) with identities 1, 5, 3, 15, and 9; and minor ninth pentad (20:24:30:36:45) with identities 5, 3, 15, 9, and 45, and no 1 identity.

Ex. 105. Five-limit chords

The second system features a two-part inverted canon, but with intervallic augmentation and saturation applied to the complement or inversion of the principal melodic material. The result, is the melodic line’s transformation from the sequence A" and F, into A", G, and E. Also, the fact that the transformed motive now occupies the upper position of the harmony (soprano and alto) means that the final sonority in the system is now D"("5) – the pitches D"), G, and F, or the ratio 32:45:80 (as opposed to the three-limit first inversion D"("5)/F sonority). In the third system, a minor alteration in the tuning scheme is required, due to the resulting dissonant Cmin7(omit 5) sonority – the pitches C, E", and B"), or the ratio 135:162:200. Analysis of the harmonic properties of the simultaneous sonority reveals an appropriate just minor third (6/5) between the bottom two sonorities, yet a grave or small fifth (40/27, or 680.449 cents) between the upper two sonorities. In order to adhere to the correct intervallic properties of the five-limit minor seventh tetrad (10:12:15:18, with identities 5-3-15-9), it is then simply a matter of retuning the minor seventh from a B"), or Pythagorean minor seventh (16/9), to a B", or acute or large minor seventh (9/5),

329 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 47. The Harmonic Consideration 243 which essentially forms a 3/2 relationship between the upper two sonorities, to produce a Cmin7(omit 5) sonority with the ratio 10:12:15. Interesting sonorities presented in the fourth system – which also features a two-part inverted canon, with the complement or inversion of the principal melodic material occupying the upper position of the harmony – include the final sonorities at the end of the system (a double inverted canon) that facilitate a modulation to G major via an E" augmented triad – the pitches E", G, and B, or the ratio 16:20:25, moving to a second inversion G major triad – the pitches D, G, and B, or the ratio 3:4:5.

Seven-Limit Just Intonation

The primary interval of seven-limit just intonation is the seventh harmonic, or septimal subminor seventh, which is represented by the frequency ratio 7/4 (968.826 cents), and notated as B;. The complement, 8/7 (231.174 cents), or septimal supermajor second, is notated as D*). According to Gayle Young, “this interval is closer to a major second than to a minor third,” yet Partch reports that it was accepted as a “consonance by Mersenne in the seventh century.”330 The notational symbols for the partial chroma ([+] and [*], lowering or raising a tone by 36/35 [one septimal comma], or 48.770 cents) are in reference to “the amount the acute or large minor seventh (9/5) exceeds the seventh harmonic” (7/4Ï36/35=9/5).331 7/4 is 31.174 cents flat from the equal minor seventh ( 6 ]2[ 5 ), while 8/7, 31.174 sharp from the equal major second ( 6 2 ). Doty states the following about the 7/4 ratio: “7/4 is a powerful consonance, whereas tempered, three-limit, and five-limit minor sevenths are all quite dissonant. Thus the harmonic use of 7/4 demands a departure from common practice. 7/4 is a necessary constituent of consonant dominant seventh chords and diminished triads.”332 Seven-limit intonation now requires a three-dimensional 2, 3, 5, 7-limit pitch lattice, capable of displaying relationships not just within seven-limit intonation, but also front and back interrelationships between the four specified limits. According to Heidi Von Gunden:

“Johnston began designing scales using three, four, and even five prime numbers. As the number of generating ratios increases, certain complications result. One is the inability to show the network of relationships on a two-dimensional plane, such as a piece of paper. A system using four generating ratios needs to be represented with a three-dimensional design. Notice that this (2, 3, 5, 7-limit pitch) lattice maintains the 3/2 ratios on the vertical axes, the 5/4 ratios on the horizontal axes, and the 7/4 ratios are the third dimensions seen as axes behind (for the seventh above) and in front (for the seventh below) of the 3/2 axes. This lattice produces natural dominant seventh chords if one reads vertically one block north (for

330 Gayle Young, “The Pitch Organization of Harmonium for James Tenney,” Perspectives of New Music 26.2 (Summer, 1988): 205. 331 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 113. 332 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 51. 244 The Harmonic Consideration

the 3/2 ratio), one block horizontally east (for the 5/4 ratio), and one block to the rear of the 1/1 ratio (for the 7/4 ratio).”333

The process adopted for the creation of seven-limit intonation pitch material involves the establishment of just major triads (4:5:6) from the seventh harmonics of the fundamental C (1/1), G (3/2), and D (9/8), which generate the six otonal pitches: D;, E;, F+(, G;, A;, and B; (21/20, 7/6, 21/16, 7/5, 14/9, and 7/4); as well as the establishment of complement minor triads (5:6:10), which generate the six utonal pitches: B*), A*, G*), F:, E*, and D*) (40/21, 12/7, 32/21, 10/7, 9/7, and 8/7). It must be noted that F+( (21/16) and G*) (32/21) will purely serve a theoretical purpose, as the work adopts the three-limit perfect fourth (4/3) and fifth (3/2) in actual scale formation.

Fig. 11. Seven-limit otonal pitch generation

G; (7/5) B; (7/4) D; (21/20) F+( (21/16) C; (28/15) E; (7/6)

E; (7/6) B; (7/4) A; (14/9)

Fig. 12. Seven-limit utonal pitch generation

A* (12/7) D*) (8/7) E* (9/7)

D*) (8/7) F: (10/7) G*) (32/21) B*) (40/21) A* (12/7) C: (15/14)

The pitch material is then sequentially arranged as:

C D; D*) E; E* F F: G; G A; A* B; B*) 1 21 8 7 9 4 10 7 3 14 12 7 40

1 20 7 6 7 3 7 5 2 9 7 4 21 F+( G*) 21 32

16 21

The series of intervals presents the septimal diatonic semitone (21/20), septimal supermajor second (8/7), septimal subminor third (7/6), septimal supermajor third (9/7), just perfect fourth (4/3), septimal tritone, or superaugmented fourth (10/7), septimal subdiminished fifth (7/5), septimal subminor sixth (14/9), septimal supermajor sixth (12/7), septimal subminor seventh (7/4), and septimal supermajor seventh

333 Von Gunden, The Music of Ben Johnston 128-29. The Harmonic Consideration 245

(40/21); as well as the non-essential tones septimal subfourth (21/16) and septimal superfifth (32/21). The following table depicts the essential tonal resources of seven-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 138. Essential seven-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ D; septimal diatonic semitone 21/20 1.050000 84.467 ÐÓ D*) septimal supermajor second 8/7 1.142857 231.174 ÐÔ E; septimal subminor third 7/6 1.166667 266.871 ÐÕ E* septimal supermajor third 9/7 1.285714 435.084 ÐÖ F+( septimal subfourth (21st harmonic) 21/16 1.312500 470.781 Ð× F: septimal tritone, or superaugmented fourth 10/7 1.428571 617.488 ÐØ G*) septimal superfifth 32/21 1.523810 729.219 ÐÙ A; septimal subminor sixth 14/9 1.555556 764.916 ÑÐ A* septimal supermajor sixth 12/7 1.714286 933.129 ÑÑ B; septimal subminor seventh (7th harmonic) 7/4 1.750000 968.826 ÑÒ B*) septimal supermajor seventh 40/21 1.904762 1115.533

ÐÑ C octave 2/1 2.000000 1200.000

The adaptation of Harrison’s pentatonic #2 to seven-limit intonation presents a collection of pitches that include: C, E;, F, G, and B; (1/1, 7/6, 4/3, 3/2, and 7/4); as well as the complements: C, A*, G, F, and D*) (2/1, 12/7, 3/2, 4/3, and 8/7).

Ex. 106. Lou Harrison pentatonic #2 (seven-limit intonation)

4 h amncConsideration Harmonic The 246 Fig. 13. 2, 3, 5, 7-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 247

The modulation to G (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

G A;( A* B; B* C C: D; D E; E* F+( F: 3 63 12 7 27 1 15 21 9 7 9 21 10

2 40 7 4 14 1 14 20 8 6 7 16 7 C7( D*) 63 8

32 7

The following two tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 139. 7-limit tuning matrix no. 1 ‘key of G’ (Program 04)

NOTE C C: D E; E* F+( F: G A;( A* B; B*

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +19 +04 ß33 +35 ß29 +17 +02 ß14 +33 ß31 +37

Table 140. 7-limit tuning matrix no. 2 ‘key of G’ (Program 05)

NOTE C C: D E; E* F+( F: G A;( A* B; B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +19 +04 ß33 +35 ß29 +17 +02 ß14 +33 ß31 ß12

The first simultaneous sonorities of section C represent the 1, 5, 3, and 7 identities that form the septimal dominant seventh tetrad (4:5:6:7). “The most important chord in the seven-limit is unquestionably the dominant seventh chord,” notes Doty. “This is the most consonant possible chord consisting of four unique identities. The difference tones of the just dominant seventh chord strongly reinforce the 1 and 3 identities of the series from which the chord derives, adding to its stability and clarity.”

248 The Harmonic Consideration

Ex. 107. Differential tones produced by the seven-limit septimal dominant seventh tetrad

The septimal dominant seventh tetrad is represented within the work in the forms of G7(omit 3), or the ratio 4:6:7; and G7/D, or the ratio 3:4:5:6:7. Norden offers the following discussion with regards to the disparity between theory and performance practice in the Western classical music tradition, and its misconception of the seven-limit septimal dominant seventh tetrad:

“We hear just intonation many times in fine performances of choral, chamber, and orchestral music, but we have not recognized it theoretically. Theory has differed widely from practice. The dominant seventh chord with the harmonic seventh (ratios 4:5:6:7), frequently sounded in performance uninfluenced by fixed-pitch instruments, is completely ignored in theory books. This is a beautiful chord, easily tuned. The chord we theorize about is the diatonic dominant-seventh chord (ratios 36:45:54:64), which in equal temperament is distorted from (386, 316, and 294 cents) to (400, 300, and 300 cents).”334

Another interesting seven-limit chord is the septimal dominant ninth pentad (with the ratio 4:5:6:7:9, and identities 1-5-3-7-9), which adds the major ninth (9/4) to the tetrad. Doty makes the following observations:

“Depending on the register in which the major ninth is sounded, beating may occur between some of the lower harmonics of a pair of tones in this relation, adding roughness to the interval. When added to the dominant seventh chord, the major ninth spawns another relatively dissonant interval, the acute minor seventh, 9/5. Nevertheless, the overall impression created by the just dominant ninth chord, at least in root position, is one of stability.”

It is important to note that the septimal dominant seventh and ninth chords additional present a series of subsets, which include the septimal diminished triad (5:6:7, with identities 5-3-7), septimal half- diminished seventh tetrad (5:6:7:9, with identities 5-3-7-9), “incomplete dominant seventh” triad (4:6:7, with identities 1-3-7), and “added-second” tetrad (8:9:10:12, with identities 1-9-5-3). The septimal diminished triad is the second most consonant chord in seven-limit just intonation (preceded by the

334 Norden, “A New Theory of Untempered Music: A Few Important Features with Special Reference to ‘A Capella’ Music,” The Musical Quarterly 232. The Harmonic Consideration 249 seven-limit septimal major triad), and due to there being no 5 identity, and hence no tritone, the sonority has little in common with the traditional function of a dominant.

Ex. 108. Differential tones produced by the seven-limit septimal dominant ninth pentad

The first system of section C also features the septimal major and minor triads (14:18:21 and 6:7:9, with identities 7-9-21 and 3-7-9). According to Doty, the subminor septimal minor triad is “quite distinct from the five-limit minor triad (10:12:15), and is, in the opinion of some listeners, more consonant.” It is interesting to note that the differential tones produced by the root position septimal minor triad imply “a fundamental a 3/2 below the root”, as opposed to the five-limit, “5/4 below the root.”335

Ex. 109. Differential tones produced by the seven-limit septimal major triad

The root position seven-limit major triad is represented by the ratio 14:18:21, identities 7-9-21, intervals 7/4, 9/8, and 21/16, and the pitches B;, D, and F+(; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of +48.770 and +0.000 cents on each count, which is a deviation of a septimal comma (36/35) for the major third. F+(5 (686.767Hz), or the septimal subfourth (21/16, or 470.781 cents), presents 0.000 beats between the third harmonic of B;4 (1373.534Hz), or the septimal subminor seventh (7/4), and the second harmonic of F+(5 (1373.534Hz), and 0.000 beats between the sixth harmonic of B;4 (2747.068Hz) and the fourth harmonic of F+(5 (2747.068Hz); while D5 (588.658Hz), or the just major tone (9/8), 65.406 beats between the fifth harmonic of B;4 (2289.224Hz) and the fourth harmonic of D5 (2354.630Hz).

335 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 56-7. 250 The Harmonic Consideration

Table 141. The beating characteristics of the seven-limit major triad

B;4 FREQUENCY D5 FREQUENCY BEATS F+(5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 457.845 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 588.658 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 686.767 – – – – 2 915.689 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1177.315 – – – – – – – – – – – – – – – – 3 1373.534 – – – – – – – – – – – – 2 1373.534 0.000 – – – – – – – – 3 1765.973 – – – – – – – – – – – – – – – – 4 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 2060.301 – – – – 5 2289.224 4 2354.630 65.406 – – – – – – – – – – – – 6 2747.068 5 – – – – – – – – 4 2747.068 0.000 – – – – – – – – – – – – 2943.288 – – – – – – – – – – – – – – – – 7 3204.913 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 3531.945 – – – – 5 3433.836 – – – – 8 3662.758 – – – – – – – – – – – – – – – – – – – – – – – –

Ex. 110. Differential tones produced by the seven-limit septimal minor triad

The root position seven-limit minor triad is represented by the ratio 6:7:9, identities 3-7-9, intervals 3/2, 7/4, and 9/8, and the pitches G, B;, and D; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß48.770 and +0.000 cents on each count, which is a deviation of a septimal comma (36/35) for the minor third. D5, or the just major tone (9/8), presents 0.000 beats between the third harmonic of G4 (1177.315Hz), or the just perfect fifth (3/2), and the second harmonic of D5 (1177.315Hz), and 0.000 beats between the sixth harmonic of G4 (2354.630Hz) and the fourth harmonic of D5; while B;4 (457.845Hz), or the septimal subminor seventh (7/4), 65.406 beats between the sixth harmonic of G4 and the fifth harmonic of B;4, and 0.000 beats between the seventh harmonic of G4 (2747.068Hz) and the sixth harmonic of B;4.

The Harmonic Consideration 251

Table 142. The beating characteristics of the seven-limit minor triad

G4 FREQUENCY B;4 FREQUENCY BEATS D5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 392.438 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 457.845 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 588.658 – – – – 2 784.877 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 915.690 – – – – – – – – – – – – – – – – 3 1177.315 – – – – – – – – – – – – 2 1177.315 0.000 – – – – – – – – 3 1373.534 – – – – – – – – – – – – – – – – 4 1569.753 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1765.973 – – – – – – – – – – – – 4 1831.379 – – – – – – – – – – – – – – – – 5 1962.192 – – – – – – – – – – – – – – – – – – – – – – – – 6 2354.630 5 2289.224 65.406 4 2354.630 0.000 7 2747.068 6 2747.068 0.000 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 2943.288 – – – – 8 3139.507 – – – – – – – – – – – – – – – – – – – – – – – –

The principal melodic material is developed further via utonal and otonal intonational reinterpretations (from quintal to septimal major and minor chromatic harmonic extensions), as well as via pitch and rhythmic recapitulation. The beginning of the second system features motivic chromatic extension applied to both the start and end of the two-part inverted canon. The sonorities at the end of the second system (bar 40) imply a Gmin9 within the pitches G, F+(, G, A*, and D (28:49:56:64:84). Although a minor third is omitted in literal terms, it does make a presence two beats earlier in the same bar. In the third system, presented are Gmin7(omit 5) within the pitches G, B;, and F+( (12:14:21); and Edim/G within the pitches G, B;, and E* (192:224:329); with the entries of the complimentary sentence transformed via intervallic diminution and motivic diatonic ornamentation on each count. In the fourth and final system, compositional techniques introduced include intervallic inclusion and rhythmic diminution. The concluding sonorities of the fourth system (bar 48) include a Dmin7(add 11)/F chord resolving to a G tonal centre – featuring a G5 chord (2:3:4:8) – within two successive 2-3 and 7-1 suspensions, and the pitches F+(, G, D, A*, and C (147:168:252:384:448).

252 The Harmonic Consideration

Eleven-Limit Just Intonation

The primary interval of eleven-limit just intonation is the eleventh harmonic, or undecimal superfourth, which is represented by the frequency ratio 11/8 (551.318 cents), and notated as F,. The complement, 16/11 (648.682 cents), or undecimal subfifth, is notated as G-. “The frequency of a sound lying exactly midway between the equal tempered F that lies just below and the equal tempered FÚ that lies just above our 11th partial can be found by multiplying (523.251) by the 11th power of the 24th root of 2 (523.251Ï 24 ]2[ 11=718.923Hz),” notes Wilford W. Berard.336 The accuracy of this numerical approximation may be stated as being within two cents of the ‘true’ frequency of 11/8. The notational symbols for the partial chroma ([,] and [-], raising or lowering a tone by 33/32 [one undecimal comma], or 53.273 cents) are in reference to the “amount the eleventh harmonic exceeds the just perfect fourth” (4/3Ï33/32=11/8). 11/8 is 51.318 cents sharp from the equal perfect fifth ( [12 ]2 7 ), while 16/11, 48.682 flat from the equal perfect fifth ( [).12 ]2 7 337 The process adopted for the creation of eleven-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the eleventh harmonics of the fundamental C (1/1), G (3/2), and F (4/3), which generate the six otonal pitches: DE), EE), F,, GE), AE, and BE) (11/10, 11/9, 11/8, 22/15, 33/20, and 11/6); as well as the establishment of complement major triads (4:5:6), which generate the six utonal pitches: B-, A-(, G-, FD(,E-, and D- (20/11, 18/11, 16/11, 15/11, 40/33, and 12/11).

Fig. 14. Eleven-limit otonal pitch generation

DE) (11/10) F, (11/8) AE (33/20) C, (33/32) GE) (22/15) BE) (11/6)

BE) (11/6) F, (11/8) EE) (11/9)

Fig. 15. Eleven-limit utonal pitch generation

D- (12/11) G- (16/11) A-( (18/11)

G- (16/11) B- (20/11) C- (64/33) E- (40/33) D- (12/11) FD( (15/11)

336 Wilford W. Berard, “The Eleventh and Thirteenth Partials,” Journal of Music Theory 5.1 (Spring, 1961): 96-7. 337 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 114. The Harmonic Consideration 253

The pitch material is then sequentially arranged as:

C DE) D- EE) E- F FD( GE) G AE A-( BE) B- 1 11 12 11 40 4 15 22 3 33 18 11 20

1 10 11 9 33 3 11 15 2 20 11 6 11 F, G- 11 16

8 11

The series of intervals presents the undecimal acute or large neutral second (11/10), undecimal grave or small neutral second (12/11), undecimal acute or large neutral third (11/9), undecimal grave or small neutral third (40/33), just perfect fourth (4/3), undecimal tritone, or augmented fourth (15/11), undecimal subdiminished fifth (22/15), just perfect fifth (3/2), undecimal acute or large neutral sixth (33/20), undecimal grave or small neutral sixth (18/11), undecimal acute or large neutral seventh (11/6), undecimal grave or small neutral seventh (20/11); as well as the non-essential tones undecimal superfourth (11/8) and undecimal subfifth (16/11). The following table depicts the essential tonal resources of eleven-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 143. Essential eleven-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ DE) undecimal acute or large neutral second 11/10 1.100000 165.004 ÐÓ D- undecimal grave or small neutral second 12/11 1.090909 150.637 ÐÔ EE) undecimal acute or large neutral third 11/9 1.222222 347.408 ÐÕ E- undecimal grave or small neutral third 40/33 1.212121 333.041 ÐÖ F, undecimal superfourth (11th harmonic) 11/8 1.375000 551.318 Ð× FD( undecimal tritone, or augmented fourth 15/11 1.363636 536.951 ÐØ G- undecimal subfifth 16/11 1.454545 648.682 ÐÙ AE undecimal acute or large neutral sixth 33/20 1.650000 866.959 ÑÐ A-( undecimal grave or small neutral sixth 18/11 1.636364 852.592 ÑÑ BE) undecimal acute or large neutral seventh 11/6 1.833333 1049.363 ÑÒ B- undecimal grave or small neutral seventh 20/11 1.818181 1034.996

ÐÑ C octave 2/1 2.000000 1200.000

5 h amncConsideration Harmonic The 254 Fig. 16. 2, 3, 5, 11-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 255

Eleven-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #2 within the pitches C, EE), F, G, and BE) (1/1, 11/9, 4/3, 3/2, and 11/6); and the complements: C, A-(, G, F, and D- (2/1, 18/11, 3/2, 4/3, and 12/11).

Ex. 111. Lou Harrison pentatonic #2 (eleven-limit intonation)

The modulation to G (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

G AE A-) BE) B- C CD( DE) D EE E-( F, FD( 3 33 18 11 20 1 45 11 9 99 27 11 15

2 20 11 6 11 1 44 10 8 80 22 8 11 C, D- 33 12

32 11

The following four tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 144. 11-limit tuning matrix no. 1 ‘key of G’ (Program 06)

NOTE C C, D – – E-( FD( F, G – – A-( BE) – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß47 +04 +00 ß45 +37 ß49 +02 +00 ß47 +49 +00

256 The Harmonic Consideration

Table 145. 11-limit tuning matrix no. 2 ‘key of G’ (Program 07)

NOTE C C, D – – E-( F+( F, G – – A( BE) B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß47 +04 +00 ß45 ß29 ß49 +02 +00 +06 +49 ß12

Table 146. 11-limit tuning matrix no. 3 ‘key of G’ (Program 08)

NOTE CD( – – D – – EE FD( F, G – – A-( B- – –

KEY C C! D D! E F F! G G! A A! B

CENTS +39 +00 +04 +00 ß31 +37 ß49 +02 +00 ß47 +35 +00

Table 147. 11-limit tuning matrix no. 4 ‘key of G’ (Program 09)

NOTE C C, D – – E-( FD( F, G – – AE BE) – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß47 +04 +00 ß45 +37 ß49 +02 +00 ß33 +49 +00

The first simultaneous sonorities of section D represent the 1, 5, 3, 7, 9, and 11 identities that form the undecimal dominant eleventh hexad (4:5:6:7:9:11). “In root position and in a reasonably high register, the addition of the 11 identity to the dominant-ninth chord doesn’t seriously injure its stability or consonance, although it does add an indescribable, piquant quality to the sonority,” notes Doty.338

Ex. 112. Differential tones produced by the eleven-limit undecimal dominant eleventh hexad

338 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 62. The Harmonic Consideration 257

The undecimal dominant eleventh hexad is represented within the work in the forms of G7(add 11/omit 3), or the ratio 4:6:7:11; and G11/D, or the ratio 3:4:5:6:7:8:11. Also introduced in the first system of section D are undecimal major and minor triads. The root position eleven-limit major triad is represented by the ratio 66:80:99, identities 33-5-99, intervals 33/32, 5/4, and 99/64, and the pitches C,, E, and G,; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß53.273 and +0.000 cents on each count, which is a deviation of an undecimal comma (33/32) for the major third. G,4 (404.702Hz), or the undecimal superfifth (99/64, or 755.228 cents), presents 0.000 beats between the third harmonic of C,4 (809.404Hz), or the undecimal comma (33/32), and the second harmonic of G,4 (809.404Hz), and 0.000 beats between the sixth harmonic of C,4 (1618.808Hz) and the fourth harmonic of G,4 (1618.808Hz); while E4, or the just major third (5/4), 40.879 beats between the fifth harmonic of C,4 (1349.007Hz) and the fourth harmonic of E4 (1308.128Hz).

