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Historically Informed Retuning of Polyphonic Vocal Performance
journal of interdisciplinary music studies spring/fall 2008, volume 2, issue 1&2, art. #0821208, pp. 121-139 Historically Informed Retuning of Polyphonic Vocal Performance Jonathan Wild and Peter Schubert Schulich School of Music, McGill University Background in history of music theory. The use of just intonation has generated heated debates ever since the first accounts of the articulation of pitch space in the scientific treatises of Ancient Greece. In the Renaissance, the discussion turned to vocal music as the locus of debate. One of the problems with using just intonation is the incommensurability of pure intervals with one another (e.g., the A four pure fifths above F is not the same pitch as the pure major third above F). Treatises and accounts of tuning systems proliferated, and our present- day understanding of what is at stake is limited by the dearth of sounding examples. Background in performance. On the one hand it is very difficult to verify precisely how modern-day singers tune, and on the other, although recent interest in historical tuning has generated several articles with electronically produced sound examples, the examples do not contribute to a direct understanding of tuning in the vocal context—synthesized sound is simply not “the real thing.” Aims. Our study aims to show how the gap between theories of tuning and the art of vocal performance may be bridged. Main contribution. By producing recordings of precisely tuned audio in rich vocal timbres (with vibrato, consonants and vowels), we make available actual repertoire examples, reliably tuned, in a human-sounding rendition. -
The Lost Harmonic Law of the Bible
The Lost Harmonic Law of the Bible Jay Kappraff New Jersey Institute of Technology Newark, NJ 07102 Email: [email protected] Abstract The ethnomusicologist Ernest McClain has shown that metaphors based on the musical scale appear throughout the great sacred and philosophical works of the ancient world. This paper will present an introduction to McClain’s harmonic system and how it sheds light on the Old Testament. 1. Introduction Forty years ago the ethnomusicologist Ernest McClain began to study musical metaphors that appeared in the great sacred and philosophical works of the ancient world. These included the Rg Veda, the dialogues of Plato, and most recently, the Old and New Testaments. I have described his harmonic system and referred to many of his papers and books in my book, Beyond Measure (World Scientific; 2001). Apart from its value in providing new meaning to ancient texts, McClain’s harmonic analysis provides valuable insight into musical theory and mathematics both ancient and modern. 2. Musical Fundamentals Figure 1. Tone circle as a Single-wheeled Chariot of the Sun (Rg Veda) Figure 2. The piano has 88 keys spanning seven octaves and twelve musical fifths. The chromatic musical scale has twelve tones, or semitone intervals, which may be pictured on the face of a clock or along the zodiac referred to in the Rg Veda as the “Single-wheeled Chariot of the Sun.” shown in Fig. 1, with the fundamental tone placed atop the tone circle and associated in ancient sacred texts with “Deity.” The tones are denoted by the first seven letters of the alphabet augmented and diminished by and sharps ( ) and flats (b). -
Cracking a Centuries-Old Tradition
Cracking a Centuries-Old Tradition hen I went to Cambridge, WEngland, on sabbatical in 2013–14, I never dreamed I would wind up conducting one of the world’s great choirs, and pos- sibly changing the way they sing early music. My project for the year was to write a follow-up to Shakespeare’s Songbook (Norton, 2004), my study of all the songs sung, quoted, or alluded to in the plays of Shakespeare. The sequel is a broader look at songs in English Equal Temperament Ruined Harmony (and by Ross W. why you should care) (Norton, 2007); one Duffin Renaissance comedy, from the on Just Intonation in the Renaissance; one 15th century through the plays of on keyboard temperament; and one on Just Intonation in the 18th century. So, in spite of Shakespeare’s contemporaries. the central purpose of my sabbatical, my tun- Being at Clare Hall at the University of ing work was getting a lot of attention, and I Cambridge allowed me easy access to the was pleased when Stephen Cleobury at King’s resources of the superb University Library College and Andrew Nethsingha at St John’s across the street, a wonderful advantage for College each asked me to coach their choral my work. But it also allowed me the option scholars (the men from the men and boys to attend choral services, virtually every choir) in Just Intonation. By coincidence, it day if I wanted, at any of the thirty-one happened that both coachings were to occur Cambridge colleges. Nowadays, colleges post on the same day, and that became a red-letter the music for all the services each term in day on my calendar—the expected high point an online “Term List,” so I could pick out in of my entire year in Cambridge. -
