SALTER, JONATHAN R., D.M.A. Chaos in Music: Historical Developments and Applications to Music Theory and Composition. (2009) Directed by Dr. Kelly Burke. 208 pp. The Doctoral Dissertation submitted by Jonathan R. Salter, in partial fulfillment of the requirements for the degree Doctor of Musical Arts at the University of North Carolina at Greensboro comprises the following: 1. Doctoral Recital I, March 24, 2007: Chausson, Andante et Allegro; Tomasi, Concerto for Clarinet; Bart´ok, Contrasts; Fitkin, Gate. 2. Doctoral Recital II, December 2, 2007: Benjamin, Le Tombeau de Ravel; Man- dat, Folk Songs; Bolcom, Concerto for Clarinet; Kov´acs, Sholem-alekhem, rov Fiedman! 3. Doctoral Recital III, May 3, 2009: Kalliwoda, Morceau du Salon; Shostakovich, Sonata, op. 94 (transcription by Kennan); Tailleferre, Arabesque; Schoenfield, Trio for Clarinet, Violin, and Piano. 4. Dissertation Document: Chaos in Music: Historical Developments and Appli- cations to Music Theory and Composition. Chaos theory, the study of nonlinear dynamical systems, has proven useful in a wide-range of applications to scientific study. Here, I analyze the application of these systems in the analysis and creation of music, and take a historical view of the musical developments of the 20th century and how they relate to similar developments in science. I analyze several 20th century works through the lens of chaos theory, and discuss how acoustical issues and our interpretation of music relate to the theory. The application of nonlinear functions to aspects of music including organization, acoustics and harmonics, and the role of chance procedures is also examined toward suggesting future possibilities in incorporating chaos theory in the act of composition. Original compositions are included, in both sheet music and recorded form. Chaos in Music: Historical Developments and Applications to Music Theory and Composition by Jonathan R. Salter A Dissertation Submitted to the Faculty of The Graduate School at The University of North Carolina at Greensboro in Partial Fulfillment of the Requirements for the Degree Doctor of Musical Arts Greensboro 2009 Approved by Dr. Kelly Burke Committee Chair c 2009 by Jonathan Salter To Anne, for equanimity in the face of chaos. ii Approval Page This dissertation has been approved by the following committee of the Faculty of The Graduate School at The University of North Carolina at Greensboro. Committee Chair Kelly Burke Committee Members Gregory Carroll Gavin Douglas Steven Stusek Date of Acceptance by Committee Date of Final Oral Examination iii Acknowledgments The contributions to this work by Dr. Kelly Burke cannot be overstated; it was primarily under her advice that I embarked upon this research. Thanks also to Dr. Gavin Douglas, Dr. Greg Carroll, and Dr. Steven Stusek for their patience, understanding, assistance in the process and suggestions for revisions. Dr. Alejandro Rutty and Dr. Mark Engebretson were incredibly helpful in guiding my compositional process. In addition, many kind colleagues and professors at the University of North Carolina at Greensboro played a role in helping me flesh out ideas and theories; alas, far too many to name individually, though they all deserve my thanks. Dr. Edward Burger, Professor of Mathematics at Williams College, served as an outside reader for this document and contributed many helpful remarks and a few corrections. I also owe a debt of gratitude to Eli Eban, who incidentally referred to chaos quite frequently in our lessons (though, unfortunately, it was in reference to my clarinet playing), as well as Alan Kay, Susan Martula, and Mich`eleGingras, who each played an essential role in my development as a clarinetist and musician. iv Preface \A journey of a thousand miles begins with a single step." | Lao-Tzu Chaos, though applied to many different areas of study, is inherently a mathe- matical idea. While this dissertation does not require an extraordinary amount of mathematical background to grasp the main concepts, experience with some of the basic tools of mathematics (functions, systems, equations, etc.) is assumed. Here, I will not shy away from using explicit mathematical formulas, and those less accus- tomed to reading mathematics should feel free to skim the mathematical sections and spend more time on the analysis and application of these mathematical ideas. Some background in music analysis