Table 148. The beating characteristics of the eleven-limit major triad

C,4 FREQUENCY E4 FREQUENCY BEATS G,4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 269.801 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 327.032 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 404.702 – – – – 2 539.603 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 654.064 – – – – – – – – – – – – – – – – 3 809.404 – – – – – – – – – – – – 2 809.404 0.000 – – – – – – – – 3 981.096 – – – – – – – – – – – – – – – – 4 1079.205 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1214.106 – – – – 5 1349.007 4 1308.128 40.879 – – – – – – – – – – – – 6 1618.808 – – – – – – – – – – – – 4 1618.808 0.000 – – – – – – – – 5 1635.160 – – – – – – – – – – – – – – – – 7 1888.610 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1962.192 – – – – 5 2023.510 – – – – 8 2158.411 – – – – – – – – – – – – – – – – – – – – – – – –

The root position eleven-limit minor triad is represented by the ratio 18:22:27, identities 9-11-27, intervals 9/8, 11/8, and 27/16, and the pitches D, F,, and A(; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of +53.273 and +0.000 cents on each count, which is a deviation of an undecimal comma (33/32) for the minor third. A(4 (441.493Hz), or the Pythagorean major sixth (27/16), presents 0.000 beats between the third harmonic of D4 (882.986Hz), or the just major tone (9/8), and 258 The Harmonic Consideration the second harmonic of A(4 (882.986Hz), and 0.000 beats between the sixth harmonic of D4 (1765.973Hz) and the fourth harmonic of A(4 (1765.973Hz); while F,4 (359.735Hz), or the undecimal superfourth (11/8), 32.703 beats between the sixth harmonic of D4 and the fifth harmonic of F,4 (1798.676Hz).

Table 149. The beating characteristics of the eleven-limit minor triad

D4 FREQUENCY F,4 FREQUENCY BEATS A(4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 294.329 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 359.735 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 441.493 – – – – 2 588.658 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 719.470 – – – – – – – – – – – – – – – – 3 882.986 – – – – – – – – – – – – 2 882.986 0.000 – – – – – – – – 3 1079.205 – – – – – – – – – – – – – – – – 4 1177.315 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1324.479 – – – – – – – – – – – – 4 1438.941 – – – – – – – – – – – – – – – – 5 1471.644 – – – – – – – – – – – – – – – – – – – – – – – – 6 1765.973 5 1798.676 32.703 4 1765.973 0.000 7 2060.301 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2158.411 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 2207.466 – – – – 8 2354.630 – – – – – – – – – – – – – – – – – – – – – – – –

The sonorities at the end of the second system (bar 56) imply a Gm9 within the pitches G, D, BE), A-(, and F, (198:297:484:864:1089); and in the third system, Gmin7(add11/omit 5) within the pitches G, BE), F,, and C, (36:44:66:99); and Edim/G within the pitches G, BE), and E-( (180:220:297). The concluding sonorities of the fourth system (bar 65) facilitate a modulation to D major via a B" augmented triad – the pitches BE), D, and FD(, or the ratio 110:135:162, moving to a second inversion D major triad – the pitches A-(, D, and FD(, or the ratio 40:55:66.

The Harmonic Consideration 259

Thirteen-Limit Just Intonation

The primary interval of thirteen-limit just intonation is the thirteenth harmonic, or tridecimal grave or small neutral, or overtone sixth, which is represented by the frequency ratio 13/8 (840.528 cents), and notated as AO. The complement, 16/13 (359.472 cents), or tridecimal acute or large neutral third, is notated as E/. The notational symbols for the partial chroma ([.] and [/], raising or lowering a tone by 65/64 [one tridecimal comma], or 26.841 cents) is in reference to the “amount the thirteenth harmonic exceeds the just minor sixth” (8/5Ï65/43=13/8).339 13/8 is 40.528 cents sharp from the equal minor sixth ( [3 ]2 2 ), while 16/13, 40.528 flat from the equal major third ( 3 2 ). The process adopted for the creation of thirteen-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the thirteen harmonics of the fundamental C (1/1), G (3/2), and F (4/3), which generate the four otonal pitches: DO), EO, GO), and AO (13/12, 39/32, 13/9, and 13/8); as well as the establishment of complement major triads (4:5:6), which generate the four utonal pitches: B/, A/, FN(, and E/ (24/13, 64/39, 18/13, and 16/13). In this particular case, the exercise omits necessary major second, perfect fourth, perfect fifth, and minor seventh pitch material.

Fig. 17. Thirteen-limit otonal pitch generation

FO (13/10) AO (13/8) CO (39/20) EO (39/32) Bg) (26/15) DO) (13/12)

DO) (13/12) AO (13/8) GO) (13/9)

Fig. 18. Thirteen-limit utonal pitch generation

B/ (24/13) E/ (16/13) FN( (18/13)

E/ (16/13) GN (20/13) A/ (64/39) CN (40/39) B/ (24/13) DN (15/13)

The necessary additional pitch material is obtained via a secondary process that involves the calculation of 5/4 relationships with DO) (13/12) and GO) (13/9), which generate the otonal pitches F. (65/48) and BO) (65/36), and utonal complements G/ (96/65) and D/ (72/65). The pitch material is then sequentially arranged as:

339 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115. 260 The Harmonic Consideration

C DO) D/ EO E/ F FN( GO) G AO A/ BO) B/ 1 13 72 39 16 4 18 13 3 13 64 65 24

1 12 65 32 13 3 13 9 2 8 39 36 13 F. G/ 65 96

48 65

The series of intervals presents the tridecimal grave or small neutral second (13/12), tridecimal acute or large neutral second (72/65), tridecimal grave or small neutral third (39/32), tridecimal acute or large neutral third (16/13), just perfect fourth (4/3), tridecimal tritone, or augmented fourth (18/13), tridecimal subdiminished fifth (13/9), just perfect fifth (3/2), tridecimal grave or small neutral, or overtone sixth (13/8), tridecimal acute or large neutral sixth (64/39), tridecimal grave or small neutral seventh (65/36), tridecimal acute or large neutral seventh (24/13); as well as the non-essential tones tridecimal superfourth (65/48) and tridecimal subfifth (96/65). The following table depicts the essential tonal resources of thirteen-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 150. Essential thirteen-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ DO) tridecimal grave or small neutral second 13/12 1.083333 138.573 ÐÓ D/ tridecimal acute or large neutral second 72/65 1.107692 177.069 ÐÔ EO tridecimal grave or small neutral third (39th harmonic) 39/32 1.218750 342.483 ÐÕ E/ tridecimal acute or large neutral third 16/13 1.230769 359.472 ÐÖ F. tridecimal superfourth 65/48 1.354167 524.886 Ð× FN( tridecimal tritone, or augmented fourth 18/13 1.384615 563.382 ÐØ G/ tridecimal subfifth 96/65 1.476923 675.114 ÐÙ AO tridecimal grave or small neutral, or overtone sixth (13th harmonic) 13/8 1.625000 840.528 ÑÐ A/ tridecimal acute or large neutral sixth 64/39 1.641026 857.517 ÑÑ BO) tridecimal grave or small neutral seventh 65/36 1.805556 1022.931 ÑÒ B/ tridecimal acute or large neutral seventh 24/13 1.846154 1061.427

ÐÑ C octave 2/1 2.000000 1200.000

Fig. 19. 2, 3, 5, 13-limit pitch lattice (notation, ratio and cent values)

T eHroi osdrto 261 Consideration Harmonic he

262 The Harmonic Consideration

The adaptation of Harrison’s pentatonic #3 to thirteen-limit intonation presents a collection of pitches that include: C, D/, EO, G, and AO (1/1, 72/65, 39/32, 3/2, and 13/8); as well as the complements: C, BO), A/, F, and E/ (2/1, 65/36, 64/39, 4/3, and 16/13).

Ex. 113. Lou Harrison pentatonic #3 (thirteen-limit intonation)

The modulation to D (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

D EO E/( F.( FN( G GN( AO A( BO B/ C. CN( 9 39 81 351 18 3 81 13 27 117 24 65 27

8 32 65 256 13 2 52 8 16 64 13 64 26 G. A/( 195 108

128 65

The following two tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 151. 13-limit tuning matrix no. 1 ‘key of D’ (Program 10)

NOTE C. CN( D EO E/( F.( FN( G GN( A( BO B/

KEY C C! D D! E F F! G G! A A! B

CENTS +27 ß35 +04 +42 ß19 +46 ß37 +02 ß33 +06 +44 ß39

The Harmonic Consideration 263

Table 152. 13-limit tuning matrix no. 2 ‘key of D’ (Program 11)

NOTE C. CN( D EO E/( F.( FN( G AO A( BO B/

KEY C C! D D! E F F! G G! A A! B

CENTS +27 ß35 +04 +42 ß19 +46 ß37 +02 +41 +06 +44 ß39

Introduced in the first system of section E (bars 65-66), are tridecimal major and minor triads. The root position thirteen-limit major triad is represented by the ratio 26:32:39, identities 13-1-39, intervals 13/8, 1/1, and 39/32, and the pitches AO, C, and EO; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of +26.841 and +0.000 cents on each count, which is a deviation of a tridecimal comma (65/64) for the major third. EO5 (637.712Hz), or the tridecimal grave or small neutral third (39/32, or 342.483 cents), presents 0.000 beats between the third harmonic of AO4 (1275.425Hz), or the tridecimal grave or small neutral sixth (13/8), and the second harmonic of EO5 (1275.425Hz), and 0.000 beats between the sixth harmonic of AO4 (2550.849Hz) and the fourth harmonic of EO5 (2550.849Hz); while C5, or the octave (2/1), 32.703 beats between the fifth harmonic of AO4 (2125.708Hz) and the fourth harmonic of C5 (2093.005Hz).

Table 153. The beating characteristics of the thirteen-limit major triad

AO 4 FREQUENCY C5 FREQUENCY BEATS EO5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 425.142 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 637.712 – – – – 2 850.283 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1046.502 – – – – – – – – – – – – – – – – 3 1275.425 – – – – – – – – – – – – 2 1275.425 0.000 – – – – – – – – 3 1569.753 – – – – – – – – – – – – – – – – 4 1700.566 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1913.137 – – – – 5 2125.708 4 2093.005 32.703 – – – – – – – – – – – – 6 2550.849 – – – – – – – – – – – – 4 2550.849 0.000 – – – – – – – – 5 2616.256 – – – – – – – – – – – – – – – – 7 2975.991 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 3139.507 – – – – 5 3188.562 – – – – 8 3401.132 – – – – – – – – – – – – – – – – – – – – – – – –

264 The Harmonic Consideration

The root position thirteen-limit minor triad is represented by the ratio 32:39:48, identities 1-39-3, intervals 1/1, 39/32, and 3/2, and the pitches C, EO, and G; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß26.841 and +0.000 cents on each count, which is a deviation of a tridecimal comma (65/64) for the minor third. G4, or the just perfect fifth (3/2), presents 0.000 beats between the third harmonic of C4 (784.877Hz), or the unison (1/1), and the second harmonic of G4, and 0.000 beats between the sixth harmonic of C4 (1569.753Hz) and the fourth harmonic of G4; while EO4 (318.856Hz), or the tridecimal grave or small neutral third (39/32), 24.527 beats between the sixth harmonic of C4 and the fifth harmonic of EO4 (1594.281Hz).

Table 154. The beating characteristics of the thirteen-limit minor triad

C4 FREQUENCY EO4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 318.856 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 392.438 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 637.712 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 784.877 0.000 – – – – – – – – 3 956.568 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1177.315 – – – – – – – – – – – – 4 1275.425 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1594.281 24.527 4 1569.753 0.000 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1913.137 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1962.192 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

The sonorities at the beginning of the second system (bars 69-70) present intonational reinterpretations of an implied thirteen-limit tridecimal dominant thirteenth heptad via the simultaneous sonorities D9(!11)/F!, featuring the pitches D, FN(, A(, C., E/(, and GN( (30420:37440:45630:54925:67392:84240); and Dmin9(!11)/F, featuring the pitches D, F.(, A(, C., E/(, and GN( (56160:68445:84240:101400:124416:155520). In striking contrast, the ‘harmonic’ root position thirteen-limit tridecimal dominant thirteenth heptad has a ratio of 4:5:6:7:9:11:13, and features The Harmonic Consideration 265 the just major third (5/4), just perfect fifth (3/2), septimal subminor seventh (7/4), undecimal superfourth (11/8), and tridecimal grave or small neutral sixth (13/8).

Ex. 114. Differential tones produced by the thirteen-limit tridecimal dominant thirteenth heptad

The simultaneous sonorities in the fourth system (bars 77-78) include a B"7/D tetrad, with the pitches D, BO, F.(, and AO (144:234:351:416). Numerous compositional techniques are simultaneously utilized in the development of the four-bar melodic sentence and complimentary sentence in thirteen-limit just intonation. Notary amongst these includes the time signature change (from quadruple, or double duple time [4/4] to compound triple time [9/8]) applied to section E (bar 65), which significantly alters the rhythmic proportions of the motivic material. Intervallic redirection and utonal and otonal harmonic extension (bars 65 and 66 essentially outlining a major/minor tonality shift) is also applied to the two-note motive. A motivic ostinato featuring chromatic expansion and rhythmic recapitulation is additionally introduced, which generates a sense of moto perpetuo, or perpetual motion.340

340 “Perpetuum mobile [Lat., perpetual motion; It. moto perpetuo]. A composition in which rhythmic motion, often in a single-note-value at rapid tempo, is continuous from beginning to end. Among composers who have used the term as title for such a piece are Paganini (op. 11), Weber (Piano Sonata op. 24, last movement), Mendelssohn (op. 119), and Johann Strauss, Jr. (op. 257). The technique is also encountered in some Chopin etudes.” For a further discussion, see Randel, ed., The New Harvard Dictionary of Music 628. 266 The Harmonic Consideration

Seventeen-Limit Just Intonation

The primary interval of seventeen-limit just intonation is the seventeenth harmonic, or septendecimal chromatic semitone, which is represented by the frequency ratio 17/16 (104.955 cents), and notated as CP. The complement, 32/17 (1095.045 cents), or septendecimal superdiminished octave, is notated as CQ. The notational symbols for the partial chroma ([0] and [1], raising or lowering a tone by 51/50 [one septendecimal comma], or 34.283 cents) is in reference to the “amount the seventeenth harmonic exceeds the grave or small just chromatic semitone” (25/24Ï51/50=17/16). 17/16 is 4.955 cents sharp from the equal semitone ( 12 2 ), while 32/17, 4.955 flat from the equal diminished octave ( [).12 ]2 11 341 The process adopted for the creation of seventeen-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the seventeenth harmonics of the fundamental C (1/1), G (3/2), and F (4/3), which generate the five otonal pitches: D0), E0, FP, A0, and B0) (17/15, 51/40, 17/12, 17/10, and 17/9); as well as the establishment of complement major triads (4:5:6), which generate the five utonal pitches: BQ, AQ, GQ, EQ, and DQ (30/17, 80/51, 24/17, 20/17, and 18/17). In this particular case, the exercise omits necessary perfect fourth and perfect fifth pitch material.

Fig. 20. Seventeen-limit otonal pitch generation

A0 (17/10) CP (17/16) E0 (51/40) GP (51/32) D0) (17/15) FP (17/12)

FP (17/12) CP (17/16) B0) (17/9)

Fig. 21. Seventeen-limit utonal pitch generation

GQ (24/17) CQ (32/17) DQ (18/17)

CQ (32/17) EQ (20/17) FQ (64/51) AQ (80/51) GQ (24/17) BQ (30/17)

The necessary additional pitch material is obtained via a secondary process that involves the calculation of an 8/5 relationship with B0) (17/9), which generates the otonal pitch G0) (68/45), and utonal complement F1( (45/34). A minor adjustment is also made with regards to E0 (51/40), and due to the fact this pitch does not represent the simplest ratio available within the seventeen-limit intonation scheme. E0) (34/27) and AQ( (27/17) are adopted as a consequence. The pitch material is then sequentially arranged as:

341 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115. The Harmonic Consideration 267

C DQ D0) EQ E0) F FP GQ G AQ( A0 BQ B0) 1 18 17 20 34 4 17 24 3 27 17 30 17

1 17 15 17 27 3 12 17 2 17 10 17 9 F1( G0) 45 68

34 45

The series of intervals presents the septendecimal diatonic semitone (18/17), septendecimal supermajor second (17/15), septendecimal subminor third (20/17), septendecimal supermajor third (34/27), just perfect fourth (4/3), septendecimal tritone, or superaugmented fourth (17/12), just perfect fifth (3/2), septendecimal subdiminished fifth (24/17), septendecimal subminor sixth (27/17), septendecimal supermajor sixth (17/10), septendecimal subminor seventh (30/17), and septendecimal supermajor seventh (17/9); as well as the non-essential tones septendecimal subfourth (45/34) and septendecimal superfifth (68/45). The following table depicts the essential tonal resources of seventeen-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 155. Essential seventeen-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ DQ septendecimal diatonic semitone 18/17 1.058824 98.955 ÐÓ D0) septendecimal supermajor second 17/15 1.133333 216.687 ÐÔ EQ septendecimal subminor third 20/17 1.176471 281.358 ÐÕ E0) septendecimal supermajor third 34/27 1.259259 399.090 ÐÖ F1( septendecimal subfourth 45/34 1.323529 485.268 Ð× FP septendecimal tritone, or superaugmented fourth 17/12 1.416667 603.000 ÐØ G0) septendecimal superfifth 68/45 1.511111 714.732 ÐÙ AQ( septendecimal subminor sixth 27/17 1.588235 800.910 ÑÐ A0 septendecimal supermajor sixth 17/10 1.700000 918.642 ÑÑ BQ septendecimal subminor seventh 30/17 1.764706 983.313 ÑÒ B0) septendecimal supermajor seventh 17/9 1.888889 1101.045

ÐÑ C octave 2/1 2.000000 1200.000

Seventeen-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #3 within the pitches C, D0), EQ, G, and AQ( (1/1, 17/15, 20/17, 3/2, and 27/17); and the complements: C, BQ, A0, F, and E0) (2/1, 30/17, 17/10, 4/3, and 34/27).

6 h amncConsideration Harmonic The 268 Fig. 22. 2, 3, 5, 17-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 269

Ex. 115. Lou Harrison pentatonic #3 (seventeen-limit intonation)

The modulation to D (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

D EQ( E0 F1( FP G GP AQ( A( BQ( B0 C1( CP 9 81 51 45 17 3 51 27 27 243 153 135 17

8 68 40 34 12 2 32 17 16 136 80 68 16 G1( A0 405 17

272 10

The following two tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 156. 17-limit tuning matrix no. 1 ‘key of D’ (Program 12)

NOTE C1( CP D EQ( E0 F1( FP G GP A( BQ( B0

KEY C C! D D! E F F! G G! A A! B

CENTS ß13 +05 +04 +03 +21 ß15 +03 +02 +07 +06 +05 +23

Table 157. 17-limit tuning matrix no. 2 ‘key of D’ (Program 13)

NOTE C1( DQ D EQ( E0 F1( FP G AQ( A( BQ( B0

KEY C C! D D! E F F! G G! A A! B

CENTS ß13 ß01 +04 +03 +21 ß15 +03 +02 +01 +06 +05 +23

270 The Harmonic Consideration

Introduced in the first system of section F (bars 81-82), are septendecimal major and minor triads. The root position seventeen-limit major triad is represented by the ratio 54:68:81, identities 27-17-81, intervals 27/16, 17/16, and 81/64, and the pitches A(, CP, and E(; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of +34.283 and +0.000 cents on each count, which is a deviation of a septendecimal comma (51/50) for the major third. E(5, or the Pythagorean major third (81/64), presents 0.000 beats between the third harmonic of A(4, or the Pythagorean major sixth (27/16), and the second harmonic of E(5, and 0.000 beats between the sixth harmonic of A(4 and the fourth harmonic of E(5 (2648.959Hz); while CP5 (555.954Hz), or the septendecimal chromatic semitone (17/16), 16.352 beats between the fifth harmonic of A(4 (2207.466Hz) and the fourth harmonic of CP5 (2223.817Hz).

Table 158. The beating characteristics of the seventeen-limit major triad

A(4 FREQUENCY CP5 FREQUENCY BEATS E(5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 441.493 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 555.954 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 662.240 – – – – 2 882.986 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1111.909 – – – – – – – – – – – – – – – – 3 1324.479 – – – – – – – – – – – – 2 1324.479 0.000 – – – – – – – – 3 1667.863 – – – – – – – – – – – – – – – – 4 1765.973 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1986.719 – – – – 5 2207.466 4 2223.817 16.352 – – – – – – – – – – – – – – – – – – – – 5 2616.256 – – – – – – – – – – – – – – – – 6 2648.959 – – – – – – – – – – – – 4 2648.959 0.000 7 3090.452 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 3335.726 – – – – 5 3311.199 – – – – 8 3531.945 – – – – – – – – – – – – – – – – – – – – – – – –

The root position seventeen-limit minor triad is represented by the ratio 34:40:51, identities 17-5-51, intervals 17/16, 5/4, and 51/32, and the pitches CP, E, and GP; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß34.283 and +0.000 cents on each count, which is a deviation of a septendecimal comma (51/50) for the minor third. GP4 (416.966Hz), or the septendecimal superaugmented fifth (51/32, or 806.910 cents), presents 0.000 beats between the third harmonic of CP4 (833.931Hz), or the septendecimal chromatic semitone (17/16), and the second The Harmonic Consideration 271 harmonic of GP4 (833.931Hz), and 0.000 beats between the sixth harmonic of CP4 (1667.863Hz) and the fourth harmonic of GP4 (1667.863Hz); while E4 (327.032Hz), or the just major third (5/4), 32.703 beats between the sixth harmonic of CP4 and the fifth harmonic of E4 (1635.160Hz).

Table 159. The beating characteristics of the seventeen-limit minor triad

CP4 FREQUENCY E4 FREQUENCY BEATS GP4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 277.977 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 327.032 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 416.966 – – – – 2 555.954 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 654.064 – – – – – – – – – – – – – – – – 3 833.931 – – – – – – – – – – – – 2 833.931 0.000 – – – – – – – – 3 981.096 – – – – – – – – – – – – – – – – 4 1111.909 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1250.897 – – – – – – – – – – – – 4 1308.128 – – – – – – – – – – – – – – – – 5 1389.886 – – – – – – – – – – – – – – – – – – – – – – – – 6 1667.863 5 1635.160 32.703 4 1667.863 0.000 7 1945.840 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1962.192 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 2084.829 – – – – 8 2223.817 – – – – – – – – – – – – – – – – – – – – – – – –

The sonorities at the beginning of the second system (bars 85-86) present an intonational reinterpretation of the simultaneous sonorities presented in the previous E section, which include hexads D9(!11)/F!, featuring the pitches D, FP, A(, C1(, E0, and GP (4590:5780:6885:8100:10404:13005); and Dmin9(!11)/F, featuring the pitches D, F1(, A(,C1(, E0, and GP (4590:5400:6885:8100:10404:13005). The motivic ostinato introduced in section E (thirteen-limit just intonation) featuring chromatic expansion and rhythmic recapitulation is now further developed in section F (seventeen-limit just intonation), with chromatic ornamentation, as well as transformation via intonational reinterpretation. The simultaneous sonorities in the fourth system (bars 93-94) include a B"7/D tetrad with the pitches D, BQ(, F1(, and AQ( (170:270:405:486). The concluding sonorities of the fourth system (bar 96) facilitate a modulation to A major (in section G) via an E"(!11) hexad – the pitches EQ(, G, BQ(,DQ, F1( and FD(, or the ratio 54:68:81:96:120:153.

272 The Harmonic Consideration

Nineteen-Limit Just Intonation

The primary interval of nineteen-limit just intonation is the nineteenth harmonic, or nonadecimal subminor, or overtone minor third, which is represented by the frequency ratio 19/16 (297.513 cents), and notated as EW. The complement, 32/19 (902.487 cents), or nonadecimal supermajor sixth, is notated as A3. The notational symbols for the partial chroma ([2] and [3], lowering or raising a tone by 96/95 [one nonadecimal comma], or 18.128 cents) is in reference to the “amount the just minor third (6/5) exceeds the nineteenth harmonic” (19/16Ï96/95=6/5).342 19/16 is 2.487 cents flat from the equal minor third ( 4 2 ), while 32/19, 2.487 sharp from the equal major sixth ( [).4 ]2 3 The process adopted for the creation of nineteen-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the nineteenth harmonics of C (1/1), G (3/2), and F (4/3), which generate the five otonal pitches: DW), EW, GW, AW, and BW (19/18, 19/16, 57/40, 19/12, and 57/32); as well as the establishment of the complement major triads (4:5:6), which generate five utonal pitches: B3, A3, FV, E3, and D3) (36/19, 32/19, 80/57, 24/19, and 64/57). In this particular case, the exercise omits necessary perfect fourth and perfect fifth pitch material.