Consonance and Dissonance in Visual Music Bill Alves Harvey Mudd College
Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 8-1-2012 Consonance and Dissonance in Visual Music Bill Alves Harvey Mudd College Recommended Citation Bill Alves (2012). Consonance and Dissonance in Visual Music. Organised Sound, 17, pp 114-119 doi:10.1017/ S1355771812000039 This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Organised Sound http://journals.cambridge.org/OSO Additional services for Organised Sound: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here Consonance and Dissonance in Visual Music Bill Alves Organised Sound / Volume 17 / Issue 02 / August 2012, pp 114 - 119 DOI: 10.1017/S1355771812000039, Published online: 19 July 2012 Link to this article: http://journals.cambridge.org/abstract_S1355771812000039 How to cite this article: Bill Alves (2012). Consonance and Dissonance in Visual Music. Organised Sound, 17, pp 114-119 doi:10.1017/ S1355771812000039 Request Permissions : Click here Downloaded from http://journals.cambridge.org/OSO, IP address: 134.173.130.244 on 24 Jul 2014 Consonance and Dissonance in Visual Music BILL ALVES Harvey Mudd College, The Claremont Colleges, 301 Platt Blvd, Claremont CA 91711 USA E-mail: [email protected] The concepts of consonance and dissonance broadly Plato found the harmony of the world in the Pythag- understood can provide structural models for creators of orean whole numbers and their ratios, abstract ideals visual music. -
Microtonality As an Expressive Device: an Approach for the Contemporary Saxophonist
Technological University Dublin ARROW@TU Dublin Dissertations Conservatory of Music and Drama 2009 Microtonality as an Expressive Device: an Approach for the Contemporary Saxophonist Seán Mac Erlaine Technological University Dublin, [email protected] Follow this and additional works at: https://arrow.tudublin.ie/aaconmusdiss Part of the Composition Commons, Musicology Commons, Music Pedagogy Commons, Music Performance Commons, and the Music Theory Commons Recommended Citation Mac Erlaine, S.: Microtonality as an Expressive Device: an Approach for the Contemporary Saxophonist. Masters Dissertation. Technological University Dublin, 2009. This Dissertation is brought to you for free and open access by the Conservatory of Music and Drama at ARROW@TU Dublin. It has been accepted for inclusion in Dissertations by an authorized administrator of ARROW@TU Dublin. For more information, please contact [email protected], [email protected]. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License Microtonality as an expressive device: An approach for the contemporary saxophonist September 2009 Seán Mac Erlaine www.sean-og.com Table of Contents Abstract i Introduction ii CHAPTER ONE 1 1.1 Tuning Theory 1 1.1.1 Tuning Discrepancies 1 1.2 Temperament for Keyboard Instruments 2 1.3 Non‐fixed Intonation Instruments 5 1.4 Dominance of Equal Temperament 7 1.5 The Evolution of Equal Temperament: Microtonality 9 CHAPTER TWO 11 2.1 Twentieth Century Tradition of Microtonality 11 2.2 Use of Microtonality -
The Unexpected Number Theory and Algebra of Musical Tuning Systems Or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72
The Unexpected Number Theory and Algebra of Musical Tuning Systems or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72 Matthew Hawthorn \Music is the pleasure the human soul experiences from counting without being aware that it is counting." -Gottfried Wilhelm von Leibniz (1646-1716) \All musicians are subconsciously mathematicians." -Thelonius Monk (1917-1982) 1 Physics In order to have music, we must have sound. In order to have sound, we must have something vibrating. Wherever there is something virbrating, there is the wave equation, be it in 1, 2, or more dimensions. The solutions to the wave equation for any given object (string, reed, metal bar, drumhead, vocal cords, etc.) with given boundary conditions can be expressed as a superposition of discrete partials, modes of vibration of which there are generally infinitely many, each with a characteristic frequency. The partials and their frequencies can be found as eigenvectors, resp. eigenvalues of the Laplace operator acting on the space of displacement functions on the object. Taken together, these frequen- cies comprise the spectrum of the object, and their relative intensities determine what in musical terms we call timbre. Something very nice occurs when our object is roughly one-dimensional (e.g. a string): the partial frequencies become harmonic. This is where, aptly, the better part of harmony traditionally takes place. For a spectrum to be harmonic means that it is comprised of a fundamental frequency, say f, and all whole number multiples of that frequency: f; 2f; 3f; 4f; : : : It is here also that number theory slips in the back door. -
Lute Tuning and Temperament in the Sixteenth and Seventeenth Centuries
