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The Origin and Well-Formedness of Tonal Pitch Structures

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The Origin and Well-Formedness of Tonal Pitch Structures The Origin and Well-Formedness of Tonal Pitch Structures Aline Honingh The Origin and Well-Formedness of Tonal Pitch Structures ILLC Dissertation Series DS-2006-05 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ The Origin and Well-Formedness of Tonal Pitch Structures Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.mr. P.F. van der Heijden ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op vrijdag 20 oktober 2006, te 12.00 uur door Aline Klazina Honingh geboren te Broek in Waterland Promotores: Prof.dr. R. Bod Prof.dr. H. Barendregt Faculteit der Natuurwetenschappen, Wiskunde en Informatica This research was supported by the Netherlands Organization for Scientific Research (NWO) in the context of the Innovation Impulse programme \Towards a Unifying Model for Linguistic, Musical and Visual Processing". Copyright c 2006 by Aline K. Honingh Printed and bound by PrintPartners Ipskamp. ISBN-10: 90-5776-156-4 ISBN-13: 978-90-5776-156-0 Contents Acknowledgments ix 1 Introduction and musical background 1 1.1 Questions to address in this thesis . 1 1.2 Perception of musical tones . 2 1.2.1 Beats . 3 1.2.2 Critical bandwidth and just noticeable difference . 4 1.2.3 Virtual pitch . 5 1.2.4 Combination tones . 6 1.3 Just intonation and the compromises of temperaments . 7 1.3.1 Harmonic series . 7 1.3.2 Temperament difficulties . 10 1.3.3 Tuning and temperament systems . 12 1.4 Consonance and dissonance . 13 1.4.1 Explanations on sensory consonance and dissonance . 13 1.4.2 Different types of consonance . 17 1.5 Tonality . 20 1.5.1 Scales . 21 1.6 What lies ahead . 22 2 Algebraic interpretation of tone systems 25 2.1 Group theory applied to music . 25 2.1.1 Cyclic groups . 26 2.1.2 Properties of groups and mappings . 27 2.2 Group theoretic and geometric description of just intonation . 29 2.2.1 Just intonation in group theoretic terms . 29 2.2.2 Different realizations of the tone space . 32 2.3 Other geometrical representations of musical pitch . 37 v 3 Equal temperament to approximate just intonation 41 3.1 Short review of techniques of deriving equal-tempered systems . 42 3.1.1 Continued fractions . 43 3.1.2 Fokker's periodicity blocks . 45 3.2 Approximating consonant intervals from just intonation . 46 3.2.1 Measures of consonance . 48 3.2.2 Goodness-of-fit model . 51 3.2.3 Resulting temperaments . 54 3.3 Limitations on fixed equal-tempered divisions . 56 3.3.1 Attaching note-names to an octave division . 57 3.3.2 Equal tempered divisions represented in the tone space . 65 3.3.3 Extended note systems . 68 3.3.4 Summary and resulting temperaments . 70 4 Well-formed or geometrically good pitch structures: (star-) con- vexity 73 4.1 Previous approaches to well-formed scale theory . 73 4.1.1 Carey and Clampitt's well-formed scales . 74 4.1.2 Balzano's group theoretical properties of scales . 76 4.2 Convexity and the well-formedness of musical objects . 79 4.2.1 Convexity on tone lattices . 80 4.2.2 Convex sets in note name space . 83 4.2.3 Convexity of scales . 86 4.2.4 Convexity of chords . 90 4.2.5 Convexity of harmonic reduction . 92 4.2.6 Discussion . 95 4.3 Concluding remarks on well-formedness . 97 5 Convexity and compactness as models for the preferred intona- tion of chords 99 5.1 Tuning of chords in isolation . 99 5.1.1 A model for intonation . 100 5.1.2 Compositions in the tone space indicating the intonation . 103 5.2 Compactness and Euler . 107 5.2.1 Compactness in 3D . 107 5.2.2 Compactness in 2D . 110 5.3 Convexity, compactness and consonance . 113 5.4 Concluding remarks on compactness and convexity . 116 6 Computational applications of convexity and compactness 119 6.1 Modulation finding . 119 6.1.1 Probability of convex sets in music . 120 vi 6.1.2 Finding modulations by means of convexity . 125 6.2 Pitch spelling . 129 6.2.1 Review of other models . 130 6.2.2 Pitch spelling using compactness . 132 6.2.3 The algorithm . 136 6.2.4 Error analysis . 139 6.2.5 Evaluation and comparison to other models . 141 7 Concluding remarks 145 A Notes on lattices and temperaments 149 3 A.1 Isomorphism between P3 and Z . 149 A.2 Alternative bases of Z2 . 150 A.3 Generating fifth condition . 