Fig. 23. Nineteen-limit otonal pitch generation

CW (19/10) EW (19/16) GW (57/40) BW (57/32) FW (19/15) AW (19/12)

AW (19/12) EW (19/16) DW) (19/18)

Fig. 24. Nineteen-limit utonal pitch generation

E3 (24/19) A3 (32/19) B3 (36/19)

A3 (32/19) CV (20/19) D3) (64/57) FV (80/57) E3 (24/19) FV (30/19)

The necessary additional pitch material is obtained via a secondary process that involves the calculation of a 5/4 relationship with DW) (19/18), which generates the otonal pitch F2 (95/72) and utonal complement G3 (144/95). A minor adjustment is also made with regards to GW (57/40), and due to the fact this pitch does not represent the simplest ratio available within the nineteen-limit intonation scheme. GW) (38/27) and FV( (27/19) are adopted as a consequence. The pitch material is then sequentially arranged as:

342 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115. The Harmonic Consideration 273

C DW) D3) EW E3 F FV( GW) G AW A3 BW B3 1 19 64 19 24 4 27 38 3 19 32 57 36

1 18 57 16 19 3 19 27 2 12 19 32 19 F2 G3 95 144

72 95

The series of intervals presents the nonadecimal diatonic semitone (19/18), nonadecimal supermajor second (64/57), nonadecimal subminor, or overtone minor third (19/16), nonadecimal supermajor third (24/19), just perfect fourth (4/3), nonadecimal tritone, or superaugmented fourth (27/19), just major fifth (3/2), nonadecimal subdiminished fifth (38/27), nonadecimal subminor sixth (19/12), nonadecimal supermajor sixth (32/19), nonadecimal subminor seventh (57/32), nonadecimal supermajor seventh (36/19); as well as the non-essential tones nonadecimal subfourth (95/72) and nonadecimal superfifth (144/95). The following table depicts the essential tonal resources of nineteen-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 160. Essential nineteen-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ DW) nonadecimal diatonic semitone 19/18 1.055556 93.603 ÐÓ D3) nonadecimal supermajor second 64/57 1.122807 200.532 ÐÔ EW nonadecimal subminor, or overtone minor third (19th harmonic) 19/16 1.187500 297.513 ÐÕ E3 nonadecimal supermajor third 24/19 1.263158 404.442 ÐÖ F2 nonadecimal subfourth 95/72 1.319444 479.917 Ð× FV( nonadecimal tritone, or superaugmented fourth 27/19 1.421053 608.352 ÐØ G3 nonadecimal superfifth 144/95 1.515789 720.083 ÐÙ AW nonadecimal subminor sixth 19/12 1.583333 795.558 ÑÐ A3 nonadecimal supermajor sixth 32/19 1.684211 902.487 ÑÑ BW nonadecimal subminor seventh (57th harmonic) 57/32 1.781250 999.468 ÑÒ B3 nonadecimal supermajor seventh 36/19 1.894737 1106.397

ÐÑ C octave 2/1 2.000000 1200.000

7 h amncConsideration Harmonic The 274 Fig. 25. 2, 3, 5, 19-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 275

The adaptation of Harrison’s pentatonic #4 to nineteen-limit intonation presents a collection of pitches that include: C, E3, F, G, and B3 (1/1, 24/19, 4/3, 3/2, and 36/19); as well as the complements: C, AW, G, F, and DW) (2/1, 19/12, 3/2, 4/3, and 19/18).

Ex. 116. Lou Harrison pentatonic #4 (nineteen-limit intonation)

The modulation to A( (presented in the tuning matrix) effectively results in the following transposition of the pitch series:

A( BW B3 C2( CV( D DV( EW E( F2( FV( G2( GV( 27 57 36 513 81 9 729 19 81 171 27 1539 243

16 32 19 512 76 8 608 16 64 128 19 1024 152 D2 E3( 285 243

256 190

The following tuning matrix contains the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 161. 19-limit tuning matrix no. 1 ‘key of A’ (Program 14)

NOTE C2( CV( D DV( E( F2( FV( G2( GV( A( BW B3

KEY C C! D D! E F F! G G! A A! B

CENTS +13 +10 +04 +14 +08 +01 +08 +05 +12 +06 ß01 +06

The root position nineteen-limit major triad is represented by the ratio 38:48:57, identities 19-3-57, intervals 19/16, 3/2, and 57/32, and the pitches EW, G, and BW; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß18.128 and +0.000 cents on each count, which is a 276 The Harmonic Consideration deviation of a nonadecimal comma (96/95) for the major third. BW4 (466.021Hz), or the nonadecimal subminor seventh (57/32, or 999.468 cents), presents 0.000 beats between the third harmonic of EW4 (932.041Hz), or the nonadecimal subminor third (19/16), and the second harmonic of BW4 (932.041Hz), and 0.000 beats between the sixth harmonic of EW4 (1864.082Hz) and the fourth harmonic of BW4 (1864.082Hz); while G4, or the just perfect fifth (3/2), 16.352 beats between the fifth harmonic of EW4 (1553.402Hz) and the fourth harmonic of G4.

Table 162. The beating characteristics of the nineteen-limit major triad

EW4 FREQUENCY G4 FREQUENCY BEATS BW4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 310.680 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 392.438 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 466.021 – – – – 2 621.361 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 784.877 – – – – – – – – – – – – – – – – 3 932.041 – – – – – – – – – – – – 2 932.041 0.000 – – – – – – – – 3 1177.315 – – – – – – – – – – – – – – – – 4 1242.721 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1398.062 – – – – 5 1553.402 4 1569.753 16.352 – – – – – – – – – – – – 6 1864.082 – – – – – – – – – – – – 4 1864.082 0.000 – – – – – – – – 5 1962.192 – – – – – – – – – – – – – – – – 7 2174.763 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2354.630 – – – – 5 2330.103 – – – – 8 2485.443 – – – – – – – – – – – – – – – – – – – – – – – –

The root position nineteen-limit minor triad is represented by the ratio 16:19:24, identities 1-19-3, intervals 1/1, 19/16, and 3/2, and the pitches C, EW, and G; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of +18.128 and +0.000 cents on each count, which is a deviation of a nonadecimal comma (96/95) for the minor third. The nonadecimal subminor third (19/16) has been proposed by a number of theorists as a possible substitute for the just minor third (6/5) in the five-limit minor triad (10:12:15), which yields identities 3, 5, and 15, but no 1 identity. The alternative septimal subminor third (7/6) in the seven-limit minor triad (6:7:9) presents identities 3, 7, and 9, and therefore also lacks 1. In striking contrast, the harmonic structure of the nineteen-limit minor triad (16:19:24) allows for the representation of the 1 identity, which coincides with the root the chord. The Harmonic Consideration 277

Doty offers the following discussion with regards to the musical significance of the nonadecimal subminor third:

“However, 19/16 is definitely not a consonance, no more so than is the tempered minor third, and therefore does not yield a consonant triad. In any case, it appears that a group of tones with relative frequencies higher than 8 or 9 is likely to produce an ambiguous sensation of periodicity pitch, so that nothing of practical value is achieved making the root of the chord the fundamental of the harmonic series.”343

G4, or the just perfect fifth (3/2), presents 0.000 beats between the third harmonic of C4, or the unison (1/1), and the second harmonic of G4, and 0.000 beats between the sixth harmonic of C4 and the fourth harmonic of G4; while EW4 (310.680Hz), or the nonadecimal subminor third (19/16), 16.352 beats between the sixth harmonic of C4 and the fifth harmonic of EW4.

Table 163. The beating characteristics of the nineteen-limit minor triad

C4 FREQUENCY EW4 FREQUENCY BEATS G4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 261.626 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 310.680 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 392.438 – – – – 2 523.251 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 621.361 – – – – – – – – – – – – – – – – 3 784.877 – – – – – – – – – – – – 2 784.877 0.000 – – – – – – – – 3 932.041 – – – – – – – – – – – – – – – – 4 1046.502 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1177.315 – – – – – – – – – – – – 4 1242.721 – – – – – – – – – – – – – – – – 5 1308.128 – – – – – – – – – – – – – – – – – – – – – – – – 6 1569.753 5 1553.402 16.352 4 1569.753 0.000 7 1831.379 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 1864.082 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 1962.192 – – – – 8 2093.005 – – – – – – – – – – – – – – – – – – – – – – – –

The end of the second system in section G presents a B"("5/omit 3) simultaneous sonority within a 7-1 suspension in the bass – the pitches BW, BW, and E(, or the ratio 19:38:54, with the following harmony

343 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 64-65. 278 The Harmonic Consideration delineating the triad B"("5)/D – the pitches D, BW, and E(, or the ratio 12:19:27. The concluding sonorities of the fourth system include a E("5)/G! triad (171:216:304) resolving to an A tonal centre within a chord inversion and a 7-1 suspension in the bass.

Twenty-Three-Limit Just Intonation

The primary interval of twenty-three-limit just intonation is the twenty-third harmonic, or trivigesimal tritone, or superaugmented fourth, which is represented by the frequency ratio 23/16 (628.274 cents), and notated as FX(. The complement, 32/23 (571.726 cents), or trivigesimal subdiminished fifth, is notated as GY). The notational symbols for the partial chroma ([4] and [5], raising or lowering a tone by 46/45 [one trivigesimal comma], or 38.051 cents) is in reference to the “amount the twenty-third harmonic exceeds the just tritone” (45/32Ï46/45=23/16). 23/16 is 28.274 cents sharp from the equal augmented fourth ( 2 2 ), while 32/23, 28.274 flat from the equal diminished fifth ( 2 2 ).344 The process adopted for the creation of twenty-three-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the twenty-third harmonics of the fundamental C (1/1), G (3/2), and F (4/3), which generate the six otonal pitches: D4, E4, FX(, G4, A4(, and B4 (23/20, 23/18, 23/16, 23/15, 69/40, and 23/12); as well as the establishment of complement major triads (4:5:6), which generate the six utonal pitches: BY), AY, GY), F5, EY), and DY) (40/23, 36/23, 32/23, 30/23, 80/69, and 24/23).

Fig. 26. Twenty-three-limit otonal pitch generation

D4 (23/20) FX( (23/16) A4( (69/40) CX( (69/64) G4 (23/15) B4 (23/12)

B4 (23/12) FX( (23/16) E4 (23/18)

Fig. 27. Twenty-three-limit utonal pitch generation

DY) (24/23) GY) (32/23) AY (36/23)

GY) (32/23) BY) (40/23) CY) (128/69) EY) (80/69) DY) (24/23) F5 (30/23)

344 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115. The Harmonic Consideration 279

A minor adjustment is also made with regards to A4( (69/40), and due to the fact this pitch does not represent the simplest ratio available within the twenty-three-limit intonation scheme. A4 (46/27) and EY (27/23) is adopted as a consequence. The pitch material is then sequentially arranged as:

C DY) D4 EY E4 F FX( GY) G AY A4 BY) B4 1 24 23 27 23 4 23 32 3 36 46 40 23

1 23 20 23 18 3 16 23 2 23 27 23 12 F5 G4 30 23

23 15

The series of intervals presents the trivigesimal diatonic semitone (24/23), trivigesimal supermajor second (23/20), trivigesimal subminor third (27/23), trivigesimal supermajor third (23/18), just perfect fourth (4/3), trivigesimal tritone, or superaugmented fourth (23/16), just major fifth (3/2), trivigesimal subdiminished fifth (32/23), trivigesimal subminor sixth (36/23), trivigesimal supermajor sixth (46/27), trivigesimal subminor seventh (40/23), trivigesimal supermajor seventh (23/12); as well as the non-essential tones trivigesimal subfourth (30/23) and trivigesimal subminor sixth (36/23). The following table depicts the essential tonal resources of twenty-three-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 164. Essential twenty-three-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ DY) trivigesimal diatonic semitone 24/23 1.043478 73.681 ÐÓ D4 trivigesimal supermajor second 23/20 1.150000 241.961 ÐÔ EY trivigesimal subminor third 27/23 1.173913 277.591 ÐÕ E4 trivigesimal supermajor third 23/18 1.277778 424.364 ÐÖ F5 trivigesimal subfourth 30/23 1.304348 459.994 Ð× FX( trivigesimal tritone, or superaugmented fourth (23rd harmonic) 23/16 1.437500 628.274 ÐØ G4 trivigesimal superfifth 23/15 1.533333 740.006 ÐÙ AY trivigesimal subminor sixth 36/23 1.565217 775.636 ÑÐ A4 trivigesimal supermajor sixth 46/27 1.703704 922.409 ÑÑ BY) trivigesimal subminor seventh 40/23 1.739130 958.039 ÑÒ B4 trivigesimal supermajor seventh 23/12 1.916667 1126.319

ÐÑ C octave 2/1 2.000000 1200.000

8 h amncConsideration Harmonic The 280 Fig. 28. 2, 3, 5, 23-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 281

Twenty-three-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #4 within the pitches C, E4, F, G, and B4 (1/1, 23/18, 4/3, 3/2, and 23/12); and the complements: C, BQ, G, F, and DY) (2/1, 30/17, 3/2, 4/3, and 24/23).

Ex. 117. Lou Harrison pentatonic #4 (twenty-three-limit intonation)

The modulation to A( (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

A( BY B4( C5( CX( D DX( EY E( F5( FX( G5 GX( 27 81 621 729 69 9 621 27 81 243 23 135 207

16 46 320 368 64 8 512 23 64 184 16 92 128 D5 E4( 405 207

368 160

The following two tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 165. 23-limit tuning matrix no. 1 ‘key of A’ (Program 15)

NOTE C5( CX( D DX( E( F5( FX( G5 GX( A( BY B4(

KEY C C! D D! E F F! G G! A A! B

CENTS ß17 +30 +04 +34 +08 ß18 +28 ß36 +32 +06 ß20 +48

282 The Harmonic Consideration

Table 166. 23-limit tuning matrix no. 2 ‘key of A’ (Program 16)

NOTE C5( CX( D DX( E( F5( FX( G5( GX( A( BY B4(

KEY C C! D D! E F F! G G! A A! B

CENTS ß17 +30 +04 +34 +08 ß18 +28 ß15 +32 +06 ß20 +48

The first system of section H features the trivigesimal major triad. The root position twenty-three-limit major triad is represented by the ratio 18:23:27, identities 9-23-27, intervals 9/8, 23/16, and 27/16, and the pitches D, FX(, and A(; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of +38.051 and +0.000 cents on each count, which is a deviation of a trivigesimal comma (46/45) for the major third. A(4, or the Pythagorean major sixth (27/16), presents 0.000 beats between the third harmonic of D4, or the just major tone (9/8), and the second harmonic of A(4, and 0.000 beats between the sixth harmonic of D4 and the fourth harmonic of A(4; while FX(4 (376.087Hz), or the trivigesimal tritone, or superaugmented fourth (23/16), 32.703 beats between the fifth harmonic of D4 (1471.644Hz) and the fourth harmonic of FX(4 (1504.347Hz).

Table 167. The beating characteristics of the twenty-three-limit major triad

D4 FREQUENCY FX(4 FREQUENCY BEATS A(4 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 294.329 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 376.087 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 441.493 – – – – 2 588.658 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 752.174 – – – – – – – – – – – – – – – – 3 882.986 – – – – – – – – – – – – 2 882.986 0.000 – – – – – – – – 3 1128.260 – – – – – – – – – – – – – – – – 4 1177.315 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1324.479 – – – – 5 1471.644 4 1504.347 32.703 – – – – – – – – – – – – 6 1765.973 – – – – – – – – – – – – 4 1765.973 0.000 – – – – – – – – 5 1880.434 – – – – – – – – – – – – – – – – 7 2060.301 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2256.521 – – – – 5 2207.466 – – – – 8 2354.630 – – – – – – – – – – – – – – – – – – – – – – – –

The Harmonic Consideration 283

The root position twenty-three-limit minor triad is represented by the ratio 46:54:69, identities 23-27-69, intervals 23/16, 27/16, and 69/64, and the pitches FX(, A(, and CX(; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß38.051 and +0.000 cents on each count, which is a deviation of a trivigesimal comma (46/45) for the minor third. CX(5 (564.130Hz), or the trivigesimal chromatic semitone (69/64, or 130.229 cents), presents 0.000 beats between the third harmonic of FX(4 (1128.260Hz), or the trivigesimal superaugmented fourth (23/16), and the second harmonic of CX(5 (1128.260Hz) and 0.000 beats between the sixth harmonic of FX(4 (2256.521Hz) and the fourth harmonic of CX(5 (2256.521Hz); while A(4, or the Pythagorean major sixth (27/16), 49.059 beats between the sixth harmonic of FX(4 and the fifth harmonic of A(4.

Table 168. The beating characteristics of the twenty-three-limit minor triad

FX(4 FREQUENCY A(4 FREQUENCY BEATS CX(5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 376.087 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 441.493 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 564.130 – – – – 2 752.174 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 882.986 – – – – – – – – – – – – – – – – 3 1128.260 – – – – – – – – – – – – 2 1128.260 0.000 – – – – – – – – 3 1324.479 – – – – – – – – – – – – – – – – 4 1504.347 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1692.390 – – – – – – – – – – – – 4 1765.973 – – – – – – – – – – – – – – – – 5 1880.434 – – – – – – – – – – – – – – – – – – – – – – – – 6 2256.521 5 2207.466 49.059 4 2256.521 0.000 7 2632.607 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2648.959 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 2820.651 – – – – 8 3008.694 – – – – – – – – – – – – – – – – – – – – – – – –

The final chord in the second system is the triad B"("5), featuring the pitches BY, E(, and D, or the ratio 144:207:368. The third system requires a minor alteration in the tuning scheme, due to the resulting dissonant Amin7(omit 5) simultaneous sonority – the pitches A(, C5(, and G5, or the ratio 23:27:40. Analysis of the harmonic properties of the simultaneous sonority reveals the appropriate trivigesimal subminor third (27/23) between the bottom two sonorities, yet a grave or small fifth (40/27) between the upper two sonorities. In order to adhere to the more aesthetically pleasing intervallic properties of the twenty-three-limit minor seventh tetrad it is then simply a matter of retuning G5 (135/92) up a 284 The Harmonic Consideration syntonic comma (81/80) to G5( (2187/1472), which essentially forms a 3/2 relationship between the upper two sonorities, and generates a Amin7(omit 5) simultaneous sonority with the ratio 46:54:81. Interesting sonorities presented in the fourth system include the final sonorities at the end of the system, which facilitate a modulation to E major via an Emaj7/D! tetrad – the pitches DX(,E(, GX(, and B4(, or the ratio 345:360:460:552, moving to a second inversion E major triad – the pitches B4(,E(, and GX(, or the ratio 207:270:345.

Twenty-Nine-Limit Just Intonation

The primary interval of twenty-nine-limit just intonation is the twenty-ninth harmonic, or grave or small nonavigesimal neutral seventh, which is represented by the frequency ratio 29/16 (1029.577 cents), and notated as B_. The complement, 32/29 (170.423 cents), or nonavigesimal acute or large neutral second, is notated as D7). The notational symbols for the partial chroma ([6] and [7], raising or lowering a tone by 145/144 [one nonavigesimal comma], or 11.981 cents) is in reference to the “amount the twenty- ninth harmonic exceeds the acute or large minor seventh” (9/5Ï145/144=29/16). 29/16 is 29.577 cents sharp from the equal minor seventh ([6 ]2 5 ), while 32/29, 29.577 flat from the equal major second ( 12 2).345 The process adopted for the creation of twenty-nine-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the twenty-ninth harmonics of the fundamental C (1/1), G (3/2), and F (4/3), which generate the six otonal pitches: D_, E_, F6(, G_, A_, and B_ (87/80, 29/24, 87/64, 29/20, 29/18, and 29/16); as well as the establishment of complement major triads (4:5:6), which generate the six utonal pitches: B7), A7, G7), F^, E7, and D7) (160/87, 48/28, 128/87, 40/29, 36/29, and 32/29).

Fig. 29. Twenty-nine-limit otonal pitch generation

G_ (29/20) B_ (29/16) D_ (87/80) F6( (87/64) C_ (29/15) E_ (29/24)

E_ (29/24) B_ (29/16) A_ (29/18)

Fig. 30. Twenty-nine-limit utonal pitch generation

A7 (48/29) D7) (32/29) E7 (36/29)

D7) (32/29) F^ (40/29) G7) (128/87) B7) (160/87) A7 (48/29) C^ (30/29)

345 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115. The Harmonic Consideration 285

A minor adjustment is also made with regards to D_ (87/80), and due to the fact this pitch does not represent the simplest ratio available within the twenty-nine-limit intonation scheme. D_) (29/27) and B7 (54/29) is adopted as a consequence. The pitch material is sequentially arranged as:

C D_) D7) E_ E7 F F^ G_ G A_ A7 B_ B7 1 29 32 29 36 4 40 29 3 29 48 29 54

1 27 29 24 29 3 29 20 2 18 29 16 29 F6( G7) 87 128

64 87

The series of intervals presents the nonavigesimal grave or small neutral second (29/27), nonavigesimal acute or large neutral second (32/29), nonavigesimal grave or small neutral third (29/24), nonavigesimal acute or large neutral third (36/29), just perfect fourth (4/3), nonavigesimal tritone, or augmented fourth (40/29), just perfect fifth (3/2), nonavigesimal subdiminished fifth (29/20), nonavigesimal grave or small neutral sixth (29/18), nonavigesimal acute or large neutral sixth (48/29), nonavigesimal grave or small neutral seventh (29/16), nonavigesimal acute or large neutral seventh (54/29); as well as the non-essential tones nonavigesimal superfourth (87/64) and nonavigesimal subfifth (128/87). The following table depicts the essential tonal resources of twenty-nine-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 169. Essential twenty-nine-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ D_) nonavigesimal grave or small neutral second 29/27 1.074074 123.712 ÐÓ D7) nonavigesimal acute or large neutral second 32/29 1.103448 170.423 ÐÔ E_ nonavigesimal grave or small neutral third 29/24 1.208333 327.622 ÐÕ E7 nonavigesimal acute or large neutral third 36/29 1.241379 374.333 ÐÖ F6( nonavigesimal superfourth (87th harmonic) 87/64 1.359375 531.532 Ð× F^ nonavigesimal tritone, or augmented fourth 40/29 1.379310 556.737 ÐØ G7) nonavigesimal subfifth 128/87 1.471264 668.468 ÐÙ A_ nonavigesimal grave or small neutral sixth 29/18 1.611111 825.667 ÑÐ A7 nonavigesimal acute or large neutral sixth 48/29 1.655172 872.378 ÑÑ B_ nonavigesimal grave or small neutral seventh (29th harmonic) 29/16 1.812500 1029.577 ÑÒ B7 nonavigesimal acute or large neutral seventh 54/29 1.862069 1076.288

ÐÑ C octave 2/1 2.000000 1200.000 8 h amncConsideration Harmonic The 286 Fig. 31. 2, 3, 5, 29-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 287

The adaptation of Harrison’s pentatonic #5 to twenty-nine-limit intonation presents a collection of pitches that include: C, D7), F, G, and B_ (1/1, 32/29, 4/3, 3/2, and 29/16); as well as the complements: C, B_, G, F, and D7) (2/1, 29/16, 3/2, 4/3, and 32/29).

