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University of California San Diego UNIVERSITY OF CALIFORNIA SAN DIEGO Music From Chaos: Nonlinear Dynamical Systems as Generators of Musical Materials A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Music by Rick Bidlack Committee in charge: Professor Roger Reynolds, Chair Professor Will Ogdon Assistant Professor Rand Steiger Professor Henry Abarbanel Professor David Antin 1990 © Copyright 1990 by Rick Bidlack The dissertation of Rick Bidlack is approved, and it is acceptable in quality and form for publication on microfilm: Chair University of California, San Diego 1990 iii for my Mother and my Father and all my Friends iv We find the order we want from the chaos we cause. inscription in a college dormitory in Texas V Contents Table of Contents . vi List of Figures . viii List of Tables . X Acknowledgements xi Vita .. xii Abstract ..... xiii 1 Initial Conditions 1 1.1 Regimes of Chaos . 1 1.2 Rudimentaxy Nonlineax Dynamics 3 1.3 Musical Rationale: Historical and Contemporaxy 16 1.4 The Experimental Appaxatus 19 2 The Henon Map ...... 22 2.1 Technical Description 22 2.2 Musical Examples 27 3 The Lorenz System 47 3.1 Technical Description 47 3.2 Musical Examples 54 4 The Standard Map 72 4.1 Technical Description 72 4.2 Musical Examples 79 5 The Henon-Heiles System 91 5.1 Technical Description 91 5.2 Musical Examples 99 6 Musical Implications 118 A Chaotic Systems in 'C' 123 A.1 The Logistic Equation 123 A.2 The Henon Map . 124 A.3 The Lorenz System ... 124 A.4 The Standard Map . 125 A.5 The Henon-Heiles System 126 vi B Generation of Music from Chaos .. ....... .. 128 C Sensitive Dependence: A Study of the Lorenz System 132 References 148 vii List of Figures 1.1 Bifurcation diagrams of the logistic equation . .. .. .. 12 1.2 Magnification sequence of bifurcation diagram of the logistic equation 13 1.3 Example of the graphical scoring system . 21 2.1 The Henon map . .. .. 24 2.2 Magnification sequence of the Henon map 25 2.3 Basin of attraction of the Henon map. .. .. .. 26 2.4 Bifurcation diagrams of the Henon map . 28 2.5 Magnification of a section of the bifurcation diagram 29 2.6 Further magnification of the Henon bifurcation diagram 30 2.7 Phase portraits of the Henon map. 31 2.8 Bifurcation diagram of the Henon attractor 32 2.9 Scores to Example l(a) and 2(a) . 36 2.10 Scores to Examples 3(a), 3(b) and 4(a). 37 2.1 1 Scores to Examples 5( a) and 5(b) . 38 2.12 Scores to Examples 6(a), 6(b) and 6(d). 39 2.13 Scores to Examples 7(a) and 7(d) . 40 2.14 Score to Example 8(a) 41 2.15 Score to Example 8( d) 42 2. 16 Score to Example 9(a) 43 2.17 Score to Example 9(b) 44 2.18 Score to Example 9(d) 45 2.19 Score to Example lO(a) 46 3.1 Phase portrait of the Lorenz system 49 3.2 Phase portraits of the Lorenz system . 50 3.3 Two orbits of the Lorenz attractor at R = 150. 52 3.4 Surfaces of section through the Lorenz attractor 53 3.5 Surfaces of section through the Lorenz attractor 53 3.6 Score to Example 1 .. .. ...... 57 3.7 Score to Example 2 .............. 62 3.8 Score to Example 3( a), section at z = 27 .. 64 3. 9 Score to Example 3(b), section at z = 27 .. 65 3.10 Score to Example 4(a), section at y = 8.485282 .. 66 3.1 1 Score to Example 4(b ), section at y = 8.485282 . 67 3.12 Score to Example 5(a), section at z = 149 .. ... 68 viii 3.13 Score to Example 5(b ), section at z = 14 9 ... 69 3.14 Score to Example 6(a), section at y = 6.103 278 70 3.15 Score to Example 6(b ), section at y = 6.103 278 71 4.1 Phase portrait of the standard map ..... 74 4.2 Sequence of points forming a cyclical orbit . 75 4.3 Six orbits of the standard map . .. .. 76 4.4 Phase plots of the standard map . .. 77 4.5 Bifurcation diagrams of the standard map 80 4.6 Scores to Examplesl(a) and l(b) . 83 4.7 Scores to Examples 2(a) and 2(b) .... 84 4.8 Score to Example 3(b) .. .. .. 85 4.9 Scores to Examples 4(a), 5(a) and 6(a) . 86 4.10 Score to Example 8( e) .. .. .. .. 87 4.11 Score to Example 9(f) .. .. 88 5.1 Phase space of the Henon-Heiles system 92 5.2 Phase portraits of the Henon-Heiles system 93 5.3 Phase portraits of the Henon-Heiles system 94 5.4 Poincare sections of the Henon-Heiles system, at E = 1/12 . 96 5.5 Poincare sections of the Henon-Heiles system, at E = 1/8 97 5.6 Poincare sections of the Henon-Heiles system, at E = 1/6 98 5.7 Score to Example 1 ............ 103 5.8 Scores to Examples 2(a), 2(b) and 3(a). 108 5.9 Scores to Examples 4(a) and 5(a) . 