is also assumed, though again, those readers less familiar with music theory may wish to pay more attention to the application of the analysis than the analysis itself. v Table of Contents Page List of Tables ..............................................................viii List of Figures ...............................................................ix Definitions ...................................................................xi Chapter Introduction ..................................................................1 Part I: History ...............................................................4 I. The Development of Chaos .........................................5 1.1 Pythagoras and the Greeks . 6 1.2 Isaac Newton . 11 1.3 George William Hill . 13 1.4 Henri Poincar´e. 15 1.5 Relativity and Quantum Mechanics . 18 1.6 Edward Lorenz . 20 1.7 Mandelbrot . 24 1.8 What is Chaos Theory? . 28 1.9 Summary . 32 II. 20th Century Music History and Chaos .........................34 2.1 Serialism . 36 2.2 Randomness or Chance Procedures . 38 2.3 Minimalism . .43 2.4 Pointillism and Independence . 44 2.5 Electronic Music . 49 2.6 Summary . 53 III. Music, Science and Society ........................................54 3.1 Metacognition . 55 3.2 Free Will . 60 3.3 Summary . 62 Summary of Part I ..........................................................63 vi Part II: Music Theory ......................................................65 IV. Understanding Harmony ...........................................66 4.1 Fourier Analysis and the Anatomy of the Ear . 67 4.2 Aperiodic Waveforms . 73 4.3 Consonance and Dissonance . .76 4.4 Fractal Dimension of Sound . .78 4.5 Summary . 79 V. Chaotic Analysis ....................................................81 5.1 Information Theory and the Mozart Concerto . 84 5.2 Serialism . 94 5.3 Olivier Messiaen . 96 5.4 John Cage . 100 5.5 Iannis Xenakis . .102 5.6 Gy¨orgyLigeti . 104 5.7 Steve Reich . .109 5.8 Summary . 112 Summary of Part II ........................................................113 Part III: Composition ......................................................114 VI. Creating Chaos in Music ..........................................115 6.1 Randomness and Chance Procedures . .116 6.2 Chaos in Form . .117 6.3 Chaotic Serialism . 125 6.4 Thematic Transformation . 127 6.5 Rules Music . 130 Summary of Part III .......................................................145 Postlude ....................................................................146 Bibliography ................................................................148 p Appendix A: Proof of the Irrationality of 2 . .160 Appendix B: Chaos Drumming ............................................163 Appendix C: Chaotic Serialism ...........................................203 Appendix D: Recorded Examples ........................................208 vii List of Tables Page 2.1 Probability Distribution of Coin Flips . .41 5.1 Mozart Clarinet Concerto, Allegro: Phrase Lengths . 86 5.2 Weber, Clarinet Concerto No. 1, Allegro: Phrase Lengths . 87 5.3 Weber Probability Distribution . 90 5.4 Mozart Probability Distribution . 91 5.5 Pattern of Accents in Ligeti, D´esordre, First 6 Cycles . 107 viii List of Figures Page 1.1 The Mandelbrot Set . 24 1.2 The Koch Curve and Snowflake . 26 1.3 An Actual Snowflake . 27 p 1.4 Hypotenuse of a Right Triangle is 2 ......................................32 2.1 Georges Seurat, Sunday Afternoon of the Island of La Grand Jatte . 47 3.1 John Cage, 4'33" . 56 3.2 Marcel Duchamp's Fountain ...............................................57 3.3 Ars Poetica by Archibald MacLeish, 1926 . 58 4.1 The Anatomy of the Ear . 68 4.2 The Harmonic Series with Approximate Frequencies (Hz) . 70 4.3 Fourier Analysis of Clarinet Sound and Generated Noise . 71 5.1 Ligeti, Etudes,´ Book 1, No. 1 D´esordre, measures 1{2 . 105 5.2 Ligeti, Etudes,´ Book 1, No. 1 D´esordre, measures 14{15 . .108 5.3 Cycles in Ligeti, D´esordre, First 6 Cycles . 109 5.4 Cycles in Ligeti, D´esordre, First 6 Cycles: Proportional . 110 5.5 Cycles in Ligeti, D´esordre, First 6 Cycles: Normalized to Beginning . 111 6.1 Points of Equilibrium on the Logistic Map . 119 6.2 Points of Equilibrium on the Logistic Map: Closeup . .120 6.3 Chaos Drumming . .122 6.4 Chaos Drumming: Sensitive Dependence . .123 6.5 First Five levels of the Cantor set . .131 6.6 The Sierpinski Triangle . ..