Ex. 118. Lou Harrison pentatonic #5 (twenty-nine-limit intonation)

The modulation to E( (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

E( F6( F^( G6( G^( A( A^( B_( B( C6( C^( D6( D^( 81 87 81 783 729 27 405 2349 243 261 243 2349 2187

64 64 58 512 464 16 232 1280 128 256 232 2048 1856 A6(( B7 7047 54

4096 29

The following tuning matrix contains the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 170. 29-limit tuning matrix no. 1 ‘key of E’ (Program 17)

NOTE C6( C^( D6( D^( E( F6( F^( G6( G^( A( A^( B(

KEY C C! D D! E F F! G G! A A! B

CENTS +33 ß20 +37 ß16 +08 +32 ß22 +35 ß18 +06 ß35 +10

The first system of section I (bars 128-29) features the nonavigesimal minor and major triads. The root position twenty-nine-limit major triad is represented by the ratio 58:72:87, identities 29-9-87, intervals 29/16, 9/8, and 87/64, and the pitches B_, D, and F6(; and presents the just major third (5/4) and just 288 The Harmonic Consideration perfect fifth (3/2) with a falsity of +11.981 and +0.000 cents on each count, which is a deviation of a nonavigesimal comma (145/144) for the major third. F6(5 (711.295Hz), or the nonavigesimal superfourth (87/64, or 531.532 cents), presents 0.000 beats between the third harmonic of B_4 (1422.589Hz), or the nonavigesimal grave or small neutral seventh (29/16), and the second harmonic of F6(5 (1422.589Hz), and 0.000 beats between the sixth harmonic of B_4 (2845.178Hz) and the fourth harmonic of F6(5 (2845.178Hz); while D5, or the just major tone (9/8), 16.352 beats between the fifth harmonic of B_4 (2370.982Hz) and the fourth harmonic of D5.

Table 171. The beating characteristics of the twenty-nine-limit major triad

B_4 FREQUENCY D5 FREQUENCY BEATS F6(5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 474.196 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 588.658 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 711.295 – – – – 2 948.393 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1177.315 – – – – – – – – – – – – – – – – 3 1422.589 – – – – – – – – – – – – 2 1422.589 0.000 – – – – – – – – 3 1765.973 – – – – – – – – – – – – – – – – 4 1896.785 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 2133.884 – – – – 5 2370.982 4 2354.630 16.352 – – – – – – – – – – – – 6 2845.178 – – – – – – – – – – – – 4 2845.178 0.000 – – – – – – – – 5 2943.288 – – – – – – – – – – – – – – – – 7 3319.374 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 3531.945 – – – – 5 3556.473 – – – – 8 3793.571 – – – – – – – – – – – – – – – – – – – – – – – –

The root position twenty-nine-limit minor triad is represented by the ratio 24:29:36, identities 3-29-9, intervals 3/2, 29/16, and 9/8, and the pitches G, B_, and D; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß11.981 and +0.000 cents on each count, which is a deviation of a nonavigesimal comma (145/144) for the minor third. D5, or the just major tone (9/8), presents 0.000 beats between the third harmonic of G4, or the just perfect fifth (3/2), and the second harmonic of D5, and 0.000 beats between the sixth harmonic of G4 and the fourth harmonic of D5; while B_4 (474.196Hz), or the nonavigesimal grave or small neutral seventh (29/16), 16.352 beats between the sixth harmonic of G4 and the fifth harmonic of B_4.

The Harmonic Consideration 289

Table 172. The beating characteristics of the twenty-nine-limit minor triad

G4 FREQUENCY B_4 FREQUENCY BEATS D5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 392.438 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 474.196 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 588.658 – – – – 2 784.877 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 948.393 – – – – – – – – – – – – – – – – 3 1177.315 – – – – – – – – – – – – 2 1177.315 0.000 – – – – – – – – 3 1422.589 – – – – – – – – – – – – – – – – 4 1569.753 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1765.973 – – – – – – – – – – – – 4 1896.785 – – – – – – – – – – – – – – – – 5 1962.192 – – – – – – – – – – – – – – – – – – – – – – – – 6 2354.630 5 2370.982 16.352 4 2354.630 0.000 7 2747.068 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 2845.178 – – – – 5 2943.288 – – – – 8 3139.507 – – – – – – – – – – – – – – – – – – – – – – – –

The sonorities at the beginning of the second system (bars 133-34) present hexads Emin9(!11)/G, featuring the pitches E(, G6(, B(, D6(, F^(, and A^( (1392:1682:2088:2523:3072:3840); and E9(!11)/G!, featuring the pitches E(, G^(, B(, D6(, F^(, and A^( (464:576:696:841:1024:1280). The simultaneous sonorities in the fourth and fifth systems (bars 141-42) include an Emin9/D pentad with the pitches E(, B(, D6(, F^(, and G6( (1392:2088:2523:3072:3364). Time signature change (from quadruple, or double duple time [4/4] to compound quadruple time [12/8]) is now applied to section I (bar 129); significantly altering not only the rhythmic proportions of the motivic material yet again, but also expanding the motivic ostinato introduced in section E (thirteen-limit just intonation. The technique of otonal and utonal harmonic extension (bars 129 and 130 essentially now outline a minor/major tonality shift) is additionally applied to the two-note motive.

Thirty-One-Limit Just Intonation

The primary interval of thirty-one-limit just intonation is the thirty-first harmonic, or untrigesimal supermajor seventh, which is represented by the frequency ratio 31/16 (1145.036 cents), and notated as B8. The complement, 32/31 (54.964 cents), or untrigesimal diatonic semitone, or Greek enharmonic quarter-tone is notated as Da). The notational symbols for the partial chroma ([8] and [9], raising or lowering a tone 290 The Harmonic Consideration by 31/30 [one untrigesimal comma], or 56.767 cents) is in reference to the “amount the thirty-first harmonic exceeds the just diatonic major seventh” (15/8Ï31/30=31/16). 31/16 is 45.036 cents sharp from the equal major seventh ( 12 ]2[ 11 , or approximately 967/512,346 while 32/31, 45.036 cents flat from the equal minor second ( 12 2 ).347 The process adopted for the creation of thirty-one-limit intonation pitch material involves the establishment of just minor triads (5:6:10) from the thirty-first harmonics of the fundamental C (1/1), G (3/2), and F (4/3), which generate the six otonal pitches: D8, E8, F`(, G8, A8, and B8 (93/80, 31/24, 93/64, 31/20, 31/18, and 31/16); as well as the establishment of complement major triads (4:5:6), which generate the six utonal pitches: Ba), Aa, Ga), F9, Ea, and Da) (160/93, 48/31, 128/93, 40/31, 36/31, and 32/31).

Fig. 32. Thirty-one-limit otonal pitch generation

G8 (31/20) B8 (31/16) D8 (93/80) F`( (93/64) C8 (31/30) E8 (31/24)

E8 (31/24) B8 (31/16) A8 (31/18)

Fig. 33. Thirty-one-limit utonal pitch generation

Aa (48/31) Da) (32/31) Ea (36/31)

Da) (32/31) F9 (40/31) Ga) (128/93) Ba) (160/93) Aa (48/31) C9 (60/31)

A minor adjustment is also made with regards to D8 (93/80), and due to the fact this pitch does not represent the simplest ratio available within the thirty-one-limit intonation scheme. D8) (31/27) and Ba) (54/31) are adopted as a consequence. The pitch material is then sequentially arranged as:

C Da) D8) Ea E8 F F`( Ga) G Aa A8 Ba B8 1 32 31 36 31 4 93 128 3 48 31 54 31

1 31 27 31 24 3 64 93 2 31 18 31 16 F9 G8 40 31

31 20

346 Daniélou, Tableau Comparatif des Intervalles Musicaux 29. 347 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115. The Harmonic Consideration 291

The series of intervals presents the nonavigesimal grave or small neutral second (29/27), nonavigesimal acute or large neutral second (32/29), nonavigesimal grave or small neutral third (29/24), nonavigesimal acute or large neutral third (36/29), just perfect fourth (4/3), nonavigesimal tritone, or augmented fourth (40/29), just perfect fifth (3/2), nonavigesimal subdiminished fifth (29/20), nonavigesimal grave or small neutral sixth (29/18), nonavigesimal acute or large neutral sixth (48/29), nonavigesimal grave or small neutral seventh (29/16), nonavigesimal acute or large neutral seventh (54/29); as well as the non-essential tones nonavigesimal superfourth (87/64) and nonavigesimal subfifth (128/87). The following table depicts the essential tonal resources of thirty-one-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 173. Essential thirty-one-limit pitch material

DEGREE NOTE INTERVAL RATIO RATIO CENTS NUMBER (FRACTION) (DECIMAL) ÐÑ C unison 1/1 1.000000 0.000 ÐÒ Da) untrigesimal diatonic semitone, or Greek enharmonic quarter-tone 32/31 1.032258 54.964 ÐÓ D8) untrigesimal supermajor second 31/27 1.148148 239.171 ÐÔ Ea untrigesimal subminor third 36/31 1.161290 258.874 ÐÕ E8 untrigesimal supermajor third 31/24 1.291667 443.081 ÐÖ F9 untrigesimal subfourth 40/31 1.290323 441.278 Ð× F`( untrigesimal tritone, or superaugmented fourth (93rd harmonic) 93/64 1.453125 646.991 ÐØ G8 untrigesimal superfifth 31/20 1.550000 758.722 ÐÙ Aa untrigesimal subminor sixth 48/31 1.548387 756.919 ÑÐ A8 untrigesimal supermajor sixth 31/18 1.722222 941.126 ÑÑ Ba untrigesimal subminor seventh 54/31 1.741935 960.829 ÑÒ B8 untrigesimal supermajor seventh (31st harmonic) 31/16 1.937500 1145.036

ÐÑ C octave 2/1 2.00000 1200.000

The thirty-one limit is the source of one of La Monte Young’s tunings for the ‘dream chord’ – a sonority utilized in a number of his ‘dream’ compositions, such as The Four Dreams of China, and The Second Dream of the High Tension Line Step-Down Transformer. The essence of the ‘dream chord’ is represented by the ratio 6:8:9, and identities 3-1-9, which is then extended via the inclusion of an additional tone generated by the division of the 9/8 interval. Three varieties of the ‘dream chord’ include the tetrads 24:32:35:36 (3-1-35-9), 42:56:62:63 (21-7-31-63), and 12:16:17:18 (3-1-17-9). The second example features the untrigesimal ratio of 31/28 (equal to 176.210 cents).348

348 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 64. 9 h amncConsideration Harmonic The 292 Fig. 34. 2, 3, 5, 31-limit pitch lattice (notation, ratio and cent values)

The Harmonic Consideration 293

Thirty-one-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #4 within the pitches C, D8), F, G, and Ba (1/1, 31/27, 4/3, 3/2, and 54/31); and the complements: C, Ba, A0, F, and D8) (2/1, 54/31, 3/2, 4/3, and 31/27).

Ex. 119. Lou Harrison pentatonic #5 (thirty-one-limit intonation)

The modulation to E( (presented in the tuning matrixes) effectively results in the following transposition of the pitch series:

E( F9( F`( G9( G`( A( A`(( Ba B( C9( C`( D9( D`( 81 81 93 729 837 27 7533 54 243 243 279 2187 2511

64 62 64 496 512 16 4096 31 128 124 256 1984 2048 A9( B8( 405 2511

248 1280

The following two tuning matrixes contain the data required to represent the system of intonation chromatically within the program memory of the Akai S3000XL.

Table 174. 31-limit tuning matrix no. 1 ‘key of E’ (Program 18)

NOTE C9( C`( D9( D`( E( F9( F`( G9( G`( A( – – B(

KEY C C! D D! E F F! G G! A A! B

CENTS ß35 +49 ß31 +53 +08 ß37 +47 ß33 +51 +06 +00 +10

294 The Harmonic Consideration

Table 175. 31-limit tuning matrix no. 2 ‘key of E’ (Program 19)

NOTE C9( C`( D9( D`( E( F9( F`( G9( G`( A( – – A`((

KEY C C! D D! E F F! G G! A A! B

CENTS ß35 +49 ß31 +53 +08 ß37 +47 ß33 +51 +06 +00 ß45

The first system of section J (bars 145-46) features the untrigesimal minor and major triads. The root position thirty-one-limit major triad is represented by the ratio 24:31:36, identities 3-31-9, intervals 3/2, 31/16, and 9/8, and the pitches G, B8, and D; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of +56.767 and +0.000 cents on each count, which is a deviation of a untrigesimal comma (31/30) for the major third. D5, or the just major tone (9/8), presents 0.000 beats between the third harmonic of G4, or the just perfect fifth (3/2), and the second harmonic of D5, and 0.000 beats between the sixth harmonic of G4 and the fourth harmonic of D5; while B84, or the untrigesimal supermajor seventh (31/16), 65.406 beats between the fifth harmonic of G4 and the fourth harmonic of B84 (2027.598Hz).

Table 176. The beating characteristics of the thirty-one-limit major triad

G4 FREQUENCY B84 FREQUENCY BEATS D5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 392.438 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 506.900 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 588.658 – – – – 2 784.877 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1013.799 – – – – – – – – – – – – – – – – 3 1177.315 – – – – – – – – – – – – 2 1177.315 0.000 – – – – – – – – 3 1520.699 – – – – – – – – – – – – – – – – 4 1569.753 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 1765.973 – – – – 5 1962.192 4 2027.598 65.406 – – – – – – – – – – – – 6 2354.630 – – – – – – – – – – – – 4 2354.630 0.000 – – – – – – – – 5 2534.498 – – – – – – – – – – – – – – – – 7 2747.068 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 6 3531.945 – – – – 5 2943.288 – – – – 8 3139.507 – – – – – – – – – – – – – – – – – – – – – – – –

The Harmonic Consideration 295

The root position thirty-one-limit minor triad is represented by the ratio 62:72:93, identities 31-9-93, intervals 31/16, 9/8, and 93/64, and the pitches B8, D, and F`(; and presents the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß56.767 and +0.000 cents on each count, which is a deviation of a untrigesimal comma (31/30) for the minor third. F`(5 (760.349Hz), or the untrigesimal tritone, or superaugmented fourth (93/64), presents 0.000 beats between the third harmonic of B84 (1520.699Hz), or the untrigesimal supermajor seventh (31/16), and the second harmonic of F`(5 (1520.699Hz), and 0.000 beats between the sixth harmonic of B84 (3041.397Hz) and the fourth harmonic of F`(5 (3041.397Hz); while D5, or the just major tone (9/8), 98.110 beats between the sixth harmonic of B84 and the fifth harmonic of D5.

Table 177. The beating characteristics of the thirty-one-limit minor triad

B84 FREQUENCY D5 FREQUENCY BEATS F`(5 FREQUENCY BEATS (PARTIAL) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) (PARTIAL) (HERTZ) (HERTZ) 1 506.900 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 588.658 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1 760.349 – – – – 2 1013.799 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2 1177.315 – – – – – – – – – – – – – – – – 3 1520.699 – – – – – – – – – – – – 2 1520.699 0.000 – – – – – – – – 3 1765.973 – – – – – – – – – – – – – – – – 4 2027.598 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3 2281.048 – – – – – – – – – – – – 4 2354.630 – – – – – – – – – – – – – – – – 5 2534.498 – – – – – – – – – – – – – – – – – – – – – – – – 6 3041.397 5 2943.288 98.110 4 3041.397 0.000 – – – – – – – – 6 3531.945 – – – – – – – – – – – – – – – – 7 3548.297 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 3801.746 – – – – 8 4055.196 – – – – – – – – – – – – – – – – – – – – – – – –

The sonorities in the second system (bars 149-50) present hexads Emin9(!11)/G, featuring the pitches E(, G9(, B(, D9(, F`(, and A`(( (26784:31104:40176:46656:61504:77841); and E9(!11)/G!, featuring the pitches E(, G`(, B(, D9(, F`(, and A`(( (26784:34596:40176:46656:61504:77841). The simultaneous sonorities in the beginning of the fifth system (bars 157-58) include an Emin9/B hexad with the pitches B(, D9(, E(, B(, F`(, and G9( (2511:2916:3348:5022:7688:7776). The work is concluded with an Emin7 tetrad, with the pitches E(, B(, D9(, and G9( (62:93:108:144, with identities 31-93-27-9), which essentially outline an untrigesimal minor seventh simultaneous sonority harmonically 296 The Harmonic Consideration constructed from the tonic (1/1), untrigesimal subminor third (36/31), just perfect fifth (3/2), and untrigesimal subminor seventh (54/31).

Johnston’s Dictum

Johnston offers the following summary of some the characteristics of extended just intonation principles, and ‘the harmonic consideration’:

“Definitely the affect is unique with each tuning. Each overtone is a unique rasa. The third partial, which generates perfect fifths and fourths, contributes stability and strength. The fifth partial, which is the third and sixth, contributes warmth of emotion; ordinary human warmth. The seventh partial creates sensuality, for example in vernacular music like the blues. The eleventh partial introduces ambiguity, because the intervals of 12/11 and 11/10, which are the overtone intervals surrounding the eleventh partial, are neutral seconds, squarely in between major and minor. The 11/9 is a neutral third. The 11/8 is in between a perfect fourth and augmented fourth. The thirteenth partial has a melancholy, dark quality. Nearly every time I’ve used it, it has something to do with death, which would square with the meaning of thirteen in numerology. The seventeenth and nineteenth don’t really bring anything new, because seventeen is almost exactly a tempered half-step and nineteen is close to a tempered minor third.”349

349 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 198. Conclusion

‘Manual’ of Microtonal Composition

The thesis, via its presentation of an articulated exposition of three ‘original’ and unique microtonal composition models individually exploring the expanded tonal resources of Pythagorean intonation, equal temperament, and just intonation was not only able to demonstrate the uniqueness, as well as musical potential of such systems, but also mark some directions for further exploration. The works – Àzàdeh for santñr and tape, Exposiciones for sampled microtonal Schoenhut toy piano, and La Homa Kanto for harmonically tuned synthesizer quartet – together with the classification of 724 intervals in the octave presented at the close (the culmination of research directly related to the compositions), stand as a testament to the limited potential of the vast resources of the microtonal paradigm; merely three grains of sand in a gargantuan ocean. The study will nevertheless have the capacity to serve as one possible ‘manual’ of microtonal composition, and hopefully inspire the composer of tomorrow to adopt the notion, if not some of the principles.

A Vast Universe of Subtle Intervallic Relationships

In conclusion, it may be stated that the virtues of the microtonal paradigm are best summed up by some of it proponents. What follows is a collection of observations that characterize the general appeal of microtonal composition, at the same time disclosing the rationale of some composers for the total abandonment of twelve-tone equally-tempered composition. In a Perspectives of New Music article entitled Six American Composers on Nonstandard Tunings, Douglas Keislar asks a selection of American composers actively involved in the application of “microtonal scales or other non-standard tunings” to describe what they find most interesting about composing with nonstandard tunings. The replies of Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William Schottstaedt follow:

Blackwood: “The aspect that intrigues me most is finding conventional harmonic progressions, or at least coherent progressions found by extension of their analogues in the more familiar tunings.”

Eaton: “Microtones permit a greater variety of harmonic and melodic motion, which in opera helps delineate and define character. My interest in microtones came from three directions. First, I wrote some of the very first pieces that involved woodwind multiphonics in the early 1960s, and I was intrigued by the ‘out-of-tuneness’ of the multiphonics Secondly, I was interested in cluster music. After a while, though, it seemed like a lot of sound and fury signifying absolutely nothing. But by changing the tuning between or within clusters, I could again generate harmonic and melodic motion and have events of some significance 298 Conclusion

occur. Finally, during this period of my life I was making a living as a jazz musician. With jazz I could get involved immediately with microtonal intervals.”

Harrison: “The reason for my interest (in nonstandard tunings) is very simple. Real intervals – the ones with whole number ratios – grab you; they’re beautiful; they draw you into the music; whereas fake intervals like those of equal temperament don’t do much.”

Johnston: “I love extending my vocabulary and trying to imagine unfamiliar sounds. You can generally imagine a melodic line, but it’s very difficult to imagine what combination of strange intervals will sound like.”

Mandelbaum: “I find that extending the consonance to the seventh partial provides a fascinating means of enrichment. It retains traditional consonance and dissonance, unlike atonal music, which has to abolish the old in order to arrive at the new.”

Schottstaedt: “I like the unusual sounds, the intense dissonances in particular, such as the squeezed minor seconds and stretched fourth in Dinosaur Music. I’ve never been much interested in getting cleaner consonances; beats don’t offend me.”350

Doty offers an interesting discussion on the ‘golden age’ of Western music, and the direct benefits that may be attributed to the adoption of equal temperament in the common practice era, which unlike the previous meantone system of tuning, facilitated unrestricted modulation, and hence the development of complex chromatic harmony. Although in spite of its benevolent influences on harmonic music, twelve- tone equal temperament is also acknowledged for ultimately leading to its “demise as a vital compositional style.” Doty offers the following conclusions:

“Twelve-tone equal temperament is a limited and closed system. Once you have modulated around the so-called circle of fifths, through its twelve major and twelve minor keys, and once you have stacked up every combination of tones that can reasonably be considered a chord, there is nowhere left to go in search of new resources. This is essentially where Western composers found themselves at the beginning of the twentieth century. Everything that could be done with the equally-tempered scale and the principles of tonal harmony had been tried, and the system was breaking down. This situation led many composers to the erroneous conclusion that consonance, tonality, and even pitch had been exhausted as organizing principles. What was really exhausted were merely the very limited resources of the tempered scale. By substituting twelve equally spaced tones for a vast universe of subtle intervallic relationships, the composers and theorists of the eighteenth and nineteenth centuries effectively painted Western music into a corner from which it has not, as yet, extradited itself.”351

350 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 184-85. 351 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 5. Conclusion 299

In the final report presented at the end of his residency at the University of Wisconsin in Madison between May 1944 and May 1947 (the setting for the completion of Genesis of a Music – published in 1949 by University of Wisconsin Press), Partch paints an even bleaker picture of the twelve-tone equally- tempered continuum when he speaks of “Western world’s current three-hundred years of twelve-tone paralysis.” Ronald V. Wiecki makes the following statement about the legacy of Partch:

“Partch’s work may remain a curiosity, a rare and esoteric treasure, even if the present glimmer of interest in microtonality grows into a significant aspect of musical practice. But his aesthetic theories, both in their origins and their expression, provide an interesting look into the sense of failure present in American music in the early 1920s, a period in which Partch’s attitudes must have taken shape. That Partch found it necessary to react in such an extreme manner to this sense of crisis is eloquent testimony to its pervasiveness. How conscious Partch was of this situation needs to be examined further, but it can hardly be disputed that Partch’s music is passionate, that it contains truths distilled from the American experience, and that it successfully addresses the essential problem of all American composers of the time – that of establishing a uniquely American musical practice, one not slavishly or unthinkingly based on an imported model. Partch’s response to this problem simply attempted to penetrate much more deeply into the historical roots of the problem, and his solution will remain to inspire others.”352

352 Ronald V. Wiecki, “Relieving ‘12-Tone Paralysis’: Harry Partch in Madison, Wisconsin, 1944-1947” American Music 9.1 (Spring, 1991): 43-60.