LUTE TUNING AND TEMPERAMENT IN THE SIXTEENTH AND SEVENTEENTH CENTURIES BY ADAM WEAD Submitted to the faculty of the Jacobs School of Music in partial fulfillment of the requirements for the degree, Doctor of Music, Indiana University August, 2014 Accepted by the faculty of the Jacobs School of Music, Indiana University, in partial fulfillment of the requirements for the degree Doctor of Music. Nigel North, Research Director & Chair Stanley Ritchie Ayana Smith Elisabeth Wright ii Contents Acknowledgments . v Introduction . 1 1 Tuning and Temperament 5 1.1 The Greeks’ Debate . 7 1.2 Temperament . 14 1.2.1 Regular Meantone and Irregular Temperaments . 16 1.2.2 Equal Division . 19 1.2.3 Equal Temperament . 25 1.3 Describing Temperaments . 29 2 Lute Fretting Systems 32 2.1 Pythagorean Tunings for Lute . 33 2.2 Gerle’s Fretting Instructions . 37 2.3 John Dowland’s Fretting Instructions . 46 2.4 Ganassi’s Regola Rubertina .......................... 53 2.4.1 Ganassi’s Non-Pythagorean Frets . 55 2.5 Spanish Vihuela Sources . 61 iii 2.6 Sources of Equal Fretting . 67 2.7 Summary . 71 3 Modern Lute Fretting 74 3.1 The Lute in Ensembles . 76 3.2 The Theorbo . 83 3.2.1 Solutions Utilizing Re-entrant Tuning . 86 3.2.2 Tastini . 89 3.2.3 Other Solutions . 95 3.3 Meantone Fretting in Tablature Sources . 98 4 Summary of Solutions 105 4.1 Frets with Fixed Semitones . 106 4.2 Enharmonic Fretting . 110 4.3 Playing with Ensembles . 113 4.4 Conclusion . 118 A Complete Fretting Diagrams 121 B Fret Placement Guide 124 C Calculations 127 C.1 Hans Gerle . -
Musical Tuning Systems As a Form of Expression
Musical Tuning Systems as a Form of Expression Project Team: Lewis Cook [email protected] Benjamin M’Sadoques [email protected] Project Advisor Professor Wilson Wong Department of Computer Science This report represents the work of WPI undergraduate students submitted to the faculty as evidence of completion of a degree requirement. WPI routinely publishes these reports on its website without editorial or peer review. For more information about the projects program at WPI, please see http://www.wpi.edu/academics/ugradstudies/project- learning.html Abstract Many cultures and time periods throughout history have used a myriad of different practices to tune their instruments, perform, and create music. However, most musicians in the western world will only experience 12-tone equal temperament a represented by the keys on a piano. We want musicians to recognize that choosing a tuning system is a form of musical expression. The goal of this project was to help musicians of any skill-level experience, perform, and create music involving tuning systems. We created software to allow musicians to experiment and implement alternative tuning systems into their own music. ii Table of Contents Abstract ................................................................................................................................... ii Table of Figures .................................................................................................................... vii 1 Introduction ........................................................................................................................ -
Flexible Tuning Software: Beyond Equal Temperament
Syracuse University SURFACE Syracuse University Honors Program Capstone Syracuse University Honors Program Capstone Projects Projects Spring 5-1-2011 Flexible Tuning Software: Beyond Equal Temperament Erica Sponsler Follow this and additional works at: https://surface.syr.edu/honors_capstone Part of the Computer and Systems Architecture Commons, and the Other Computer Engineering Commons Recommended Citation Sponsler, Erica, "Flexible Tuning Software: Beyond Equal Temperament" (2011). Syracuse University Honors Program Capstone Projects. 239. https://surface.syr.edu/honors_capstone/239 This Honors Capstone Project is brought to you for free and open access by the Syracuse University Honors Program Capstone Projects at SURFACE. It has been accepted for inclusion in Syracuse University Honors Program Capstone Projects by an authorized administrator of SURFACE. For more information, please contact [email protected]. Flexible Tuning Software: Beyond Equal Temperament A Capstone Project Submitted in Partial Fulfillment of the Requirements of the Renée Crown University Honors Program at Syracuse University Erica Sponsler Candidate for B.S. Degree and Renée Crown University Honors May/2011 Honors Capstone Project in _____Computer Science___ Capstone Project Advisor: __________________________ Dr. Shiu-Kai Chin Honors Reader: __________________________________ Dr. James Royer Honors Director: __________________________________ James Spencer, Interim Director Date:___________________________________________ Flexible Tuning Software: Beyond Equal -
On the Notation and Performance Practice of Extended Just Intonation
