Chapter 17 Further Reading: Fractals, Self-Similarity & Musical Patterns

Primary Sources

Voss, Richard F., and John Clarke. "1/f noise in music and speech." Nature 258 (1975): 317–318. The original paper. Short (two pages) and accessible without advanced mathematics. Freely available through many library databases. Reading the original is worthwhile for understanding exactly what was claimed and what was not.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. W.H. Freeman, 1982. The book that brought fractals to a wide audience. Mandelbrot's prose is distinctive and his visual examples are stunning. Chapters 1-5 provide the foundational concepts. The book is physically large (coffee-table format) with many illustrations that make the concepts concrete.

Mandelbrot, Benoit B. "How Long Is the Coast of Britain?" Science 156, no. 3775 (1967): 636–638. The paper that introduced the coastline paradox and the concept of fractal dimension to a scientific audience. Short and accessible. A landmark in the history of science.

Fractals and Music

Hsü, Kenneth J., and Andreas Hsü. "Fractal Geometry of Music." Proceedings of the National Academy of Sciences 87, no. 3 (1990): 938–941. An important early paper extending Voss's analysis to melodic fractal dimension. Analyzes Bach, Mozart, and other composers. Available through PNAS online archive.

Boon, Jean-Pierre, and Olivier Decroly. "Dynamical Systems Theory for Music Dynamics." Chaos 5, no. 3 (1995): 501–508. A more technical treatment applying dynamical systems concepts (including strange attractors) to music. Requires some background in dynamical systems theory but the introduction is accessible.

Pressing, Jeff. "Nonlinear Maps as Generators of Musical Design." Computer Music Journal 12, no. 2 (1988): 35–46. A practical treatment of how chaotic maps and fractal generators can be used for algorithmic composition. Many concrete examples.

Leach, Jeremy, and John Fitch. "Nature, Music, and Algorithmic Composition." Computer Music Journal 19, no. 2 (1995): 23–33. A clear and accessible treatment of L-systems and other algorithmic methods applied to music composition, with musical examples.

Nancarrow and Extreme Rhythm

Gann, Kyle. The Music of Conlon Nancarrow. Cambridge University Press, 1995. The definitive study of Nancarrow's player piano studies. Comprehensive, detailed, and analytically sophisticated. Gann provides mathematical analysis of each study's tempo ratios and discusses their perceptual and structural implications.

Tenney, James. "Conlon Nancarrow's Studies for Player Piano." In Conlon Nancarrow: Selected Studies for Player Piano, Soundings, 1977. An earlier analytical essay by composer-theorist James Tenney, one of Nancarrow's most perceptive analysts.

Ligeti, György. "On My Études for Piano." Sonus 9, no. 1 (1988): 3–7. Ligeti discusses Nancarrow's influence on his own work. Provides insight into how a major composer absorbed and transformed Nancarrow's rhythmic ideas.

Recordings: Nancarrow's studies are recorded on the Wergo label (Conlon Nancarrow: Studies for Player Piano, Wergo 6907-2). Listening while reading Gann's analytical descriptions is strongly recommended.

Self-Organized Criticality

Bak, Per. How Nature Works: The Science of Self-Organized Criticality. Copernicus, 1996. Bak's popular account of self-organized criticality and the 1/f phenomenon across physical, biological, and social systems. Accessible to non-scientists and directly relevant to the chapter's discussion of why 1/f statistics are universal.

Jensen, Henrik Jeldtoft. Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems. Cambridge University Press, 1998. More technical treatment of SOC theory. Chapter 1 and the introduction give an accessible overview; later chapters require physics and mathematics background.

Music Cognition and Fractal Perception

Large, Edward W., and Mari Riess Jones. "The Dynamics of Attending: How People Track Time-Varying Events." Psychological Review 106, no. 1 (1999): 119–159. A foundational paper in music cognition showing that listeners track music using an attentional system that naturally produces multi-scale correlations — directly relevant to why 1/f music is cognitively engaging.

Todd, Neil P. McAngus. "The Kinematics of Musical Expression." Journal of the Acoustical Society of America 97, no. 3 (1995): 1940–1949. Analysis of expressive timing and dynamics in music performance showing that these vary in ways that have fractal character — a study of the "human" contribution to 1/f statistics in performance.

Algorithmic Composition

Prusinkiewicz, Przemyslaw, and Aristid Lindenmayer. The Algorithmic Beauty of Plants. Springer, 1990. The definitive reference on L-systems, with beautiful graphics of plant structures generated by simple rules. Chapter 1 is accessible to non-mathematicians; the book as a whole gives the mathematical foundation needed for musical L-systems.

Roads, Curtis. The Computer Music Tutorial. MIT Press, 1996. Comprehensive reference covering algorithmic composition, fractal music generation, stochastic processes, and many other technical topics in computer music. Relevant chapters: 9 (Stochastic methods) and 10 (Algorithmic composition).

Miranda, Eduardo Reck. Composing Music with Computers. Focal Press, 2001. Accessible introduction to algorithmic composition including fractal methods, L-systems, and cellular automata. Many worked examples.

Online Resources

The Fractal Foundation (fractalfoundation.org): Interactive fractal generators and educational materials, including the ability to generate and listen to fractal music.

Nancarrow Audio Archive: Recordings of Nancarrow's player piano studies are available on streaming platforms (Spotify, Apple Music) under the Wergo label.

Voss's 1/f Noise Demos: Several academic websites host demonstrations of white, pink, and brown noise for comparison, as well as tools for computing the power spectrum of musical recordings.