Bibliography

Books

Backus, John. The Acoustical Foundations of Music. 2nd ed. New York: Norton, 1977. Barbour, J. Murray. Tuning and Temperament: A Historical Survey. New York: Dover Publications, 2004. Benade, Arthur H. Fundamentals of Musical Acoustics. 2nd ed. New York: Dover Publications, 1990. — . Horn, Strings, and Harmony. New York: Dover Publications, 1992. Bineš, Taqi. The Short History of Persian Music. Tehran: Àrvin Publication, 1995. Blackwood, Easley. The Structure of Recognizable Diatonic Tunings. Princeton, NJ: Princeton U. Press, 1985. Bosanquet, R. H. M. An Elementary Treatise on Musical Intervals and Temperament. Ed. Rudolf Rasch. Utrecht, The Netherlands: Diapason Press, 1987. Brinner, Benjamin. Knowing Music, Making Music: Javanese Gamelan and the Theory of Musical Competence and Interaction. Chicago: U. of Chicago Press, 1995. Chalmers, John H. Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales. Hanover, NH: Frog Peak Music, 1993. Collier, Graham. “Compositional Devices.” Vol. 1. Boston, MA: Berklee Press, 1975. Cope, David. Techniques of the Contemporary Composer. New York: Schirmer Books, 1997. Cowell, Henry. New Musical Resources. Cambridge: Cambridge U. Press, 1996. Daniélou, Alain. Introduction to the Study of Musical Scales. London: India Society, 1943. — . Tableau Comparatif des Intervalles Musicaux. Pondichéry, India: Institut Français d'Indologie, 1958. — . Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. Rochester, VT: Inner Traditions, 1995. Davies, Hugh. “Toy Piano.” The New Grove Dictionary of Music and Musicians. Ed. Stanley Sadie and John Tyrrell. 2nd ed. Vol. 12. London: Macmillan Reference, 2001. 615. Deva, B. Chaitanya. Indian Music. New Delhi: Indian Council for Cultural Relations, 1974. Doty, David D. The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation. 3rd ed. San Francisco: Other Music, 2002. During, Jean, Scheherazade Q Hassan, and Alastair Dick. “Santñr.” The New Grove Dictionary of Musical Instruments. Ed. Stanley Sadie. Vol. 3. London: Macmillan Reference, 1984. 291-92. Farhat, Hormoz. The Dastgàh Concept in Persian Music. New York: Cambridge U. Press, 1990. — . “Iran: Classical Traditions.” The New Grove Dictionary of Music and Musicians. Ed. Stanley Sadie and John Tyrrell. 2nd ed. Vol. 12. London: Macmillan Reference, 2001. 530-37. — . “The Music of Islam.” Ancient and Oriental Music. Ed. Egon Wellesz. London: Oxford U. Press, 1957. 421-77. 302 Bibliography

Farmer, Henry George. A History of Arabian Music (to the XIIIth Century). London: Luzac & Co., 1967. — . Historical Facts for the Arabian Musical Influence. London: William Reeves, 1930. Gilmore, Bob. Harry Partch: A Biography. New Haven, CT: Yale U. Press, 1998. Gordon, Roderick D. The World of Musical Sound. Dubuque, IA: Kendall/Hunt, 1979. Harrison, Lou. Lou Harrison’s Music Primer: Various Items About Music to 1970. New York: C. F. Peters, 1971. Helmholtz, Hermann L. F. On the Sensations of Tone: As a Physiological Basis for the Theory of Music. 2nd ed. New York: Dover Publications, 1954. Henderson, Isobel. “Ancient Greek Music.” Ancient and Oriental Music. Ed. Egon Wellesz. London: Oxford U. Press, 1957. 336-403. Isacoff, Stuart. Temperament: How Music Became a Battleground for the Great Minds of Western Civilization. New York: Vintage, 2003. Johnston, Ian. Measured Tones: The Interplay of Physics and Music. Bristol; Philadelphia: Adam Hilger, 1989. Katzner, Kenneth. The Languages of the World. London: Routledge, 1986. Kostka, Stefan. Materials and Techniques of Twentieth-Century Music. 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. Kunst, Jaap. Music in Java: Its History, its Theory and its Technique. Ed. E. L. Heins. 3rd ed. Vol. 1. The Hague: Martinus Nijhoff, 1973. Lindley, Mark. Mathematical Models of Musical Scales: A New Approach. Bonn: Verlag für Systematische Musikwissenschaft, 1993. Lloyd, Llewelyn Southworth, and Hugh Boyle. Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation. London: McDonald and Jane’s, 1978. Lloyd-Watts, Valery, Carole L. Bigler, and Willard A. Palmer. Ornamentation: A Question and Answer Manual. Van Nuys, CA: Alfred Publishing Co., 1995. Macpherson, Stewart, and Anthony Payne. Rudiments of Music. London: Steiner & Bell, 1994. Malm, William P. Music Cultures of the Pacific, the Near East, and Asia. Englewood Cliffs, NJ: Prentice-Hall, 1967. McCombie, Ian. The Piano Handbook. London: David & Charles, 1980. Miller, Leta E., and Fredric Lieberman. Lou Harrison: Composing a World. New York: Oxford U. Press, 1998. Moin, Moäamad. An Intermediate Iranian Dictionary. 6 vols. Tehran: Amirkabir Publication, 1974. Marñfi, Mñsà. Les Systems de la Musique Traditionelle de L'Iran (Radif). Tehran: Iranian Music Association Publisher, 1995. Bibliography 303

Nakanishi, Akira. Writing Systems of the World: Alphabets, Syllabaries, Pictograms. Rutland, VT: Charles E. Tuttle Co., 1980. Nketia, J. H. Kwabena. The Music of Africa. New York: W. W. Norton & Co., 1974. Olson, Harry F. Music, Physics and Engineering. 2nd ed. New York: Dover Publications, 1967. Pacholczyk, Josef M. “Secular Classical Music in the Arabic Near East.” Musics of Many Cultures. Berkeley, CA: U. of California Press, 1980. 253-68. Partch, Harry. Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments. 2nd ed. New York: Da Capo, 1974. Perlman, Marc. Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory. Berkeley: U. of California Press, 2004. Persichetti, Vincent. Twentieth-Century Harmony: Creative Aspects and Practice. New York: W. W. Norton & Co., 1961. Picken, Laurence. “The Music of Far Eastern Asia: 2. Other Countries.” Ancient and Oriental Music. Ed. Egon Wellesz. London: Oxford U. Press, 1957. 135-94. Piston, Walter. Counterpoint. 4th ed. London: Victor Gollancz, 1964. Rameau, Jean-Philippe. Treatise on Harmony. Trans. Philip Gossett. New York: Dover Publications, 1971. Randel, Don Michael, ed. The New Harvard Dictionary of Music. Cambridge, MA: Belknap Press of Harvard U Press, 1986. Read, Gardner. 20th-Century Microtonal Notation. Westport, CT: Greenwood Press, 1990. Romanowski, Patricia, Holly George-Warren, and Jon Pareles, ed. The New Rolling Stone Encyclopedia of Rock & Roll. New York: Fireside, 1995. Sadie, Stanley, ed. The Grove Concise Dictionary of Music. London: Macmillan, 1994. — . The New Grove Dictionary of Musical Instruments. 3 vols. London: Macmillan, 1984. Salvador-Daniel, Francesco, and Henry George Farmer. The Music and Musical Instruments of the Arab: With Introduction on How to Appreciate Arab Music. Portland, ME: Longwood Press, 1976. Seashore, Carl E. Psychology of Music. New York: Dover Publications, 1967. Sethares, William A. Tuning, Timbre, Spectrum, Scale. 2nd ed. London: Springer-Verlag, 2005. Slonimsky, Nicolas. Music of South America. New York: Da Capo Press, 1972. Soanes, Catherine, ed. Oxford Dictionary of Current English. 3rd ed. London: Oxford U. Press, 2001. Steiglitz, Ken. A DSP Primer: with Applications to Digital Audio and Computer Music. Menlo Park, CA: Addison-Wesley Pub., 1995. Talai, Dariush. “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System.” Ed. Virginia Danielson, Scott Marcus and Dwight Reynolds. The Garland Encyclopedia of World Music: The Middle East. Vol. 6. New York: Routledge, 2002. 865-74. 304 Bibliography

— . Traditional Persian Art Music: The Radif of Mirza Abdollah. Trans. Manoochehr Sadeghi. Coasta Mesa, CA: Mazda, 2000. Tenney, James. A History of ‘Consonance’ and ‘Dissonance’. New York: Excelsior Music Publishing Co., 1988. Tenzer, Michael. Balinese Music. Singapore: Periplus Editions, 1991. Thomson, Virgil. Virgil Thomson. New York: Da Capo Press, 1966. Touma, Habib Hassan. The Music of the Arabs. Trans. Laurie Schwartz. Portland, OR: Amadeus Press, 1957. Zonis, Ella. “Classical Iranian Music.” Musics of Many Cultures. Berkeley, CA: U. of California Press, 1980. 269-83. — . Classical Persian Music: An Introduction. Cambridge, MA: Harvard U. Press, 1973. Von Gunden, Heidi. The Music of Ben Johnston. Metuchen, N.J.:The Scarecrow Press, 1986. Wright, Owen. “Arab Music: Art Music.” The New Grove Dictionary of Music and Musicians. Ed. Stanley Sadie and John Tyrrell. 2nd ed. Vol. 12. London: Macmillan Reference, 2001. 797-833. — . The Modal System of Arab and Persian Music A.D. 1250-1300. London Oriental Series. Vol. 28. Oxford: Oxford U. Press, 1978.

Discography

Blackwood, Easley. Microtonal Compositions by Easley Blackwood. Perf. Easley Blackwood (polyfusion synthesizer), and Jeffrey Kust (guitar). Rec. 16 Sep. 1990. Liner notes by Easley Blackwood. Cedille, 1994. CDR 90000 018.

Dissertations

Ayers, Lydia. “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional Applications.” DMA diss. U. of Illinois, Urbana-Champaign, 1994. PA 9512292. Piraglu, Qmars. “Faràmarz Pàyvar and His Place in Iranian Music.” Diss. Melbourne U., Austral., 2002. — . “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties.” Diss. Göteborg U., Swed., 1998.

Electronic Publications

Antares Audio Technologies. 2006. Antares Audio Technologies. 15 Dec. 2006. . Bibliography 305

Barrett, Kevin. “Understanding Polyrhythms.” Funkster’s Groove Theory. 23 Mar. 2004. 18 Jan. 2006. . “BBC Weather Centre: World Weather.” BBC Home Page. 6 Apr. 2006. British Broadcasting Corporation. 6 Apr. 2006. Braun, Martin. “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic Musical Scale.” Neuroscience of Music. 2 Aug. 2002: S-671 95 Klässbol, Sweden, 10 Apr. 2006. . dB Audioware: Professional Audio Software. 2006. dB Audioware Limited. 15 Dec. 2006. . Dhomont, Francis. “Acousmatic Update.” Contact! 8.2 Spring, 1995. CEC – Communauté Électroacoustique Canadienne / Canadian Electroacoustic Community. 27 Jan. 2006. DSound. 2005. DSound. 15 Dec. 2006. . “Encyclopedia of Microtonal Music Theory.” Microtonal, Just Intonation Electronic Music Software. 2005. Tonalsoft. 17 Nov. 2006. . Extensible Toy Piano Project, The. Ed. David Claman and Matt Malsky. 1 Jan. 2005. Clark U., Worcester, MA. 21 Aug. 2005. . Gann, Kyle. “Anatomy of an Octave.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005. . — . “An Introduction to Historical Tunings.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005. . — . “Just Intonation Explained.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005. . — . “My Idiosyncratic Reasons for Using Just Intonation.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005. . Keenan, David C. “A Note on the Naming of Musical Intervals.” David Keenan’s Home Page. 3 Nov. 2001. 22 Nov. 2006. . Lewis, Martha Beth. “Tuning Your Piano: Why Pianos Go Out of Tune.” Martha Beth Lewis’ Home Page. 1999. 8 Apr. 2006. . Op de Coul, Manuel. “Huygens-Fokker: List of intervals.” Huygens-Fokker Foundation: Centre for Microtonal Music. 2006. Huygens-Fokker Foundation. 22 Nov. 2006. . Peterson, James B. “Names of Bases.” The Math Forum: Ask Dr. Math. 15 Apr. 2002. Drexel U., Philadelphia, PA. 22 Nov. 2006. . 306 Bibliography

PSP Audioware: Audio Processors and Effects Plug-ins. 2006. PSP Audioware. 15 Dec. 2006. . Schoenhut Toy Piano Company: Over 130 Years. 2005. Schoenhut Toy Piano Company. 21 Aug. 2005. . Sony Media Software: Home for Vegas, Sound Forge and Acid. 2006. Sony Corporation of America. 15 Dec. 2006. . Spin Audio Software. 2006. Spin Audio Software. 15 Dec. 2006. . Stearns, Dan. “Some Thoughts on an Alternative Definition of Equal Temperament.” Kronosonic. 2006. The International Society for Creative Guitar and String Music. 1 Aug. 2006. . Tatlow, Ruth. “Golden Number (Golden Section).” Grove Music Online. Ed. L. Macy. 2004. 16 Dec. 2004. . Vai, Steve. “Little Black Dots: Tempo Metal.” The Official Steve Vai Website. 1983. 18 Jan. 2006. . Waves: The World’s Leading Developer of Audio Signal Processing Software. 2006. Waves Incorporated. 15 Dec. 2006. .

Multimedia

Encyclopaedia Britannica. “Humidity.” Encyclopaedia Britannica 2001. Deluxe ed. CD-ROM. Chicago: Encyclopaedia Britannica, 2001. — . “Sound.” Encyclopaedia Britannica 2001. Deluxe ed. CD-ROM. Chicago: Encyclopaedia Britannica, 2001. — . “Statistics.” Encyclopaedia Britannica 2001. Deluxe ed. CD-ROM. Chicago: Encyclopaedia Britannica, 2001.

Periodicals

Adkins, Cecil. “The Technique of the Monochord.” Acta Musicologica 39 (Jan.-Jun., 1967): 34-43. Barbour, J. Murray. “Irregular Systems of Temperament.” Journal of the American Musicological Society 1.3 (Autumn, 1948): 20-26. Benjamin, Gerald R. “Julian Carrillo and ‘Sonido Trece’ (Dedicated to the Memory of Nabor Carrillo).” Anuario 3 (1967): 33-68. Berard, Wilford W. “The Eleventh and Thirteenth Partials.” Journal of Music Theory 5.1 (Spring, 1961): 95- 107. Bibliography 307

Blackwood, Easley. “Modes and Chord Progressions in Equal Tunings.” Perspectives of New Music 29.2 (Summer, 1991): 166-200. Bobbitt, Richard. “The Physical Basis of Intervallic Quality and its Application to the Problem of Dissonance.” Journal of Music Theory 3.2 (Nov., 1959): 173-207. Bosanquet, R. H. M. “Temperament; Or, the Division of the Octave (Part I).” Proceedings of the Musical Association. 1st Sess. (1874-75): 4-17. — . “Temperament; Or, the Division of the Octave (Part II).” Proceedings of the Musical Association. 2nd Sess. (1874-75): 112-58. — . “The Theory of the Division of the Octave, and the Practical Treatment of the Musical Systems Thus Obtained.” Proceedings of the Royal Society of London 23 (1874-1875): 390-408. Brown, J. P., and William B. Rose. “Humidity and Moisture in Historic Buildings: The Origins of Buildings and Object Conservation.” APT Bulletin 27.3 (1996): 12-23. Desantos, Sandra. “Acousmatic Morphology: An Interview with François Bayle.” Computer Music Journal 21.3 (Fall, 1997): 11-19. De Klerk, Dirk. “Equal Temperament.” Acta Musicologica 51.1 (Jan.-Jun., 1979): 140-50. De Vitry, Philippe. “Philippe de Vitry's ‘Ars Nova’: A Translation.” Journal of Music Theory 5.2 (Winter, 1961): 204-23. Dunne, Edward, and Mark McConnell. “Pianos and Continued Fractions.” Mathematics Magazine 72.2 (Apr., 1999): 104-15. Ellis, Alexander J. “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments wit Fixed Tones.” Proceedings of the Royal Society of London 13 (1863-64): 93-108. Ellis, Alexander J., and Alfred J. Hipkins. “Tonometrical Observations on Some Existing Non-Harmonic Musical Scales.” Proceedings of the Royal Society of London 37 (1884): 368-85. Fokker, A. D. “Equal Temperament and the Thirty-One-Keyed Organ.” The Scientific Monthly 81.4 (Oct., 1955): 161-66. Fonville, John. ”Ben Johnston’s Extended Just Intonation: A Guide for Interpreters.” Perspectives of New Music 29.2 (Summer, 1991): 106-37. Fuller, Ramon. “A Study of Microtonal Equal Temperaments.” Journal of Music Theory 35.1/2 (Spring- Autumn, 1991): 211-37. Gilmore, Bob. “Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben Johnston, and James Tenney.” Perspectives of New Music 33.1/2 (Winter-Summer, 1995): 458- 503. — . “On Harry Partch's Seventeen Lyrics by Li Po.” Perspectives of New Music 30.2 (Summer, 1992): 22- 58. 308 Bibliography

Jairazbhoy, N. A., and A. W. Stone. “Intonation in Present-Day North Indian Classical Music.” Bulletin of the School of Oriental and African Studies, University of London 26.1 (1963): 119-32. Johnston, Ben. ”Scalar Order as a Compositional Resource.” Perspectives of New Music 2.2 (Summer, 1964): 56-76. Keislar, Douglas. ”Introduction.” Perspectives of New Music 29.1 (Winter, 1991): 173-75. Keislar, Douglas, Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William Schottstaedt. “Six American Composers on Nonstandard Tunings.” Perspectives of New Music 29.1 (Winter, 1991): 176-211. Kelley, Truman L. “How Many Figures are Significant.” Science 60.1562 (Dec. 5, 1924): 524. Kuttner, Fritz A. “A Musicological Interpretation of the Twelve Lüs in China's Traditional Tone System.” Ethnomusicology 9.1 (Jan., 1965): 22-38. — . “Prince Chu Tsai-Yu's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory.” Ethnomusicology 19.2 (May, 1975): 163-206. Leedy, Douglas. “A Venerable Temperament Rediscovered.” Perspectives of New Music 29.2 (Summer, 1991): 201-11. — . “Selected Musical Compositions (1948-1972).” Notes 46.1 (Sep., 1989): 224-26. Mandelbaum, Joel. “Toward the Expansion of Our Concepts of Intonation.” Perspectives of New Music 13.1 (Autumn-Winter, 1974): 216-26. Marcus, Scott. “The Interface Between Theory and Practice: Intonation in Arab Music.” Asian Music 24.2 (Spring-Summer, 1993): 39-58. McClure, A. R. “Studies in Keyboard Temperaments.” The Galpin Society Journal 1 (Mar., 1948): 28-40. Miller, Terry E., and Sam-ang Sam. “The Classical Musics of Cambodia and Thailand: A Study of Distinctions.” Ethnomusicology 39.2 (Spring-Summer, 1995): 229-43. Nooshin, Laudan. “The Song of the Nightingale: Processes of Improvisation in Dastgàh Segàh (Iranian Classical Music).” British Journal of Ethnomusicology 7 (1998): 69-116. Norden, N. Lindsay. “A New Theory of Untempered Music: A Few Important Features with Special Reference to ‘A Capella’ Music.” The Musical Quarterly 22.2 (Apr., 1936): 217-33. Orton, Richard. “The 31-Note Organ.” The Musical Times 107.1478 (Apr., 1966): 342-43. Pearson, E. S., and Joan Haines. “The Use of Range in Place of Standard Deviation in Small Samples.” Supplement in the Journal of the Royal Statistical Society 2.1 (1935): 83-98. Pertout, Andrián. “Siamak Noory: Pardis.” Interview with Siamak Noory. Australian Musician 30 (Winter, Jun. 2002): 35. — . “Siamak Noory: Pardis.” Interview with Siamak Noory. Mixdown 104 (Dec. 2002): 38. — . “Siamak Noory: The Santurist – Part 1.” Interview with Siamak Noory. Mixdown 97 (May 2002): 25. — . “Siamak Noory: The Santurist – Part 2.” Interview with Siamak Noory. Mixdown 98 (Jun. 2002): 25. Bibliography 309

— . “Siamak Noory: The Santurist – Part 3.” Interview with Siamak Noory. Mixdown 101 (Sep. 2002): 26. — . “Siamak Noory: The Santurist – Part 4.” Interview with Siamak Noory. Mixdown 103 (Nov. 2002): 32. Rapoport, Paul. “The Notation of Equal Temperaments.” Xenharmonikôn: An Informal Journal of Experimental Music 16 (Autumn, 1995): 61-84. — . “The Structural Relationships of Fifths and Thirds in Equal Temperaments.” Journal of Music Theory 37.2 (Autumn, 1993): 351-89. — . “Towards the Infinite Expansion of Tonal Resources.” Tempo 144 (Mar., 1983): 7-11. Reider, Joseph. “Jewish and Arabic Music.” The Jewish Quarterly Review 7.4 (Apr., 1917): 635-44. Sacks, Oliver, G. Schlaug, L. Jäncke, Y. Huang, and H. Steinmetz. “Musical Ability.” Science 268.5211 (May 5, 1995): 621-22. Stanford, Charles Villiers. “On Some Recent Tendencies in Composition.” Proceedings of the Musical Association, 47th Sess. (1920): 39-53. Swinburne, James. “The Ideal Scale: Its AEtiology, Lysis and SequelAE.” Proceedings of the Musical Association, 63rd sess. (1936-1937): 39-64. Tischler, Hans. “Musica Ficta in the Thirteenth Century.” Music & Letters 54.1 (Jan., 1973): 38-56. Touma, Habib Hassan. “The Maqam Phenomenon: An Improvisation Technique in the Music of the Middle East.” Ethnomusicology 15.1 (Jan., 1971): 38-48. Wiecki, Ronald V. “Relieving ‘12-Tone Paralysis’: Harry Partch in Madison, Wisconsin, 1944-1947.” American Music 9.1 (Spring, 1991): 43-66. Winnington-Ingram, R. P. “Aristoxenus and the Intervals of Greek Music.” The Classical Quarterly 26.3/4 (Jul.-Oct., 1932): 195-208. Wolf, Daniel James. “Alternative Tunings, Alternative Tonalities.” Contemporary Music Review 22.1-2 (2003): 3-14. Young, Gayle. “The Pitch Organization of Harmonium for James Tenney.” Perspectives of New Music 26.2 (Summer, 1988): 204-12. Zonis, Ella. “Contemporary Art Music in Persia.” The Music Quarterly 51.4 (Oct., 1965): 636-48.

Photographs

1967 William Dowd French Double Harpsichord. Personal photograph of Alex Pertout. 3 Feb. 2007. Qmars Piraglu. Personal photograph of Andrián Pertout. 22 Oct. 2006. 310 Bibliography

Malsky, Matt. “Schoenhut Model 6625: 25-Key Toy Piano.” Feb. 2005. The Extensible Toy Piano Project. Ed. David Claman and Matt Malsky. 1 Mar. 2005. Clark U., Worcester, MA. 21 Aug. 2005.