151 Samenvatting 153 Index 171 vii Acknowledgments Een woord van dank aan de mensen die er aan bijgedragen hebben dat dit proef- schrift nu is zoals het is. Ik wil graag mijn promotor en begeleider Rens Bod bedanken, allereerst voor het feit dat hij mij aangenomen heeft voor deze AIO baan met daarbij het vertrouwen dat ik iets kon bijdragen aan een tot dat mo- ment mij nog onbekend wetenschapsgebied. Ik heb veel vrijheid gekregen zodat ik me kon richten op het onderwerp van mijn interesse, maar ook kreeg ik, op cruciale momenten wanneer ik door de bomen het bos niet meer zag, de sturing die ik nodig had. Rens heeft een enorm aanstekelijk enthousiasme dat me altijd weer kon motiveren. Rens, heel erg bedankt voor de fijne samenwerking, begelei- ding en inspiratie. Mijn tweede promotor, Henk Barendregt, ben ik in de eerste plaats dankbaar voor het accepteren van het promotorschap, iets wat geenszins vanzelfsprekend was aangezien ik dat hem een jaar geleden pas gevraagd heb. De intensieve reeks afspraken die we gehad hebben, zijn heel waardevol geweest voor het uiteindelijke resultaat. Bedankt. Besides my two supervisors, also several other people have taught me a lot and influenced my work. I am grateful to Thomas Noll, for the private course on mathematical music theory during a bus trip in Italy; for the discussions on 19-tone equal temperament in Paris; for the discussions during my visit to Berlin; and for all the feedback given on my work. I am grateful to David Meredith, who read and commented on my whole thesis, and whose work was of great inspiration over the last four years. Especially, my chapter on pitch spelling had benefited a lot from his dissertation on the subject. I want to thank Kamil Adiloglu, Elaine Chew, Nick Collins, Jan van de Craats, Peter van Emde Boas, J¨org Garbers, Dion Gijswijt, Henkjan Honing, Benedikt L¨owe, Fr´ed´eric Maintenant, Michael McIntyre, Wim van der Meer, Dirk-Jan Povel, Remko Scha, Stefan Schlobach, Michiel Schuijer, Leigh Smith, Paul Tegelaar, Dan Tidhar, Leen Torenvliet, Henk Visser, Anja Volk, Frans Wiering and Menno van Zaanen for helpful suggestions and discussions about my work. My research has furthermore benefited from the kind correspondence with David Benson, Peter Cariani, Paul Erlich, Ernst Terhardt, and many people from the `Alternate Tunings Mailing List'. Neta, I have enjoyed working together in the project during the time that we spent together in Amsterdam and Cambridge. I am grateful to Alan Blackwell for inviting me as a visiting scholar to Cambridge, and to Ian Cross and all the people of the science and music group for contributing to the unforgettable time I had in Cambridge. ix I have very much enjoyed our monthly reading group, where people with various backgrounds came to discuss topics within the common field of interest: music. I want to thank all members of this reading group for their contributions and enthusiasm. Het ILLC is altijd een heel plezierig instituut geweest om te werken. Mijn dank gaat uit naar Frank Veltman, Ingrid van Loon, Marjan Veldhuisen, Tanja Kassenaar, Jessica Pogorzelski en Ren´e Goedman voor alle hulp en de fijne werk- omgeving. Yoav en Jelle, dank jullie wel, voor de gezelligheid, maar ook voor het geduld bij het uitleggen van lastige mathematische en computationele problemen. Ik ben Neta, Merlijn, Yoav, Sieuwert en Olivia, mijn kamergenoten in chronolo- gische volgorde, dankbaar voor de sfeer op de gezelligste kamer van het instituut. Een paar speciale woorden voor Merlijn: heel erg bedankt voor de gezellige tijd die we hier samen hebben doorgebracht, voor al het `gekwebbel' maar ook voor de fijne gesprekken over complexiteit, muziek en het leven. De hele werkvloer droeg bij aan de goede sfeer: Brian, Clemens, Eric, Fenrong, Joost, Leigh, Nick, Olivier, Reut, Stefan, Ulle, en alle anderen: heel erg bedankt! Ook buiten de werkkring is er een aantal mensen geweest die me de afgelopen vier jaar ge¨ınspireerd en gemotiveerd hebben. Als grote bron van inspirate wil ik als eerste Bas Pollard noemen, van wie ik veel geleerd heb over intonatie- problematiek maar ook over muziek in het algemeen, al lang voordat ik met dit onderzoek begon. Ik bedank Nick Devons voor de fijne vioollessen die vaak een therapeutische werking hadden en ervoor zorgden dat ik muziek ´altijd leuk ben blijven vinden, ook al vlotte het onderzoek op dat moment misschien niet zo. En dan zijn er nog de mensen die me gesteund hebben en voor prettige afleid- ing hebben gezorgd in de avonduren en weekenden. Bedankt lieve NSO vriendin- nen Elske, Margriet en Janneke; CREA vriendinnen Simone, Jeantine en Nienke; Winston kwartet Tessa, Matthijs en Maurice (\o, wat speel ik gevoelig"); Etain trio, Marjolein en Roeselien - dank jullie wel voor alle gezellige etentjes, fijne gesprekken en muzikale hoogtepuntjes.
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