On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation by Marc Sabat 1. Introduction: Two Different E’s Like the metric system, the modern tempered tuning which divides an octave into 12 equal but irrational proportions was a product of a time obsessed with industrial standardization and mass production. In Schönberg’s words: a reduction of natural relations to manageable ones. Its ubiquity in Western musical thinking, epitomized by the pianos which were once present in every home, and transferred by default to fixed-pitch percussion, modern organs and synthesizers, belies its own history as well as everyday musical experience. As a young musician, I studied composition, piano and violin. Early on, I began to learn about musical intervals, the sound of two tones in relation to each other. Without any technical intervention other than a pitch-pipe, I learned to tune my open strings to the notes G - D - A - E by playing two notes at once, listening carefully to eliminate beating between overtone-unisons and seeking a stable, resonant sound-pattern called a “perfect fifth”. At the time, I did not know or need to know that this consonance was the result of a simple mathematical relationship, that the lower string was vibrating twice for every three vibrations of the upper one. However, when I began to learn about placing my fingers on the strings to tune other pitches, the difficulties began. To find the lower E which lies one whole step above the D string, I needed to place my first finger down. -
Shadings in the Chromatic Field: Intonations After Morton Feldman
Shadings in the Chromatic Field: Intonations after Morton Feldman Marc Sabat ... this could be an element of the aural plane, where I'm trying to balance, a kind of coexistence between the chromatic field and those notes selected from the chromatic field that are not in the chromatic series.1 Harmony, or how pitched sounds combine, implies microtonality, enharmonic variations of tuning. Historically, these came to be reflected in written music by having various ways of spelling pitches. A harmonic series over E leads to the notes B and G#, forming a just major triad. Writing Ab instead of G# implies a different structure, but in what way? How may such differences of notation be realized as differences of sound? The notion of enharmonic "equivalence," which smooths away such shadings, belongs to a 20th century atonal model: twelve-tone equal temperament. This system rasters the frequency glissando by constructing equal steps in the irrational proportion of vibration 1:12√2. Twelve successive steps divide an octave, at which interval the "pitch- classes" repeat their names. Their vertical combinations have been exhaustively demonstrated, most notably in Tom Johnson's Chord Catalogue. However, the actual sounding of pitches, tempered or not, always reveals a microtonally articulated sound continuum. Hearing out the complex tonal relations within it suggests a new exploration of harmony: composing intonations in writing, playing and hearing music. Morton Feldman recognized that this opening for composition is fundamentally a question of rethinking the notational image. In works composed for the most part between 1977 and 1985, inspired by his collaboration with violinist Paul Zukofsky, Feldman chose to distinguish between enharmonically spelled pitches. -
MTO 12.3: Duffin, Just Intonation in Renaissance Theory and Practice
Volume 12, Number 3, October 2006 Copyright © 2006 Society for Music Theory Ross W. Duffin ABSTRACT: Just intonation has a reputation as a chimerical, theoretical system that simply cannot work in practice. This is based on the assessment of most modern authorities and supported by misgivings expressed during the Renaissance when the practice was supposedly at its height. Looming large among such misgivings are tuning puzzles printed by the 16th-century mathematician, Giovanni Battista Benedetti. However, Renaissance music theorists are so unanimous in advocating the simple acoustical ratios of Just intonation that it seems clear that some reconciliation must have occurred between the theory and practice of it. This article explores the basic theory of Just intonation as well as problematic passages used to deny its practicability, and proposes solutions that attempt to satisfy both the theory and the ear. Ultimately, a resource is offered to help modern performers approach this valuable art. Received June 2006 Introduction | Theoretical Background | Benedetti's Puzzles | Problematic Passages | Is Just Tuning Possible? A New Approach | Problem Spots in the Exercises | Rehearsal Usage | Conclusion Introduction The idea . that one can understand the ratios of musical consonances without experiencing them with the senses is wrong. Nor can one know the theory of music without being versed in its practice. [1] So begins the first of two letters sent by the mathematician Giovanni Battista Benedetti to the composer Cipriano de Rore in 1563. Subsequently publishing the letters in 1585,(1) Benedetti was attempting to demonstrate that adhering to principles of Just intonation, as championed most famously by Gioseffo Zarlino,(2) would, in certain cases, cause the pitch of the ensemble to migrate.