Appendix A

Comparative Table of Musical Intervals

1 Appendices 312 Table 178. Comparative table of musical intervals

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÐÑ unison (1st harmonic) 1/1 1.000000 261.626 0.000 ÐÒ equal hundredth-semitone 1200 2 , or approximately 1731/1730 1.000578 261.777 1.000

11 81 ÐÓ one-eleventh syntonic comma, or skhisma 80 , or 32805/32768 1.001130 261.921 1.955 6 81 ÐÔ one-sixth syntonic comma 80 1.002073 262.168 3.584 ÐÕ cyclic octave (A) LIII 3õó/2øô 1.002090 262.172 3.615

5 81 ÐÖ one-fifth syntonic comma 80 1.002488 262.276 4.301 4.5 81 Ð× two-ninth syntonic comma 80 1.002764 262.349 4.779 4 81 ÐØ one-quarter syntonic comma 80 1.003110 262.439 5.377 3.5 81 ÐÙ two-seventh syntonic comma 80 1.003556 262.556 6.145 3 81 ÑÐ one-third syntonic comma 80 1.004149 262.711 7.169 2 81 ÑÑ one-half syntonic comma 80 1.006231 263.256 10.753 ÑÒ nonavigesimal comma 145/144 1.006944 263.442 11.981 ÑÓ equal sixteenth-tone 96 2 1.007246 263.521 12.500

1.333333 81 ÑÔ three-quarter syntonic comma 80 1.009360 264.074 16.130 ÑÕ equal twelfth-tone 72 2 1.009674 264.156 16.667 ÑÖ nonadecimal comma 96/95 1.010526 264.380 18.128 Ñ× subdiminished second, or diaskhisma 2048/2025 1.011358 264.597 19.553 ÑØ syntonic comma 81/80 1.012500 264.896 21.506 ÑÙ 53-et syntonic comma 53 2 1.013164 265.070 22.642

ÒÐ Pythagorean comma (A) XII 3ñò/2ñù, or 531441/524288 1.013643 265.195 23.460 ÒÑ equal eighth-tone 48 2 1.014545 265.431 25.000 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÒÒ tridecimal comma (65th harmonic) 65/64 1.015625 265.713 26.841

0.8 81 ÒÓ one and one-quarter syntonic comma 80 1.015649 265.720 26.883 ÒÔ minimal diesis 20000/19683 1.016105 265.839 27.660 ÒÕ 43-et diminished second 43 2 1.016250 265.877 27.907 ÒÖ grave or small diesis 3125/3072 1.017253 266.139 29.614

0.666667 81 Ò× one and one-half syntonic comma 80 1.018808 266.546 32.259 ÒØ equal sixth-tone 36 2 1.019441 266.712 33.333 ÒÙ septendecimal comma 51/50 1.020000 266.858 34.283

0.571429 81 ÓÐ one and three-quarter syntonic comma 80 1.021977 267.375 37.636 ÓÑ trivigesimal comma 46/45 1.022222 267.439 38.051 ÓÒ 31-et superoctave, or diminished second 31 2 1.022611 267.541 38.710 ÓÓ undecimal grave or small chromatic semitone 45/44 1.022727 267.572 38.906 ÓÔ equal fifth-tone 30 2 1.023374 267.741 40.000 ÓÕ diminished second, or great diesis 128/125 1.024000 267.905 41.059

0.5 81 ÓÖ two syntonic commas, or Mathieu superdiesis 80 , or 6561/6400 1.025156 268.207 43.013 Ó× 53-et great diesis (53 2)2 1.026502 268.559 45.283

ÓØ great diesis (A) XXIV 3òô/2óø 1.027473 268.813 46.920

0.444444 81 ÓÙ two and one-quarter syntonic comma 80 1.028345 269.041 48.389 A ÔÐ septimal comma 36/35 1.028571 269.101 48.770 313 ppendices ÔÑ equal quarter-tone 24 2 , or approximately 527/512 1.029302 269.292 50.000 ÔÒ 23-et Greek enharmonic or septimal quarter-tone 23 2 1.030596 269.630 52.174 ÔÓ undecimal comma (33rd harmonic) 33/32 1.031250 269.801 53.273 1 Appendices 314 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

0.4 81 ÔÔ two and one-half syntonic comma 80 1.031544 269.878 53.766 ÔÕ 22-et Greek enharmonic or septimal quarter-tone 22 2 1.032008 270.000 54.545 ÔÖ untrigesimal diatonic semitone, or Greek enharmonic quarter-tone 32/31 1.032258 270.065 54.964 Ô× 43-et double augmented seventh (43 2)2 1.032765 270.198 55.814 ÔØ Greek enharmonic quarter-tone, or untrigesimal comma 31/30 1.033333 270.346 56.767 ÔÙ 21-et Greek enharmonic or septimal quarter-tone 21 2 1.033558 270.405 57.143

0.363636 81 ÕÐ two and three-quarter syntonic comma 80 1.034752 270.718 59.142 ÕÑ 20-et Greek enharmonic or septimal quarter-tone 20 2 1.035265 270.852 60.000 ÕÒ 19-et just diatonic semitone, or major half-tone 19 2 1.037155 271.346 63.158

0.333333 81 ÕÓ three syntonic commas 80 , or 531441/512000 1.037971 271.560 64.519 ÕÔ tridecimal grave or small chromatic semitone 27/26 1.038462 271.688 65.337 ÕÕ 18-et grave or small chromatic semitone, or equal third-tone 18 2 1.039259 271.897 66.667 ÕÖ Pythagorean double diminished third 134217728/129140163 1.039318 271.912 66.765 Õ× 53-et grave or small chromatic semitone, or minor half-tone (53 2)3 1.040015 272.094 67.925

0.307692 81 ÕØ three and one-quarter syntonic comma 80 1.041199 272.404 69.895 ÕÙ cyclic grave or small chromatic semitone, or minor half-tone (A) XXXVI 3óö/2õ÷ 1.041491 272.481 70.380 ÖÐ 17-et grave or small chromatic semitone, or minor half-tone 17 2 1.041616 272.513 70.588 ÖÑ grave or small just chromatic semitone, or minor half-tone 25/24 1.041667 272.527 70.672 ÖÒ trivigesimal diatonic semitone 24/23 1.043478 273.001 73.681 ÖÓ 16-et grave or small chromatic semitone, or minor half-tone 16 2 1.044274 273.209 75.000 0.285714 81 75.272 ÖÔ three and one-half syntonic comma 80 1.044438 273.252 (A) VII ß13 0.571429 80 76.049 ÖÕ meantone chromatic semitone, or minor half-tone 4 2187/2048× 81 1.044907 273.374 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÖÖ 31-et augmented octave, or subminor second (31 2) 2 1.045734 273.591 77.419 Ö× 67th harmonic 67/64 1.046875 273.889 79.070 ÖØ 15-et grave or small chromatic semitone, or minor half-tone 15 2 1.047294 273.999 80.000 ÖÙ nonavigesimal grave or small chromatic semitone 243/232 1.047414 274.030 80.198

0.266667 81 ×Ð three and three-quarter syntonic comma 80 1.047687 274.102 80.649 ×Ñ 43-et chromatic semitone, or minor half-tone (43 2) 3 1.049547 274.588 83.721 ×Ò septimal diatonic semitone 21/20 1.050000 274.707 84.467 ×Ó 14-et Pythagorean limma 14 2 1.050757 274.905 85.714

×Ô Pythagorean limma, or diatonic semitone (D) V 256/243 1.053498 275.622 90.225 ×Õ 53-et Pythagorean limma (53 2)4 1.053705 275.676 90.566

×Ö acute or large Pythagorean limma 135/128 1.054688 275.933 92.179 ×× 13-et Pythagorean limma 13 2 1.054766 275.954 92.308 ×Ø nonadecimal diatonic semitone 19/18 1.055556 276.160 93.603

×Ù cyclic Pythagorean limma (A) XLVIII 3ôø/2÷ö 1.055700 276.198 93.840 ØÐ septendecimal diatonic semitone 18/17 1.058824 277.015 98.955 ØÑ equal semitone 12 2 , or approximately 1024/967 1.059463 277.183 100.000 ØÒ 23-et just diatonic semitone, or major half-tone (23 2) 2 1.062127 277.880 104.348 ØÓ septendecimal chromatic semitone (17th harmonic) 17/16 1.062500 277.977 104.955 A ØÔ 11-et just diatonic semitone, or major half-tone 11 2 1.065041 278.642 109.091 315 ppendices ØÕ nonadecimal chromatic semitone 81/76 1.065789 278.838 110.307 ØÖ 43-et minor second (43 2) 4 1.066603 279.051 111.628 Ø× just diatonic semitone, or major half-tone 16/15 1.066667 279.067 111.731 1 Appendices 316 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ØØ 53-et just diatonic semitone, or major half-tone (53 2)5 1.067577 279.305 113.208

ØÙ Pythagorean apotome, or chromatic semitone (A) VII 3÷/2ññ, or 2187/2048 1.067871 279.382 113.685 ÙÐ 21-et just diatonic semitone, or major half-tone (21 2) 2 1.068242 279.479 114.286 ÙÑ 31-et superaugmented octave, or minor second (31 2) 3 1.069380 279.777 116.129

(D) V 1 1 0.8 81 ÙÒ meantone minor second 4 256/243× 80 1.069984 279.935 117.108 ÙÓ septimal chromatic semitone 15/14 1.071429 280.313 119.443 ÙÔ 10-et just diatonic semitone, or major half-tone 10 2 1.071773 280.403 120.000 ÙÕ nonavigesimal grave or small neutral second 29/27 1.074074 281.005 123.712

2 ÙÖ 19-et great limma, or large half-tone (19 2) 1.075691 281.428 126.316 Ù× trivigesimal chromatic semitone (69th harmonic) 69/64 1.078125 282.065 130.229 ÙØ great limma, acute or large half-tone 27/25 1.080000 282.556 133.238 ÙÙ 9-et great limma, or large half-tone 9 2 1.080060 282.571 133.333 ÑÐÐ 53-et great limma, acute or large half-tone (53 2)6 1.081630 282.897 135.849

ÑÐÑ cyclic great limma, acute or large half-tone (A) XIX 3ñù/2óð 1.082440 283.194 137.145 ÑÐÒ tridecimal grave or small neutral second 13/12 1.083333 283.428 138.573 ÑÐÓ 43-et double diminished third (43 2)5 1.083936 283.585 139.535 ÑÐÔ 17-et three-quarter-tone (17 2)2 1.084964 283.854 147.143 ÑÐÕ three-quarter-tone 135/124 1.088710 284.834 141.176 ÑÐÖ untrigesimal chromatic semitone 279/256 1.089844 285.131 148.946 ÑÐ× equal three-quarter-tone 8 2 , or approximately 1024/939 1.090508 285.305 150.000 ÑÐØ undecimal grave or small neutral second 12/11 1.090909 285.410 150.637

(A) XIV ß3 1 0.285714 80 152.098 ÑÐÙ meantone double augmented octave 2 4782969/4194304× 81 1.091830 285.651 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

4 ÑÑÐ 31-et double augmented octave, or neutral second (31 2) 1.093560 286.103 154.839 ÑÑÑ septimal neutral second (35th harmonic) 35/32 1.093750 286.153 155.140 ÑÑÒ 23-et grave or small tone (23 2) 3 1.094624 286.382 156.522 ÑÑÓ 53-et grave or small tone (53 2)7 1.095869 286.707 158.491 ÑÑÔ 15-et grave or small tone (15 2) 2 1.096825 286.957 160.000

ÑÑÕ cyclic grave or small tone (A) XXXI 3óñ/2ôù 1.097208 287.058 160.605 ÑÑÖ grave or small tone 800/729 1.097394 287.106 160.897 ÑÑ× acute or large double augmented octave 1125/1024 1.098633 287.430 162.851 ÑÑØ 22-et grave or small tone (22 2) 3 1.099131 287.561 163.636 ÑÑÙ undecimal acute or large neutral second 11/10 1.100000 287.788 165.004 ÑÒÐ 43-et double augmented octave (43 2) 6 1.101550 288.194 167.442 ÑÒÑ nonavigesimal acute or large neutral second 32/29 1.103448 288.690 170.423 ÑÒÒ 7-et grave or small tone 7 2 1.104090 288.858 171.429 ÑÒÓ tridecimal acute or large neutral second 72/65 1.107692 289.801 177.069 ÑÒÔ 71st harmonic 71/64 1.109375 290.241 179.697 ÑÒÕ 20-et just minor tone (20 2) 3 1.109569 290.292 180.000

ÑÒÖ Pythagorean diminished third (D) X 65536/59049 1.109858 290.367 180.450 ÑÒ× 53-et just minor tone (53 2)8 1.110295 290.482 181.132 A ÑÒØ just minor tone 10/9 1.111111 290.695 182.404 317 ppendices

ÑÒÙ cyclic minor tone (A) XLIII 3ôó/2öø 1.112178 290.974 184.065 ÑÓÐ acute or large double superaugmented octave 18225/16384 1.112366 291.023 184.357 ÑÓÑ 13-et just minor tone (13 2) 2 1.112531 291.067 184.616 1 Appendices 318 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÑÓÒ 19-et just minor tone (19 2)3 1.115658 291.885 189.474

(A) IIß2 1.5 80 ÑÓÓ third-comma meantone major tone 3 9/8× 81 1.115722 291.901 189.572

(A) II ß4 1.75 80 ÑÓÔ two-seventh-comma meantone major tone 7 9/8× 81 1.117042 292.247 191.621

(A) IIß 1 2 80 ÑÓÕ meantone major tone 2 9/8× 81 1.118034 292.506 193.157 ÑÓÖ 31-et just and Pythagorean major tone (31 2) 5 1.118287 292.572 193.548

(A) II ß4 2.25 80 ÑÓ× two-ninth-comma meantone major tone 9 9/8× 81 1.118806 292.708 194.352

(A) IIß2 2.5 80 ÑÓØ fifth-comma meantone major tone 5 9/8× 81 1.119424 292.870 195.307 ÑÓÙ 43-et just and Pythagorean major tone (43 2)7 1.119450 292.877 195.349

(A) II ß2 3 80 ÑÔÐ sixth-comma meantone major tone 6 9/8× 81 1.120351 293.113 196.741 ÑÔÑ equal tone 6 2 , or approximately 55/49 1.122462 293.665 200.000 ÑÔÒ nonadecimal supermajor second 64/57 1.122807 293.755 200.532 ÑÔÓ 65-et just and Pythagorean major tone (65 2)11 1.124459 294.187 203.077 ÑÔÔ 118-et just and Pythagorean major tone (118 2) 20 1.124662 294.240 203.390 ÑÔÕ 53-et just and Pythagorean major tone (53 2)9 1.124911 294.306 203.774

ÑÔÖ just and Pythagorean major tone (A) II (9th harmonic) 9/8 1.125000 294.329 203.910 ÑÔ× 41-et just and Pythagorean major tone (41 2) 7 1.125629 294.493 204.878 ÑÔØ 140-et just and Pythagorean major tone (140 2)24 1.126173 294.636 205.714 ÑÔÙ 99-et just and Pythagorean major tone (99 2)17 1.126398 294.695 206.061 ÑÕÐ 87-et just and Pythagorean major tone (87 2)15 1.126942 294.837 206.897

ÑÕÑ 55th cyclic fifth (A) LV 3õõ/2ø÷ 1.127352 294.944 207.525 ÑÕÒ 23-et just and Pythagorean major tone (23 2) 4 1.128114 295.143 208.696 ÑÕÓ 17-et just and Pythagorean major tone (17 2) 3 1.130116 295.667 211.765 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÑÕÔ septendecimal supermajor second 17/15 1.133333 296.509 216.687 ÑÕÕ 11-et acute or large tone (11 2)2 1.134313 296.765 218.182 ÑÕÖ 43-et diminished third (43 2)8 1.137642 297.636 223.256 ÑÕ× subdiminished third, or acute or large tone 256/225 1.137778 297.672 223.463 ÑÕØ 16-et acute or large tone (16 2)3 1.138789 297.936 225.000 ÑÕÙ acute or large tone 729/640 1.139063 298.008 225.416 ÑÖÐ 53-et acute or large tone (53 2)10 1.139720 298.180 226.415

ÑÖÑ Pythagorean double augmented octave, or cyclic acute or large tone (A) XIV 3ñô/2òò, or 4782969/4194304 1.140349 298.344 227.370 ÑÖÒ 73rd harmonic 73/64 1.140625 298.417 227.789 ÑÖÓ 21-et acute or large tone (21 2) 4 1.141140 298.551 228.571 ÑÖÔ septimal supermajor second 8/7 1.142857 299.001 231.174 ÑÖÕ 31-et supermajor second, or diminished third (31 2) 6 1.143573 299.188 232.258

(D) X 2 1 0.4 81 ÑÖÖ meantone diminished third 2 65536/59049× 80 1.144867 299.526 234.216 ÑÖ× untrigesimal supermajor second 31/27 1.148148 300.385 239.171 ÑÖØ 5-et supermajor second 5 2 1.148698 300.529 240.000 ÑÖÙ trivigesimal supermajor second 23/20 1.150000 300.869 241.961 Ñ×Ð undecimal grave or small augmented second 405/352 1.150568 301.018 242.816 Ñ×Ñ supermajor second 59049/51300 1.151053 301.145 243.545 A Ñ×Ò diminished third 144/125 1.152000 301.393 244.969 319 ppendices Ñ×Ó tridecimal grave or small augmented second 15/13 1.153846 301.876 247.741 Ñ×Ô 53-et supermajor second (53 2)11 1.154723 302.105 249.057 Ñ×Õ five equal quarter-tones (24 2) 5 , or approximately 52/45 1.155353 302.270 250.000 2 Appendices 320 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

Ñ×Ö cyclic supermajor second (A) XXVI 3òö/2ôñ 1.155907 302.415 250.830 Ñ×× 43-et triple augmented octave (43 2)9 1.156129 302.473 251.163 Ñ×Ø 37th harmonic 37/32 1.156250 302.505 251.344 Ñ×Ù 19-et five quarter-tones (19 2) 4 1.157110 302.730 252.632 ÑØÐ 14-et five quarter-tones (14 2)3 1.160129 303.520 257.143 ÑØÑ untrigesimal subminor third 36/31 1.161290 303.823 258.874 ÑØÒ five quarter-tones 93/80 1.162500 304.140 260.677 ÑØÓ 23-et five quarter-tones (23 2) 5 1.162629 304.174 260.870 ÑØÔ nonavigesimal grave or small augmented second 135/116 1.163793 304.478 262.602 ÑØÕ 9-et five quarter-tones (9 2)2 1.166529 305.194 266.667 ÑØÖ septimal subminor third 7/6 1.166667 305.230 266.871

(A) IXß2 1 0.444444 80 ÑØ× meantone augmented second 4 19683/16384× 81 1.168241 305.642 269.206 7 ÑØØ 31-et augmented second, or subminor third (31 2) 1.169431 305.953 270.968 ÑØÙ 53-et augmented second (53 2)12 1.169924 306.082 271.698 ÑÙÐ 22-et augmented second (22 2) 5 1.170620 306.264 272.727

ÑÙÑ cyclic augmented second (A) XXXVIII 3óø/2öð 1.171677 306.541 274.290 ÑÙÒ augmented second (75th harmonic) 75/64 1.171875 306.592 274.582 ÑÙÓ 13-et augmented second (13 2) 3 1.173460 307.007 276.923 ÑÙÔ trivigesimal subminor third 27/23 1.173913 307.126 277.591 ÑÙÕ 43-et augmented second (43 2)10 1.174916 307.388 279.070 ÑÙÖ septendecimal subminor third 20/17 1.176471 307.795 281.358 ÑÙ× 17-et augmented second (17 2) 4 1.177147 307.972 282.353 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÑÙØ 21-et just minor third (21 2) 5 1.179434 308.570 285.714 ÑÙÙ nonadecimal superaugmented second 45/38 1.184211 309.820 292.711

ÒÐÐ Pythagorean minor third, or trihemitone (D) III 32/27 1.185185 310.075 294.135 ÒÐÑ 53-et Pythagorean minor third, or trihemitone (53 2)13 1.185325 310.111 294.340

ÒÐÒ acute or large augmented second 1215/1024 1.186523 310.425 296.089 ÒÐÓ nonadecimal subminor, or overtone minor third (19th harmonic) 19/16 1.187500 310.680 297.513

ÒÐÔ cyclic Pythagorean minor third, or trihemitone (A) L 3õð/2÷ù 1.187663 310.723 297.750 ÒÐÕ equal minor third 4 2 , or approximately 44/37 1.189207 311.127 300.000

(D) III  3 2 81 ÒÐÖ sixth-comma meantone minor third 6 32/27× 80 1.192570 312.007 304.888 ÒÐ× 43-et just minor third (43 2)11 1.194009 312.383 306.977

(D) III  3 1.666667 81 307.039 ÒÐØ fifth-comma meantone minor third 5 32/27× 80 1.194052 312.395 (D) III  6 1.5 81 308.473 ÒÐÙ two-ninth-comma meantone minor third 9 32/27× 80 1.195041 312.653 ÒÑÐ septendecimal superaugmented second 153/128 1.195313 312.724 308.865 ÒÑÑ 31-et superaugmented second, or just minor third (31 2) 8 1.195873 312.871 309.677

(D) III 3 3 1.333333 81 ÒÑÒ meantone minor third 4 32/27× 80 1.196279 312.977 310.265 (D) III  6 1.166667 81 312.569 ÒÑÓ two-seventh-comma meantone minor third 7 32/27× 80 1.197872 313.395 ÒÑÔ trivigesimal superaugmented second 115/96 1.197917 313.406 312.633 ÒÑÕ 23-et just minor third (23 2) 6 1.198201 313.480 313.043 A ÒÑÖ 65-et just minor third (65 2)17 1.198756 313.625 313.846 321 ppendices ÒÑ× 99-et just minor third (99 2) 26 1.199661 313.862 315.152 ÒÑØ 118-et just minor third (118 2) 31 1.199732 313.880 315.254

ÒÑÙ third-comma meantone and just minor third (D) III 1 32/27×81/80, or 6/5 1.200000 313.951 315.641 2 Appendices 322 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÒÒÐ 19-et just minor third (19 2) 5 1.200103 313.978 315.789 ÒÒÑ 53-et just minor third (53 2)14 1.200929 314.194 316.981 ÒÒÒ 140-et just minor third (140 2) 37 1.201041 314.223 317.143 ÒÒÓ 87-et just minor third (87 2) 23 1.201110 314.241 317.241

ÒÒÔ Pythagorean augmented second, or cyclic minor third (A) IX 3ù/2ñô, or 19683/16384 1.201355 314.305 317.595 ÒÒÕ 15-et just minor third (15 2) 4 1.203025 314.742 320.000 ÒÒÖ undecimal neutral third (77th harmonic) 77/64 1.203125 314.768 320.144 ÒÒ× 41-et just minor third (41 2)11 1.204382 315.097 321.951 ÒÒØ septimal superaugmented second 135/112 1.205357 315.352 323.353 ÒÒÙ 11-et seven quarter-tones (11 2) 3 1.208089 316.067 327.273 ÒÓÐ nonavigesimal grave or small neutral third 29/24 1.208333 316.131 327.622 ÒÓÑ seven quarter-tones 75/62 1.209677 316.483 329.547 ÒÓÒ untrigesimal superaugmented second 155/128 1.210938 316.812 331.349 ÒÓÓ undecimal grave or small neutral third 40/33 1.212121 317.122 333.041 ÒÓÔ 18-et seven quarter-tones (18 2) 5 1.212326 317.175 333.333 ÒÓÕ 43-et double diminished fourth (43 2)12 1.213412 317.460 334.884 ÒÓÖ grave or small neutral third 243/200 1.215000 317.875 337.148 ÒÓ× 53-et neutral third (53 2)15 1.216738 318.330 339.623

ÒÓØ cyclic neutral third (A) XXI 3òñ/2óó 1.217745 318.593 341.055 ÒÓÙ tridecimal grave or small neutral third (39th harmonic) 39/32 1.218750 318.856 342.483 ÒÔÐ 7-et neutral third (7 2)2 1.219014 318.925 342.857 ÒÔÑ acute or large neutral third 8000/6561 1.219326 319.007 343.301 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÒÔÒ double augmented second 625/512 1.220703 319.367 345.255 ÒÔÓ undecimal acute or large neutral third 11/9 1.222222 319.765 347.408 ÒÔÔ 31-et double augmented second, or neutral third (31 2) 9 1.222914 319.945 348.387 ÒÔÕ seven equal quarter-tones (24 2) 7 , or approximately 60/49 1.224054 320.244 350.000 ÒÔÖ 17-et neutral third (17 2) 5 1.226135 320.788 352.941 ÒÔ× tridecimal acute or large neutral third 16/13 1.230769 322.001 359.472 ÒÔØ 10-et grave or small major third (10 2) 3 1.231144 322.099 360.000 ÒÔÙ 53-et grave or small major third (53 2)16 1.232756 322.520 362.264 ÒÕÐ 43-et double augmented second (43 2)13 1.233131 322.618 362.791

ÒÕÑ cyclic grave or small major third (A) XXXIII 3óó/2õò 1.234359 322.940 364.515 ÒÕÒ 79th harmonic 79/64 1.234375 322.944 364.537 ÒÕÓ grave or small major third 100/81 1.234568 322.995 364.807 ÒÕÔ 23-et grave or small major third (23 2) 7 1.234860 323.071 365.217 ÒÕÕ 13-et grave or small major third (13 2) 4 1.237726 323.821 369.231 ÒÕÖ nonavigesimal acute or large neutral third 36/29 1.241379 324.777 374.333 ÒÕ× 16-et grave or small major third (16 2) 5 1.241858 324.902 375.000 ÒÕØ 19-et just major third (19 2) 6 1.244693 325.643 378.947

(A) IVß1 1 0.75 80 379.145 ÒÕÙ third-comma meantone major third 3 81/64× 81 1.244835 325.681 A ÒÖÐ 41-et just major third (41 2)13 1.245801 325.933 380.488 323 ppendices ÒÖÑ 22-et just major third (22 2) 7 1.246758 326.184 381.818

(A) IV ß1 1 0.875 80 383.241 ÒÖÒ two-seventh-comma meantone major third 7 81/64× 81 1.247784 326.452 ÒÖÓ Pythagorean diminished fourth (D) VIII 8192/6561 1.248590 326.663 384.360 2 Appendices 324 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÒÖÔ 53-et just major third (53 2)17 1.248984 326.766 384.906 ÒÖÕ 140-et just major third (140 2) 45 1.249567 326.919 385.714

