Gustavo_Diaz_Jerez_DMA_Thesis_MSM - ALGORITHMIC MUSIC USING MATHEMATICAL MODELS IN MUSIC COMPOSITION by GUSTAVO DIAZ-JEREZ Submitted to The Manhattan

Gustavo_Diaz_Jerez_DMA_Thesis_MSM - ALGORITHMIC MUSIC USING...

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ALGORITHMIC MUSIC: USING MATHEMATICAL MODELS IN MUSIC COMPOSITION by GUSTAVO DIAZ-JEREZ Submitted to The Manhattan School of Music in partial fulfillment of the requirements for the degree of Doctor of Musical Arts and approved by August 2000
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ii ABSTRACT OF THE THESIS Algorithmic Music: Using Mathematical Models in Music Composition by GUSTAVO DIAZ-JEREZ Thesis advisor: Nils Vigeland The intimate connection between music and mathematics has been acknowledged throughout history. This paper is a survey of the most relevant methods employed today in algorithmic composition. It covers stochastic processes (randomness, probability functions, Markov chains); one-, two-, and three-dimensional chaotic systems; fractals; algorithms derived from noise spectra (1/f noise); number theory (Morse-Thue sequence, 3n+1/Hailstone numbers, prime numbers); cellular automata (one- and two-dimensional cellular automata); genetic algorithms; formal grammars (L- systems.) It includes a brief historical survey, an exhaustive list of available computer programs which implement most of the processes discussed, and C++ source code listings of routines valuable for algorithmic composition. Numerous musical examples are provided to demonstrate each process. These examples are included in an accompanying audio CD.
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iii CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . vi PREFACE . . . . . . . . . . . . . . . . . . . . . . . . vii Chapter I. INTRODUCTION . . . . . . . . . . . . . . . . . 1 II. A BRIEF HISTORY OF ALGORITHMIC MUSIC . . . . . 12 The Pythagorean . . . . . . . . . . . . . . 12 The Middle Ages . . . . . . . . . . . . . . 17 The Classical Period . . . . . . . . . . . . 21 The Golden Mean . . . . . . . . . . . . . . 23 The Twentieth Century . . . . . . . . . . . 27 The Schillinger System of Musical Composition . . . . . . . . . . . 27 Edgard Varèse . . . . . . . . . . . . . . 30 Iannis Xenakis . . . . . . . . . . . . . 32 Lejaren Hiller . . . . . . . . . . . . . 37 Contemporary Techniques . . . . . . . . . . 39 Chaos Theory . . . . . . . . . . . . . . 39 Cellular Automata . . . . . . . . . . . . 41 L-Systems . . . . . . . . . . . . . . . . 43
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iv Genetic Algorithms . . . . . . . . . . . 45 Other Techniques . . . . . . . . . . . . 47 III. TECHNIQUES AND APPLICATION OF MATHEMATICAL MODELS IN MUSIC . . . . . . . . . . . . . . . . 49 Mapping Techniques . . . . . . . . . . . . . 49 Modulo-based Mapping . . . . . . . . . . 50 Normalized Mapping . . . . . . . . . . . 51 Stochastic Algorithms . . . . . . . . . . . . 54 Randomness and Probability Distributions . . . . . . . . . . . . . . 54 Markov Chains . . . . . . . . . . . . . . 65 Music from Chaos . . . . . . . . . . . . . . 76 One-dimensional Chaotic Systems . . . . . 78 The Logistic Equation . . . . . . . . . 78 Attractors of Two-dimensional Chaotic Maps . . . . . . . . . . . . . . . 85 Attractors of Three-dimensional Chaotic Maps . . . . . . . . . . . . . . . 112 Fractals . . . . . . . . . . . . . . . . . 126 Noise . . . . . . . . . . . . . . . . . . . . 135 Number-Theory Algorithms . . . . . . . . . . 142 The Morse-Thue Sequence . . . . . . . . . 143 3n + 1 Numbers . . . . . . . . . . . . . . 153 The Prime Number Series . . . . . . . . . 152
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v Cellular Automata . . . . . . . . . . . . . . 165 One-dimensional Cellular Automata . . . . 169 Two-dimensional Cellular Automata . . . . 178 Genetic Algorithms . . . . . . . . . . . . . 182 L-Systems . . . . . . . . . . . . . . . . . . 197 IV. SOFTWARE AND COMPUTER RESOURCES FOR ALGORITHMIC COMPOSITION . . . . . . . . . . . . . . . . . . 217 V. SOURCE CODE LISTINGS . . . . . . . . . . . . . 238 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . 244 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 276 VITA . . . . . . . . . . . . . . . . . . . . . . . . . 284
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