109 5.10 Scores to Examples 5(b) and 6(a) . .. 110 5.11 Scores to Examples 7( a) and 8(a) .... 111 5.12 Scores to Examples 8(b), 9( a) and 9( b). 112 5.13 Scores to Examples lO(a) and lO(b) 113 5.14 Score to Examplell(a) ....... 114 5.15 Scores to Examples ll(b) and 12(a) 115 5.16 Scores to Examples 13(a) and 14(b) 116 5.17 Score to Example 15(b) ....... 117 ix List of Tables 1.1 Eight orbits of the logistic equation, in numerical form 7 1.2 Eight orbits of the logistic equation, in numerical form 9 1.3 Two orbits of the logistic equation ..... 14 2.1 Orbit of the Henon map in numerical form. 23 2.2 Mapping configurationsof the musical examples . 34 2.3 Format of the musical examples . 35 3.1 Parameters and coordinate space mappings 56 3.2 Format of the musical examples . 56 4.1 Mapping configurations of the musical examples . 81 4.2 Format of the musical examples . 82 5.1 Mapping configurationsof the musical examples . 101 5.2 Format of the musical examples . 102 X Acknowledgements A work such as this is inevitably the product of manyindividuals. I would like to extend special thanks to my chairman,Roger Reynolds, and to Henry Abarbanel, Director of the Institute for Nonlinear Science at UCSD, for their invaluable advice and careful readings of the early drafts; to Gary Cottrell, Dave Demers and the UCSD chaos group fortheir patience with my many questions; to Sylvie Gibet for her lucid explanations of the laws of physics; and to Alan Finke, John Stevens, Michael Tracey, Carol Vernallis and Bob Willey for their constant support and encouragement. While this work might have come into being without the input of these persons, it would not, undoubtedly, have been nearly as rewarding an undertaking. xi Vita November 14, 1958 Born, Tallahassee, Florida 1980 B.A. (German), Bowling Green State University, Ohio 1981 English language instructor in Hamburg, Germany 1982-1983 Teaching Assistant, Undergraduate Writing Program, UCSD 1983-1985 Research Assistant, Center for Music Experiment, UCSD 1985-1987 Research Assistant, Music Education Laboratory, UCSD 1986 M.A. (Music), UCSD 1988-1990 Software Consultant 1990 Ph.D. (Music), UCSD 1990-1991 Assistant Professor (visiting), Department of Music, State University of New York at Buffalo Fields of Study Composition Professors Roger Reynolds, Joji Yuasa and Will Ogdon Analysis Professors Bernard Rands and Roger Reynolds Theory Professors Gerald Balzano and Will Ogdon Computer Music Professors F. Richard Moore and Joji Yuasa History Professor Will Ogdon xii ABSTRACT OF THE DISSERTATION Music From Chaos: Nonlinear Dynamical Systems as Generators of Musical Materials by Rick Bidlack Doctor of Philosophy in Music University of California, San Diego, 1990 Professor Roger Reynolds, Chair A body of scientific/mathematical theory arising from a description of the be­ havior of complex dynamical systems is explored in terms of its pertinence to and utility in musical schemes for the generation of melodic lines and textures. Such systems are known to model significant behavioral features of real-world phenomena, including tur­ bulent or chaotic behavior. Many of the features of nonlinear dynamical �ystems that are intriguing from a mathematical point of view, especially the properties and nature of chaos, are likewise suggestive of musical qualities. A variety of musical behaviors may be elicited from chaotic systems, some quite novel, while others are evocative of already well-known practices. Typical is a type of continuous variation form, in which motivic materials are spun out within a constantly evolving context. Four archetypal chaotic systems, the Henon and Lorenz systems, the standard map, and the Henon-Heiles system, are explored in terms of their potentialfor the gener­ ation of melodic lines, textures and rhythms-raw materials suitable for further musical exploration and elaboration. Approximately one hour of recorded examples accompany the dissertation. xiii Chapter 1 Initial Conditions 1.1 Regimes of Chaos In recent decades, a great deal of attention from the scientific and mathemati­ cal communities has been focused on the exploration of a class of mathematical objects known as nonlinear dynamical systems. Although the symbolic specification of many of these systems is simple to the point of apparent triviality, their behavior can often be extremely rich and complex, even unpredictable.
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