28 ÒÖÖ 87-et just major third (87 2) 1.249923 327.012 386.207

ÒÖ× meantone and just major third (A) IVß 1 (5th harmonic) 81/64×80/81, or 5/4 1.250000 327.032 386.314 ÒÖØ 118-et just major third (118 2)38 1.250092 327.056 386.441 ÒÖÙ 31-et just major third (31 2)10 1.250566 327.180 387.097 Ò×Ð 65-et just major third (65 2) 21 1.250996 327.292 387.692 Ò×Ñ 99-et just major third (99 2)32 1.251131 327.328 387.879

Ò×Ò cyclic major third (A) XLV 3ôõ/2÷ñ 1.251200 327.346 387.975

(A) IV  8 1.125 80 Ò×Ó two-ninth-comma meantone major third 9 81/64× 81 1.251727 327.484 388.703

(A) IVß4 1.25 80 Ò×Ô fifth-comma meantone major third 5 81/64× 81 1.253109 327.845 390.615 Ò×Õ 43-et just major third (43 2)14 1.253169 327.861 390.698

(A) IVß4 1.5 80 393.482 Ò×Ö sixth-comma meantone major third 6 81/64× 81 1.255187 328.389 Ò×× septendecimal supermajor third 34/27 1.259259 329.454 399.090 Ò×Ø equal major third 3 2 , or approximately 63/50 1.259921 329.628 400.000 Ò×Ù nonadecimal supermajor third 24/19 1.263158 330.474 404.442 ÒØÐ grave or small diminished fourth 512/405 1.264198 330.746 405.866 ÒØÑ 53-et Pythagorean major third, or ditone (53 2)18 1.265426 331.068 407.547

ÒØÒ Pythagorean major third, or ditone (A) IV (81st harmonic) 81/64 1.265625 331.120 407.820

ÒØÓ 57th cyclic fifth (A) LVII 3õ÷/2ùð 1.268271 331.812 411.435 ÒØÔ 23-et Pythagorean major third, or ditone (23 2) 8 1.272642 332.956 417.391 ÒØÕ undecimal diminished fourth 14/11 1.272727 332.978 417.508 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÒØÖ 43-et diminished fourth (43 2)15 1.273534 333.189 418.605 ÒØ× 20-et acute or large major third (20 2) 7 1.274561 333.458 420.000 ÒØØ 17-et acute or large major third (17 2) 6 1.277162 334.138 423.529 ÒØÙ trivigesimal supermajor third 23/18 1.277778 334.299 424.364 ÒÙÐ 31-et supermajor third, or diminished fourth (31 2)11 1.278843 334.578 425.806

(D) VIII 2 0.5 81 ÒÙÑ meantone and diminished fourth, or acute or large major third 8192/6561× 80 , or 32/25 1.280000 334.881 427.373 ÒÙÒ 14-et acute or large major third (14 2) 5 1.280887 335.113 428.571 ÒÙÓ 41st harmonic 41/32 1.281250 335.208 429.062 ÒÙÔ 53-et acute or large major third (53 2)19 1.282084 335.426 430.189

ÒÙÕ cyclic acute or large major third (A) XVI 3ñö/2òõ 1.282892 335.637 431.280 ÒÙÖ septimal supermajor third 9/7 1.285714 336.376 435.084 ÒÙ× 11-et nine quarter-tones (11 2) 4 1.286665 336.624 436.364 ÒÙØ untrigesimal subfourth 40/31 1.290323 337.581 441.278 ÒÙÙ 19-et nine quarter-tones (19 2)7 1.290939 337.743 442.105 ÓÐÐ nine quarter-tones, or untrigesimal supermajor third 31/24 1.291667 337.933 443.081 ÓÐÑ 43-et triple diminished fifth (43 2)16 1.294229 338.603 446.512 ÓÐÒ nine equal quarter-tones (8 2) 3 , or approximately 83/64 1.296840 339.286 450.000 ÓÐÓ 83rd harmonic 83/64 1.296875 339.296 450.047 A ÓÐÔ 53-et subfourth (53 2)20 1.298961 339.841 452.830 325 ppendices

ÓÐÕ cyclic subfourth (A) XXVIII 3òø/2ôô 1.300395 340.217 454.740 ÓÐÖ augmented third, or subfourth 125/96 1.302083 340.658 456.986 ÓÐ× 21-et subfourth (21 2) 8 1.302201 340.689 457.143 2 Appendices 326 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÓÐØ trivigesimal subfourth 30/23 1.304348 341.251 459.994 ÓÐÙ 13-et subfourth (13 2) 5 1.305512 341.555 461.538

(A) XI ß2 3 0.363636 80 ÓÑÐ meantone augmented third 4 177147/131072× 81 1.306133 341.718 462.363 ÓÑÑ 31-et augmented third, or subfourth (31 2)12 1.307759 342.143 464.516 ÓÑÒ 18-et grave or small fourth (18 2)7 1.309385 342.568 466.667 ÓÑÓ 23-et grave or small fourth (23 2) 9 1.311579 343.143 469.565 ÓÑÔ septimal subfourth (21st harmonic) 21/16 1.312500 343.384 470.781 ÓÑÕ 43-et augmented third (43 2)17 1.315261 344.106 474.419 ÓÑÖ 53-et grave or small fourth (53 2)21 1.316061 344.315 475.472 ÓÑ× grave or small fourth 320/243 1.316872 344.527 476.539

ÓÑØ cyclic grave or small fourth (A) XL 3ôð/2öó 1.318137 344.858 478.200

ÓÑÙ acute or large augmented third 675/512 1.318359 344.917 478.492 ÓÒÐ nonadecimal subfourth 95/72 1.319444 345.200 479.917 ÓÒÑ 5-et grave or small fourth (5 2)2 1.319508 345.217 480.000 ÓÒÒ septendecimal subfourth 45/34 1.323529 346.269 485.268 ÓÒÓ 22-et just and Pythagorean perfect fourth (22 2) 9 1.327849 347.399 490.909 ÓÒÔ septendecimal superaugmented third (85th harmonic) 85/64 1.328125 347.471 491.269 ÓÒÕ 17-et just and Pythagorean perfect fourth (17 2) 7 1.330312 348.044 494.118 ÓÒÖ 87-et just and Pythagorean perfect fourth (87 2) 36 1.332184 348.533 496.552 ÓÒ× 99-et just and Pythagorean perfect fourth (99 2) 41 1.332505 348.617 496.970 ÓÒØ 140-et just and Pythagorean perfect fourth (140 2) 58 1.332639 348.652 497.143 ÓÒÙ 41-et just and Pythagorean perfect fourth (41 2)17 1.332961 348.737 497.561 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÓÓÐ just and Pythagorean perfect fourth (D) I 4/3 1.333333 348.834 498.045 ÓÓÑ 53-et just and Pythagorean perfect fourth (53 2)22 1.333386 348.848 498.113 ÓÓÒ 118-et just and Pythagorean perfect fourth (118 2) 49 1.333534 348.886 498.305 ÓÓÓ 65-et just and Pythagorean perfect fourth (65 2)27 1.333654 348.918 498.462 ÓÓÔ equal perfect fourth (12 2) 5 , or approximately 1024/767 1.334840 349.228 500.000

(D) I  1 6 81 ÓÓÕ sixth-comma meantone perfect fourth 6 4/3× 80 1.336097 349.557 501.629 ÓÓÖ cyclic perfect fourth (A) LII 3õò/2øò 1.336120 349.563 501.660 43 18 ÓÓ× 43-et just and Pythagorean perfect fourth ( 2) 1.336634 349.698 502.326

(D) I  1 5 81 ÓÓØ fifth-comma meantone perfect fourth 5 4/3× 80 1.336650 349.702 502.346

(D) I  2 4.5 81 ÓÓÙ two-ninth-comma meantone perfect fourth 9 4/3× 80 1.337019 349.798 502.824 ÓÔÐ 31-et just and Pythagorean perfect fourth (31 2)13 1.337329 349.880 503.226

(D) I  1 4 81 503.422 ÓÔÑ meantone perfect fourth 4 4/3× 80 1.337481 349.919

(D) I  2 3.5 81 504.190 ÓÔÒ two-seventh-comma meantone perfect fourth 7 4/3× 80 1.338074 350.074 (D) I  1 3 81 505.214 ÓÔÓ third-comma meantone perfect fourth 3 4/3× 80 1.338866 350.282 ÓÔÔ 19-et just and Pythagorean perfect fourth (19 2)8 1.338904 350.292 505.263 ÓÔÕ 43rd harmonic 43/32 1.343750 351.559 511.518 ÓÔÖ 7-et acute or large fourth (7 2)3 1.345900 352.122 514.286 ÓÔ× subdiminished fifth 8192/6075 1.348477 352.796 517.598 A ÓÔØ acute or large fourth 27/20 1.350000 353.195 519.551 327 ppendices ÓÔÙ 53-et acute or large fourth (53 2)23 1.350939 353.440 520.755

ÓÕÐ Pythagorean augmented third, or cyclic acute or large fourth (A) XI 3ññ/2ñ÷, or 177147/131072 1.351524 353.593 521.505 ÓÕÑ 23-et acute or large fourth (23 2)10 1.351707 353.641 521.739 2 Appendices 328 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÓÕÒ tridecimal superfourth 65/48 1.354167 354.285 524.886 ÓÕÓ 16-et acute or large fourth (16 2)7 1.354256 354.308 525.000 ÓÕÔ 43-et double diminished fifth (43 2)19 1.358355 355.380 530.233 ÓÕÕ nonavigesimal superfourth (87th harmonic) 87/64 1.359375 355.647 531.532 ÓÕÖ 9-et superfourth (9 2) 4 1.360790 356.017 533.333 ÓÕ× undecimal tritone, or augmented fourth 15/11 1.363636 356.762 536.951 ÓÕØ grave or small augmented fourth, or superfourth 512/375 1.365333 357.206 539.104 ÓÕÙ 20-et superfourth (20 2) 9 1.366040 357.391 540.000 ÓÖÐ 31-et superfourth, or diminished fifth (31 2)14 1.367568 357.791 541.935 ÓÖÑ 53-et superfourth (53 2)24 1.368723 358.093 543.396

ÓÖÒ cyclic superfourth (A) XXIII 3òó/2óö 1.369964 358.417 544.965 ÓÖÓ 11-et superfourth (11 2) 5 1.370351 358.519 545.455 ÓÖÔ eleven equal quarter-tones (24 2)11 , or approximately 1024/745 1.373954 359.461 550.000 ÓÖÕ undecimal superfourth (11th harmonic) 11/8 1.375000 359.735 551.318 ÓÖÖ untrigesimal subdiminished fifth 128/93 1.376344 360.087 553.009 ÓÖ× 13-et eleven quarter-tones (13 2) 6 1.377009 360.261 553.846 ÓÖØ eleven quarter-tones 62/45 1.377778 360.462 554.812 ÓÖÙ nonavigesimal tritone, or augmented fourth 40/29 1.379310 360.863 556.737 Ó×Ð 43-et double augmented third (43 2)20 1.380429 361.155 558.140 Ó×Ñ 15-et eleven quarter-tones (15 2)7 1.381913 361.544 560.000 Ó×Ò tridecimal tritone, or augmented fourth 18/13 1.384615 362.251 563.382

8 Ó×Ó 17-et grave or small augmented fourth (17 2) 1.385674 362.528 564.706 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

Ó×Ô 53-et grave or small augmented fourth (53 2)25 1.386741 362.807 566.038 Ó×Õ 19-et grave or small augmented fourth (19 2) 9 1.388651 363.307 568.421

Ó×Ö cyclic grave or small augmented fourth (A) XXXV 3óõ/2õõ 1.388654 363.307 568.425 Ó×× grave or small augmented fourth 25/18 1.388889 363.369 568.717 Ó×Ø 89th harmonic 89/64 1.390625 363.823 570.880 Ó×Ù 21-et grave or small augmented fourth (21 2)10 1.391066 363.938 571.429 ÓØÐ trivigesimal subdiminished fifth 32/23 1.391304 364.001 571.726 ÓØÑ 23-et grave or small augmented fourth (23 2)11 1.393063 364.461 573.913

(A) VI ß11 0.666667 80 ÓØÒ meantone tritone, or augmented fourth 2 729/512× 81 1.397542 365.633 579.471 ÓØÓ 31-et augmented fourth, or subdiminished fifth (31 2)15 1.398491 365.881 580.645 ÓØÔ septimal subdiminished fifth 7/5 1.400000 366.276 582.512 ÓØÕ 43-et just tritone, or augmented fourth (43 2)21 1.402861 367.024 586.047

ÓØÖ Pythagorean diminished fifth (D) VI 1024/729 1.404664 367.496 588.270 ÓØ× 53-et just tritone, or augmented fourth (53 2)26 1.404996 367.583 588.679 ÓØØ just tritone, or augmented fourth (45th harmonic) 45/32 1.406250 367.911 590.224 ÓØÙ nonadecimal subdiminished fifth 38/27 1.407407 368.214 591.648

ÓÙÐ cyclic tritone, or augmented fourth (A) XLVII 3ô÷/2÷ô 1.407600 368.264 591.885 ÓÙÑ septendecimal subdiminished fifth 24/17 1.411765 369.354 597.000 A ÓÙÒ equal tritone, or augmented fourth 2 2 , or approximately 181/128 1.414214 369.994 600.000 329 ppendices ÓÙÓ septendecimal tritone, or superaugmented fourth 17/12 1.416667 370.636 603.000 ÓÙÔ nonadecimal tritone, or superaugmented fourth 27/19 1.421053 371.784 608.352 ÓÙÕ tridecimal diminished fifth (91st harmonic) 91/64 1.421875 371.999 609.354 3 Appendices 330 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÓÙÖ diminished fifth, or acute or large tritone, or augmented fourth 64/45 1.422222 372.090 609.776 ÓÙ× 53-et acute or large tritone, or augmented fourth (53 2)27 1.423492 372.422 611.321

ÓÙØ Pythagorean tritone, or augmented fourth (A) VI 3ö/2ù, or 729/512 1.423828 372.510 611.730 43 22 ÓÙÙ 43-et diminished fifth ( 2) 1.425658 372.989 613.953

ÔÐÐ 59th cyclic fifth (A) LIX 3õù/2ùó 1.426804 373.289 615.345 ÔÐÑ septimal tritone, or superaugmented fourth 10/7 1.428571 373.751 617.488 ÔÐÒ 31-et superaugmented fourth, or diminished fifth (31 2)16 1.430113 374.154 619.355

(D) VI 11 0.666667 81 ÔÐÓ meantone diminished fifth 2 1024/729× 80 1.431084 374.408 620.529 ÔÐÔ 23-et acute or large diminished fifth (23 2)12 1.435685 375.612 626.087

ÔÐÕ trivigesimal superaugmented fourth (23rd harmonic) 23/16 1.437500 376.087 628.274 ÔÐÖ 21-et acute or large diminished fifth (21 2)11 1.437747 376.151 628.571 ÔÐ× acute or large diminished fifth 36/25 1.440000 376.741 631.283 ÔÐØ 19-et acute or large diminished fifth (19 2)10 1.440247 376.805 631.579 ÔÐÙ 53-et acute or large diminished fifth (53 2)28 1.442231 377.324 633.962

ÔÑÐ cyclic acute or large diminished fifth (A) XVIII 3ñø/2òø 1.443254 377.592 635.190 ÔÑÑ 17-et acute or large diminished fifth (17 2) 9 1.443341 377.615 635.294 ÔÑÒ tridecimal subdiminished fifth 13/9 1.444444 377.904 636.618 ÔÑÓ 15-et thirteen quarter-tones (15 2) 8 1.447269 378.643 640.000 ÔÑÔ 43-et double diminished sixth (43 2)23 1.448825 379.050 641.860 ÔÑÕ nonavigesimal subdiminished fifth 29/20 1.450000 379.357 643.263 ÔÑÖ thirteen quarter-tones 90/62 1.451613 379.779 645.188

7 ÔÑ× 13-et thirteen quarter-tones (13 2) 1.452423 379.991 646.154 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÔÑØ untrigesimal superaugmented fourth (93rd harmonic) 93/64 1.453125 380.175 646.991 ÔÑÙ undecimal subfifth 16/11 1.454545 380.546 648.682 ÔÒÐ thirteen equal quarter-tones (24 2)13 , or approximately 745/512 1.455653 380.836 650.000 ÔÒÑ 11-et subfifth (11 2) 6 1.459480 381.837 654.545

(A) XIII ß3 1 0.307692 80 ÔÒÒ meantone double augmented fourth 4 1594323/1048576× 81 1.460302 382.052 655.536 ÔÒÓ 53-et subfifth (53 2)29 1.461216 382.292 656.604

17 ÔÒÔ 31-et double augmented fourth, or subfifth (31 2) 1.462450 382.614 658.065

ÔÒÕ cyclic subfifth (A) XXX 3óð/2ô÷ 1.462944 382.744 658.650 ÔÒÖ 20-et subfifth (20 2)11 1.464086 383.042 660.000 ÔÒ× acute or large double augmented fourth, or subfifth 375/256 1.464844 383.241 660.896 ÔÒØ undecimal subdiminished fifth 22/15 1.466667 383.717 663.049 ÔÒÙ 47th harmonic 47/32 1.468750 384.263 665.507 ÔÓÐ 9-et subfifth (9 2) 5 1.469734 384.520 666.667 ÔÓÑ nonavigesimal subfifth 128/87 1.471264 384.920 668.468 ÔÓÒ 43-et double augmented fourth (43 2)24 1.472369 385.209 669.767 ÔÓÓ 16-et grave or small fifth (16 2) 9 1.476826 386.375 675.000 ÔÓÔ tridecimal subfifth 96/65 1.476923 386.401 675.114 ÔÓÕ 23-et grave or small fifth (23 2)13 1.479610 387.104 678.261 A ÔÓÖ Pythagorean diminished sixth (D) XI 262144/177147 1.479811 387.156 678.495 331 ppendices ÔÓ× 53-et grave or small fifth (53 2)30 1.480452 387.324 679.245 ÔÓØ grave or small fifth 40/27 1.481481 387.593 680.449

ÔÓÙ cyclic grave or small fifth (A) XLII 3ôò/2öö 1.482904 387.966 682.110 3 Appendices 332 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÔÔÐ acute or large double superaugmented fourth 6075/4096 1.483154 388.031 682.402 ÔÔÑ nonadecimal subfifth (95th harmonic) 95/64 1.484375 388.350 683.827 ÔÔÒ 7-et grave or small fifth (7 2) 4 1.485994 388.774 685.714 ÔÔÓ 19-et just and Pythagorean perfect fifth (19 2)11 1.493759 390.806 694.737

(A) I ß 1 3 80 ÔÔÔ third-comma meantone perfect fifth 3 3/2× 81 1.493802 390.817 694.786

(A) I ß2 3.5 80 ÔÔÕ two-seventh-comma meantone perfect fifth 7 3/2× 81 1.494686 391.048 695.810

(A) I ß 1 4 80 ÔÔÖ meantone perfect fifth 4 3/2× 81 1.495349 391.221 696.578 ÔÔ× 31-et just and Pythagorean perfect fifth (31 2)18 1.495518 391.266 696.774

(A) Iß2 4.5 80 ÔÔØ two-ninth-comma meantone perfect fifth 9 3/2× 81 1.495865 391.356 697.176

(A) I ß 1 5 80 ÔÔÙ fifth-comma meantone perfect fifth 5 3/2× 81 1.496278 393.415 697.654 ÔÕÐ 43-et just and Pythagorean perfect fifth (43 2)25 1.496296 391.469 697.674

(A) I ß 1 6 80 ÔÕÑ sixth-comma meantone perfect fifth 6 3/2× 81 1.496898 391.627 698.371 ÔÕÒ equal perfect fifth (12 2) 7 , or approximately 767/512 1.498307 391.995 700.000 ÔÕÓ 65-et just and Pythagorean perfect fifth (65 2)38 1.499639 392.344 701.538 ÔÕÔ 118-et just and Pythagorean perfect fifth (118 2) 69 1.499775 392.379 701.695 ÔÕÕ 53-et just and Pythagorean perfect fifth (53 2)31 1.499941 392.423 701.887

ÔÕÖ just and Pythagorean perfect fifth (A) I (3rd harmonic) 3/2 1.500000 392.438 701.955 ÔÕ× 41-et just and Pythagorean perfect fifth (41 2)24 1.500419 392.548 702.439 ÔÕØ 140-et just and Pythagorean perfect fifth (140 2) 82 1.500782 392.643 702.857 ÔÕÙ 99-et just and Pythagorean perfect fifth (99 2) 58 1.500932 392.682 703.030 ÔÖÐ 87-et just and Pythagorean perfect fifth (87 2) 51 1.501294 392.777 703.448

ÔÖÑ 54th cyclic fifth (A) LIV 3õô/2øõ 1.503135 393.259 705.570 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÔÖÒ 17-et just and Pythagorean perfect fifth (17 2)10 1.503407 393.330 705.882 ÔÖÓ 22-et just and Pythagorean perfect fifth (22 2)13 1.506196 394.059 709.091 ÔÖÔ septendecimal superfifth 68/45 1.511111 395.345 714.732 ÔÖÕ undecimal grave or small augmented fifth 50/33 1.515152 396.402 719.354 ÔÖÖ 97th harmonic 97/64 1.515625 396.526 719.895 ÔÖ× 5-et acute or large fifth (5 2)3 1.515717 396.550 720.000 ÔÖØ nonadecimal superfifth 144/95 1.515789 396.569 720.083 ÔÖÙ subdiminished sixth 1024/675 1.517037 396.896 721.508 723.014 Ô×Ð acute or large fifth 243/160 1.518750 397.344 724.528 Ô×Ñ 53-et acute or large fifth (53 2)32 1.519686 397.589 725.415 Ô×Ò Pythagorean double augmented fourth, or cyclic acute or large fifth (A) XIII 3ñó/2òð, or 1594323/1048576 1.520465 397.792 43 26 725.581 Ô×Ó 43-et diminished sixth ( 2) 1.520611 397.831 729.219 Ô×Ô septimal superfifth 32/21 1.523810 398.104 730.435 Ô×Õ 23-et acute or large fifth (23 2)14 1.524880 398.948 733.333 Ô×Ö 18-et superfifth (18 2)11 1.527435 399.616 735.484 Ô×× 31-et superfifth, or diminished sixth (31 2)19 1.529334 400.113 737.637 Ô×Ø meantone diminished sixth and wolf fifth (D) XI 2 3 262144/177147× 0.363636 81 1.531237 400.611 4 80 737.652 Ô×Ù septimal diminished sixth (49th harmonic) 49/32 1.531250 400.614

738.462 A ÔØÐ 13-et superfifth (13 2) 8 1.531966 400.802 333 ppendices 740.006 ÔØÑ trivigesimal superfifth 23/15 1.533333 401.159 742.857 ÔØÒ 21-et superfifth (21 2)13 1.535861 401.820 743.014 ÔØÓ diminished sixth, or superfifth 192/125 1.536000 401.857 3 Appendices 334 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÔØÔ tridecimal grave or small augmented fifth 20/13 1.538462 402.501 745.786 ÔØÕ 53-et superfifth (53 2)33 1.539692 402.823 747.170

ÔØÖ cyclic superfifth (A) XXV 3òõ/2óù 1.541209 403.220 748.875 ÔØ× fifteen equal quarter-tones (8 2) 5 , or approximately 128/83 1.542211 403.482 750.000 ÔØØ 43-et triple augmented fourth (43 2)27 1.545321 404.296 753.488 ÔØÙ undecimal superfifth (99th harmonic) 99/64 1.546875 404.702 755.228 ÔÙÐ untrigesimal subminor sixth 48/31 1.548387 405.098 756.919 ÔÙÑ 19-et Pythagorean minor sixth (19 2)12 1.549260 405.326 757.895 ÔÙÒ fifteen quarter-tones, or untrigesimal superfifth 31/20 1.550000 405.520 758.722 ÔÙÓ nonavigesimal grave or small augmented sixth 45/29 1.551724 405.971 760.647 ÔÙÔ 11-et fifteen quarter-tones (11 2)7 1.554406 406.672 763.636 ÔÙÕ septimal subminor sixth 14/9 1.555556 406.973 764.916 ÔÙÖ 53-et augmented fifth (53 2)34 1.559960 408.126 769.811 ÔÙ× 14-et augmented fifth (14 2) 9 1.561418 408.507 771.429

ÔÙØ cyclic augmented fifth (A) XXXVII 3ó÷/2õø 1.562236 408.721 772.335

(A) VIII ß2 0.5 80 ÔÙÙ meantone augmented and augmented fifth (25th harmonic) 6561/4096× 81 , or 25/16 1.562500 408.790 772.627 ÕÐÐ 31-et augmented fifth, or subminor sixth (31 2)20 1.563914 409.160 774.194 ÕÐÑ trivigesimal subminor sixth 36/23 1.565217 409.501 775.636 ÕÐÒ 17-et augmented fifth (17 2)11 1.565972 409.698 776.471 ÕÐÓ 20-et augmented fifth (20 2)13 1.569168 410.535 780.000 ÕÐÔ 43-et augmented fifth (43 2)28 1.570433 410.866 781.395 ÕÐÕ undecimal augmented fifth 11/7 1.571429 411.126 782.492 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÕÐÖ 23-et Pythagorean minor sixth (23 2)15 1.571534 411.154 782.609 ÕÐ× septendecimal superaugmented fifth 85/54 1.574074 411.818 785.404 ÕÐØ 101st harmonic 101/64 1.578125 412.878 789.854 ÕÐÙ nonadecimal superaugmented fifth 30/19 1.578947 413.093 790.756

ÕÑÐ Pythagorean minor sixth (D) IV 128/81 1.580247 413.433 792.180 ÕÑÑ 53-et Pythagorean minor sixth (53 2)35 1.580496 413.498 792.453

ÕÑÒ acute or large augmented fifth 405/256 1.582031 413.900 794.134 ÕÑÓ nonadecimal subminor sixth 19/12 1.583333 414.240 795.558

ÕÑÔ cyclic minor sixth (A) XLIX 3ôù/2÷÷ 1.583550 414.297 795.795 ÕÑÕ equal minor sixth (3 2) 2 , or approximately 100/63 1.587401 415.305 800.000 ÕÑÖ septendecimal subminor sixth 27/17 1.588235 415.523 800.910 ÕÑ× septendecimal superaugmented fifth (51st harmonic) 51/32 1.593750 416.966 806.910 ÕÑØ 43-et just minor sixth (43 2)29 1.595953 417.542 809.302 ÕÑÙ trivigesimal superaugmented fifth 115/72 1.597222 417.874 810.678 ÕÒÐ 31-et superaugmented fifth, or just minor sixth (31 2) 21 1.599276 418.412 812.903

ÕÒÑ meantone and just minor sixth (D) IV 1 128/81×81/80, or 8/5 1.600000 418.601 813.686

ÕÒÒ 53-et just minor sixth (53 2)36 1.601302 418.942 815.094

ÕÒÓ Pythagorean augmented fifth (A) VIII 3ø/2ñò, or 6561/4096 1.601807 419.074 815.640 A ÕÒÔ 22-et just minor sixth (22 2)15 1.604160 419.689 818.182 335 ppendices ÕÒÕ 19-et seventeen quarter-tones (19 2)13 1.606822 420.386 821.053 ÕÒÖ septimal superaugmented sixth 45/28 1.607143 420.470 821.398 ÕÒ× 103rd harmonic 103/64 1.609375 421.054 823.801 3 Appendices 336 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÕÒØ 16-et seventeen quarter-tones (16 2)11 1.610490 421.345 825.000 ÕÒÙ nonavigesimal grave or small neutral sixth 29/18 1.611111 421.508 825.667 ÕÓÐ seventeen quarter-tones 50/31 1.612903 421.977 827.592 ÕÓÑ untrigesimal superaugmented fifth 155/96 1.614583 422.416 829.394 ÕÓÒ 13-et seventeen quarter-tones (13 2) 9 1.615866 422.752 830.769 ÕÓÓ 23-et neutral sixth (23 2)16 1.619616 423.733 834.783 ÕÓÔ neutral sixth 81/50 1.620000 423.833 835.193 ÕÓÕ 43-et double diminished seventh (43 2)30 1.621888 424.327 837.209 ÕÓÖ 53-et neutral sixth (53 2)37 1.622382 424.457 837.736

ÕÓ× cyclic neutral sixth (A) XX 3òð/2óñ 1.623661 424.791 839.100 ÕÓØ 10-et neutral sixth (10 2)7 1.624505 425.012 840.000 ÕÓÙ tridecimal grave or small neutral, or overtone sixth (13th harmonic) 13/8 1.625000 425.142 840.528 ÕÔÐ double augmented fifth 625/384 1.627604 425.823 843.300 ÕÔÑ 17-et neutral sixth (17 2)12 1.631142 426.748 847.059

(A) XV ß3 3 0.266667 80 ÕÔÒ meantone double augmented fifth 4 14348907/8388608× 81 1.632667 427.147 848.676 ÕÔÓ seventeen equal quarter-tones (24 2)17 , or approximately 49/30 1.633915 427.474 850.000 ÕÔÔ 31-et double augmented fifth, or neutral sixth (31 2)22 1.635438 427.872 851.613 ÕÔÕ undecimal grave or small neutral sixth 18/11 1.636364 428.115 852.592 ÕÔÖ septimal neutral sixth (105th harmonic) 105/64 1.640625 429.229 857.095 ÕÔ× 7-et grave or small major sixth (7 2) 5 1.640671 429.241 857.143 ÕÔØ tridecimal acute or large neutral sixth 64/39 1.641026 429.334 857.517

38 ÕÔÙ 53-et grave or small major sixth (53 2) 1.643739 430.044 860.377 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÕÕÐ cyclic grave or small major sixth (A) XXXII 3óò/2õð 1.645813 430.587 862.560 ÕÕÑ grave or small major sixth 400/243 1.646091 430.659 862.852 ÕÕÒ 43-et double augmented fifth (43 2)31 1.648244 431.223 865.116 ÕÕÓ 18-et grave or small major sixth (18 2)13 1.649721 431.609 866.667 ÕÕÔ undecimal acute or large neutral sixth 33/20 1.650000 431.682 866.959 ÕÕÕ nonavigesimal acute or large neutral sixth 48/29 1.655172 433.035 872.378 ÕÕÖ 11-et grave or small major sixth (11 2)8 1.655507 433.123 872.727 ÕÕ× 53rd harmonic 53/32 1.656250 433.317 873.505 ÕÕØ 15-et just major sixth (15 2)11 1.662476 434.946 880.000

ÕÕÙ Pythagorean diminished seventh (D) IX 32768/19683 1.664787 435.551 882.405 ÕÖÐ 53-et just major sixth (53 2)39 1.665377 435.705 883.019 ÕÖÑ 19-et just major sixth (19 2)14 1.666524 436.005 884.211 ÕÖÒ just major sixth 5/3 1.666667 436.043 884.359

ÕÖÓ cyclic major sixth (A) XLIV 3ôô/2öù 1.668267 436.461 886.020 ÕÖÔ 23-et just major sixth (23 2)17 1.669169 436.697 886.957

(A) III ß3 1.333333 80 ÕÖÕ meantone major sixth 4 27/16× 81 1.671851 437.399 889.735 ÕÖÖ 107th harmonic 107/64 1.671875 437.405 889.760 ÕÖ× 31-et just major sixth (31 2) 23 1.672418 437.547 890.323 A ÕÖØ 43-et just major sixth (43 2)32 1.675029 438.230 893.023 337 ppendices ÕÖÙ equal major sixth (4 2) 3 , or approximately 37/22 1.681793 440.000 900.000 Õ×Ð nonadecimal supermajor sixth 32/19 1.684211 440.633 902.487 Õ×Ñ grave or small diminished seventh 2048/1215 1.685597 440.995 903.911 3 Appendices 338 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

Õ×Ò 53-et Pythagorean major sixth (53 2)40 1.687301 441.441 905.660

Õ×Ó Pythagorean major sixth (A) III (27th harmonic) 3ó/2ô, or 27/16 1.687500 441.493 905.865

Õ×Ô 56th cyclic fifth (A) LVI 3õö/2øø 1.691027 442.416 909.480 Õ×Õ 21-et Pythagorean major sixth (21 2)16 1.695728 443.646 914.286 Õ×Ö 17-et acute or large major sixth (17 2)13 1.699024 444.508 917.647 Õ×× septendecimal supermajor sixth 17/10 1.700000 444.763 918.642 Õ×Ø 43-et diminished seventh (43 2) 33 1.702249 445.352 920.930 Õ×Ù 109th harmonic 109/64 1.703125 445.581 921.821 ÕØÐ trivigesimal supermajor sixth 46/27 1.703704 445.732 922.409 ÕØÑ 13-et acute or large major sixth (13 2)10 1.704361 445.904 923.077 ÕØÒ undecimal grave or small augmented sixth 75/44 1.704545 445.953 923.264 ÕØÓ diminished seventh, or acute or large major sixth 128/75 1.706667 446.508 925.418 ÕØÔ 22-et acute or large major sixth (22 2)17 1.708496 446.986 927.273 ÕØÕ 53-et acute or large major sixth (53 2)41 1.709512 447.252 928.302 ÕØÖ 31-et supermajor sixth, or diminished seventh (31 2) 24 1.710234 447.441 929.032

ÕØ× Pythagorean double augmented fifth, or cyclic acute or large major sixth (A) XV 3ñõ/2òó, or 14348907/8388608 1.710523 447.517 929.325

(D) IX 2 1 0.444444 81 ÕØØ meantone diminished seventh 4 32768/19683× 80 1.711975 447.896 930.794 ÕØÙ septimal supermajor sixth 12/7 1.714286 448.501 933.129 ÕÙÐ 9-et nineteen quarter-tones (9 2)7 1.714488 448.554 933.333 ÕÙÑ undecimal supermajor sixth (55th harmonic) 55/32 1.718750 449.669 937.632 ÕÙÒ 23-et nineteen quarter-tones (23 2)18 1.720239 450.058 939.130 ÕÙÓ nineteen quarter-tones, or untrigesimal supermajor sixth 31/18 1.722222 450.577 941.126 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÕÙÔ 14-et augmented sixth (14 2)11 1.723946 451.028 942.857 ÕÙÕ nonavigesimal grave or small augmented sixth 50/29 1.724138 451.079 943.050 ÕÙÖ 19-et nineteen quarter-tones (19 2)15 1.728444 452.205 947.368 ÕÙ× 43-et triple diminished octave (43 2)34 1.729911 452.589 948.837 ÕÙØ tridecimal grave or small augmented sixth 45/26 1.730769 452.813 949.696 ÕÙÙ nineteen equal quarter-tones (24 2)19 , or approximately 45/26 1.731073 452.893 950.000 ÖÐÐ 53-et augmented sixth (53 2)42 1.732017 453.140 950.943

ÖÐÑ cyclic augmented sixth (A) XXVII 3ò÷/2ôò 1.733860 453.622 952.785 ÖÐÒ 111th harmonic 111/64 1.734375 453.757 953.299 ÖÐÓ augmented sixth 125/72 1.736111 454.211 955.031 ÖÐÔ trivigesimal subminor seventh 40/23 1.739130 455.001 958.039 ÖÐÕ 5-et augmented sixth (5 2) 4 1.741101 455.517 960.000 ÖÐÖ untrigesimal subminor seventh 54/31 1.741935 455.735 960.829

(A) X ß2 1 0.4 80 ÖÐ× meantone augmented sixth 2 59049/32768× 81 1.746928 457.041 965.784 25 ÖÐØ 31-et augmented sixth, or subminor seventh (31 2) 1.748905 457.558 967.742 ÖÐÙ septimal subminor seventh (7th harmonic) 7/4 1.750000 457.845 968.826 ÖÑÐ 21-et grave or small minor seventh (21 2)17 1.752633 458.534 971.429 ÖÑÑ nonadecimal superaugmented sixth 100/57 1.754386 458.992 973.159 A ÖÑÒ 53-et grave or small minor seventh (53 2)43 1.754817 459.105 973.585 339 ppendices ÖÑÓ 16-et grave or small minor seventh (16 2)13 1.756252 459.480 975.000

ÖÑÔ cyclic grave or small minor seventh (A) XXXIX 3óù/2öñ 1.757516 459.811 976.245 ÖÑÕ acute or large augmented sixth, or grave or small minor seventh 225/128 1.757813 459.889 976.537 4 Appendices 340 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÖÑÖ 43-et augmented sixth (43 2) 35 1.758022 459.944 976.744 ÖÑ× 11-et grave or small minor seventh (11 2) 9 1.763183 461.294 981.818 ÖÑØ septendecimal subminor seventh 30/17 1.764706 461.692 983.313 ÖÑÙ 113th harmonic 113/64 1.765625 461.933 984.215 ÖÒÐ 17-et Pythagorean minor seventh (17 2)14 1.769730 463.007 988.235 ÖÒÑ septendecimal superaugmented sixth 85/48 1.770833 463.295 989.314 ÖÒÒ 23-et Pythagorean minor seventh (23 2)19 1.772870 463.828 991.304

ÖÒÓ Pythagorean minor seventh (D) II 16/9 1.777778 465.112 996.090 ÖÒÔ 53-et Pythagorean minor seventh (53 2)44 1.777918 465.149 996.226 ÖÒÕ superaugmented sixth 3645/2048 1.779785 465.637 998.044 ÖÒÖ nonadecimal subminor seventh (57th harmonic) 57/32 1.781250 466.021 999.468

ÖÒ× cyclic minor seventh (A) LI 3õñ/2øð 1.781494 466.084 999.705 ÖÒØ equal minor seventh (6 2) 5 , or approximately 98/55 1.781797 466.164 1000.000 ÖÒÙ septimal superaugmented sixth 25/14 1.785714 467.885 1003.802 ÖÓÐ 43-et Pythagorean minor seventh (43 2)36 1.786591 467.418 1004.651 ÖÓÑ 31-et superaugmented sixth, or minor seventh (31 2) 26 1.788450 467.904 1006.452

(D)  1 2 81 ÖÓÒ meantone minor seventh 2 16/9× 80 1.788854 468.010 1006.843 ÖÓÓ 19-et acute or large minor seventh (19 2)16 1.792664 469.007 1010.526 ÖÓÔ untrigesimal superaugmented sixth 775/432 1.793981 469.351 1011.798 ÖÓÕ trivigesimal superaugmented sixth (115th harmonic) 115/64 1.796875 470.108 1014.588 ÖÓÖ 13-et Pythagorean minor seventh (13 2)11 1.797702 470.325 1015.385 ÖÓ× acute or large minor seventh 9/5 1.800000 470.926 1017.596 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ)

ÖÓØ 53-et acute or large minor seventh (53 2)45 1.801323 471.272 1018.868

ÖÓÙ Pythagorean augmented sixth, or cyclic acute or large minor seventh (A) X 3ñð/2ñõ, or 59049/32768 1.802032 471.458 1019.550 ÖÔÐ 20-et acute or large minor seventh (20 2)17 1.802501 471.580 1020.000 ÖÔÑ tridecimal grave or small neutral seventh 65/36 1.805556 472.379 1022.931 ÖÔÒ 7-et twenty-one quarter-tones (7 2) 6 1.811447 473.921 1028.571 ÖÔÓ nonavigesimal grave or small neutral seventh (29th harmonic) 29/16 1.812500 474.196 1029.577 ÖÔÔ 43-et double diminished octave (43 2)37 1.815624 475.014 1032.558 ÖÔÕ undecimal grave or small neutral seventh 20/11 1.818181 475.683 1034.996 ÖÔÖ 22-et twenty-one quarter-tones (22 2)19 1.819619 476.059 1036.364 ÖÔ× neutral seventh 729/400 1.822500 476.813 1039.103 ÖÔØ 15-et neutral seventh (15 2)13 1.823445 477.060 1040.000 ÖÔÙ 53-et neutral seventh (53 2)46 1.825036 477.476 1041.509

ÖÕÐ cyclic neutral seventh (A) XXII 3òò/2óô 1.826618 477.890 1043.010 ÖÕÑ 23-et neutral seventh (23 2) 20 1.827112 478.019 1043.478 ÖÕÒ tridecimal neutral seventh (117th harmonic) 117/64 1.828125 478.284 1044.438 ÖÕÓ 31-et double augmented sixth, or neutral seventh (31 2) 27 1.828889 478.484 1045.161 ÖÕÔ neutral seventh 4000/2187 1.828989 478.510 1045.256 ÖÕÕ acute or large double augmented sixth 1875/1024 1.831055 479.051 1047.210 A ÖÕÖ undecimal acute or large neutral seventh 11/6 1.833333 479.647 1049.363 341 ppendices ÖÕ× twenty-one equal quarter-tones (8 2)7 , or approximately 939/512 1.834008 479.823 1050.000 ÖÕØ 17-et grave or small major seventh (17 2)15 1.843379 482.275 1058.824 ÖÕÙ 59th harmonic 59/32 1.843750 482.372 1059.172 4 Appendices 342 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÖÖÐ 43-et double augmented sixth (43 2)38 1.845128 482.733 1060.465 ÖÖÑ tridecimal acute or large neutral seventh 24/13 1.846154 483.001 1061.427 ÖÖÒ 53-et grave or small major seventh (53 2)47 1.849061 483.762 1064.151

ÖÖÓ cyclic grave or small major seventh (A) XXXIV 3óô/2õó 1.851539 484.410 1066.470 ÖÖÔ 9-et grave or small major seventh (9 2)8 1.851749 484.465 1066.667 ÖÖÕ grave or small major seventh 50/27 1.851852 484.492 1066.762 ÖÖÖ 19-et grave or small major seventh (19 2)17 1.859271 486.433 1073.684 ÖÖ× septendecimal neutral seventh (119th harmonic) 119/64 1.859375 486.460 1073.781 ÖÖØ nonavigesimal acute or large neutral seventh 54/29 1.862069 487.165 1076.288 ÖÖÙ 10-et just diatonic major seventh (10 2) 9 1.866066 488.211 1080.000

(A) V ß11 0.8 80 Ö×Ð meantone major seventh 4 243/128× 81 1.869186 489.027 1082.892 Ö×Ñ 31-et just diatonic major seventh (31 2) 28 1.870243 489.303 1083.871 Ö×Ò 21-et just diatonic major seventh (21 2)19 1.872235 489.825 1085.714

Ö×Ó Pythagorean diminished octave (D) VII 4096/2187 1.872885 489.995 1086.315 Ö×Ô 53-et just diatonic major seventh (53 2)48 1.873402 490.130 1086.792 Ö×Õ just diatonic major seventh (15th harmonic) 15/8 1.875000 490.548 1088.269 Ö×Ö 43-et just diatonic major seventh (43 2)39 1.875112 490.577 1088.372

Ö×× cyclic diatonic major seventh (A) XLVI 3ôö/2÷ò 1.876800 491.019 1089.930 Ö×Ø 11-et just diatonic major seventh (11 2)10 1.877862 491.297 1090.909 Ö×Ù 23-et just diatonic major seventh (23 2)21 1.883014 492.645 1095.652 ÖØÐ equal major seventh (12 2)11 , or approximately 967/512 1.887749 493.883 1100.000 ÖØÑ septendecimal supermajor seventh 17/9 1.888889 494.182 1101.045 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ÖØÒ grave or small neutral seventh (121st harmonic) 121/64 1.890625 494.636 1102.636 ÖØÓ nonadecimal supermajor seventh 36/19 1.894737 495.712 1106.397 ÖØÔ 13-et Pythagorean major seventh (13 2)12 1.896155 496.083 1107.692 ÖØÕ grave or small diminished octave 256/135 1.896296 496.120 1107.821 ÖØÖ 53-et Pythagorean major seventh (53 2)49 1.898064 496.582 1109.434

ÖØ× Pythagorean major seventh (A) V 3õ/2÷, or 243/128 1.898438 496.680 1109.775

ÖØØ 58th cyclic fifth (A) LVIII 3õø/2ùñ 1.902406 497.718 1113.390 ÖØÙ 14-et Pythagorean major seventh (14 2)13 1.903390 497.976 1114.286 ÖÙÐ septimal supermajor seventh 40/21 1.904762 498.334 1115.533 ÖÙÑ 43-et diminished octave (43 2) 40 1.905583 498.549 1116.279 ÖÙÒ 61st harmonic 61/32 1.906250 498.724 1116.885 ÖÙÓ 15-et just diatonic major seventh (15 2)14 1.909683 499.622 1120.000 ÖÙÔ 31-et supermajor seventh, or diminished octave (31 2) 29 1.912532 500.367 1122.581

(D) VII 13 0.571429 81 ÖÙÕ meantone diminished octave 4 4096/2187× 80 1.914046 500.763 1123.951 ÖÙÖ 16-et acute or large major seventh (16 2)15 1.915207 501.067 1125.000 ÖÙ× trivigesimal supermajor seventh 23/12 1.916667 501.449 1126.319 ÖÙØ diminished octave, or acute or large major seventh 48/25 1.920000 502.321 1129.328 ÖÙÙ 17-et acute or large major seventh (17 2)16 1.920093 502.346 1129.412 A ×ÐÐ 123rd harmonic 123/64 1.921875 502.812 1131.017 343 ppendices ×ÐÑ 53-et acute or large major seventh (53 2)50 1.923050 503.119 1132.075

×ÐÒ cyclic acute or large major seventh (A) XVII 3ñ÷/2òö 1.924338 503.456 1133.235 ×ÐÓ 18-et acute or large major seventh (18 2)17 1.924448 503.485 1133.333 4 Appendices 344 DEGREE INTERVAL FACTOR RATIO FREQUENCY CENTS NUMBER (DECIMAL) (HERTZ) ×ÐÔ 19-et twenty-three quarter-tones (19 2)18 1.928352 504.506 1136.842 ×ÐÕ 20-et twenty-three quarter-tones (20 2)19 1.931873 505.427 1140.000 ×ÐÖ 21-et twenty-three quarter-tones (21 2)20 1.935064 506.262 1142.857 ×Ð× twenty-three quarter-tones 60/31 1.935484 506.372 1143.233 ×ÐØ 43-et double diminished second (43 2) 41 1.936549 506.651 1144.186 ×ÐÙ untrigesimal supermajor seventh (31st harmonic) 31/16 1.937500 506.900 1145.036 ×ÑÐ 22-et twenty-three quarter-tones (22 2)21 1.937969 507.022 1145.455 ×ÑÑ undecimal subdiminished octave 64/33 1.939394 507.395 1146.727 ×ÑÒ 23-et twenty-three quarter-tones (23 2)22 1.940626 507.717 1147.826 ×ÑÓ twenty-three equal quarter-tones (24 2)23 , or approximately 1024/527 1.943064 508.355 1150.000 ×ÑÔ 53-et suboctave (53 2)51 1.948365 509.742 1154.717

×ÑÕ cyclic suboctave (A) XXIX 3òù/2ôõ 1.950593 510.325 1156.695

(A) XIIß 3 0.333333 80 ×ÑÖ meantone augmented seventh, or suboctave (125th harmonic) 531441/524288× 81 , or 125/64 1.953125 510.987 1158.941 ×Ñ× 31-et augmented seventh, suboctave (31 2)30 1.955777 511.681 1161.290 ×ÑØ 43-et augmented seventh (43 2) 42 1.968019 514.884 1172.093 ×ÑÙ septimal subdiminished octave (63rd harmonic) 63/32 1.968750 515.075 1172.736 ×ÒÐ 53-et grave or small octave (53 2)52 1.974014 516.452 1177.358 ×ÒÑ grave or small octave 160/81 1.975309 516.791 1178.494

×ÒÒ cyclic grave or small octave (A) XLI 3ôñ/2öô 1.977205 517.287 1180.155 ×ÒÓ acute or large meantone augmented seventh 2025/1024 1.977539 517.375 1180.447 ×ÒÔ 127th harmonic 127/64 1.984375 519.163 1186.422

ÐÑ octave (2nd harmonic) 2/1 2.000000 523.251 1200.000 Appendix B

Microtonal Notation Font

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h i j k l m n o

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Appendices 347

‘ ’ “ ” • – —

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À Á Â Ã Ä Å Æ Ç

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348 Appendices

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