Chapter 20 Further Reading: Mathematical Patterns in Composition — From Bach to Messiaen

Historical and Philosophical Foundations

Godwin, Joscelyn. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Inner Traditions, 1993. A magnificent anthology of primary sources from Pythagoras through the twentieth century, tracing the mathematical-musical tradition in Western thought. Includes Boethius, Kepler, Descartes, Rameau, and Hindemith. Essential historical context for Chapter 20.

Fauvel, John, Raymond Flood, and Robin Wilson, eds. Music and Mathematics: From Pythagoras to Fractals. Oxford University Press, 2003. An accessible scholarly collection covering the mathematics-music relationship across history. Chapters on Pythagorean theory, counterpoint as mathematics, tuning systems, Fourier analysis, and chaos. An excellent overview of the field.

Lippman, Edward. A History of Western Musical Aesthetics. University of Nebraska Press, 1992. Comprehensive history of the philosophical discourse about music, including the mathematical tradition from antiquity through the twentieth century. Essential for understanding the intellectual context of the composers and theorists in this chapter.

Bach and Counterpoint

Mann, Alfred. The Study of Fugue. Dover, 1987 (reprint). The standard introduction to fugue technique, combining historical writing about fugue (Marpurg, Kirnberger, Albrechtsberger) with analytical examples. Explains the contrapuntal operations and their musical applications with clarity.

Williams, Peter. J.S. Bach: A Life in Music. Cambridge University Press, 2007. The most comprehensive scholarly biography of Bach currently available. Excellent on the mathematical aspects of Bach's practice, including his use of number symbolism and proportional design.

Tatlow, Ruth. Bach's Numbers: Compositional Proportion and Significance. Cambridge University Press, 2015. A revisionist scholarly study that challenges many claims about numerical/proportional design in Bach while establishing on rigorous grounds what can actually be demonstrated. Essential reading for anyone interested in separating fact from mythology in Bach number analysis.

Fibonacci and Golden Ratio

Lendvaï, Ernő. Béla Bartók: An Analysis of His Music. Kahn & Averill, 1971. The foundational study of Fibonacci and golden-ratio proportions in Bartók, by the analyst most responsible for establishing this line of inquiry. Read alongside subsequent skeptical scholarship.

Howat, Roy. Debussy in Proportion: A Musical Analysis. Cambridge University Press, 1983. The most careful and statistically rigorous study of golden-ratio proportions in any composer's works. Howat applies genuine statistical methodology and distinguishes between coincidental and purposive proportioning. A model of how this kind of analysis should be done.

Livio, Mario. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. Broadway Books, 2002. A popular but rigorous debunking of many Fibonacci/golden-ratio myths in art, architecture, and music, alongside a genuine account of where the golden ratio does and does not appear. Excellent corrective to popular mythology.

Messiaen

Messiaen, Olivier. Technique of My Musical Language. Translated by John Satterfield. Alphonse Leduc, 1956. Messiaen's own theoretical treatise, describing his modes of limited transposition, non-retrogradable rhythms, and other compositional innovations. Essential primary source. Dense but remarkably clear.

Hill, Peter, and Nigel Simeone. Messiaen. Yale University Press, 2005. The definitive English-language biography, with detailed coverage of the Quartet's composition and premiere at Stalag VIII-A, based on extensive archival research and interviews.

Shenton, Andrew, ed. Messiaen the Theologian. Ashgate, 2010. A collection of essays examining the relationship between Messiaen's Catholic theology and his musical practice, including analyses of how theological concepts are embodied in specific musical structures. Directly relevant to Case Study 20-2.

Schoenberg and Serialism

Schoenberg, Arnold. Style and Idea: Selected Writings of Arnold Schoenberg. University of California Press, 1984. Schoenberg's own essays on his compositional methods, including his account of why he developed twelve-tone technique. Essential primary source.

Perle, George. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern. University of California Press, 1991 (6th ed.). The standard analytical introduction to twelve-tone music and its compositional systems. Rigorous and thorough.

Shreffler, Anne C. "Mein Weg geht jetzt vorüber: The Vocal Origins of Webern's Twelve-Tone Composition." Journal of the American Musicological Society 47, no. 2 (1994): 275–339. Detailed study of how Webern applied twelve-tone technique differently from Schoenberg, with more extreme symmetry and a different relationship between the row and musical surface. Illuminates the range of possibility within a shared compositional system.

Xenakis

Xenakis, Iannis. Formalized Music: Thought and Mathematics in Composition. Pendragon Press, 1992 (revised ed.). Xenakis's theoretical magnum opus — massively demanding mathematically, but the essential primary source for his stochastic methods. Chapters on Metastasis, Pithoprakta, stochastic music, and later symbolic and set-theoretical approaches. Not for the faint of heart, but indispensable.

Varga, Bálint András. Conversations with Iannis Xenakis. Faber, 1996. Interview-based introduction to Xenakis's life and work. Provides the biographical and intellectual context for his compositions in accessible form.

Solomos, Makis. From Music to Sound: The Emergence of Sound in 20th- and 21st-Century Music. Routledge, 2020. Scholarly study of the transformation of compositional thinking from "notes" to "sounds" in 20th-century music, with extensive coverage of Xenakis and spectral composition.

Spectral Music

Murail, Tristan. "The Revolution of Complex Sounds." Contemporary Music Review 24, nos. 2–3 (2005): 121–135. Murail's own account of spectral composition's origins, methods, and aesthetic aims. Clear and authoritative.

Fineberg, Joshua, ed. "Spectral Music." Special issue, Contemporary Music Review 19, nos. 2–3 (2000). A comprehensive scholarly survey of spectral composition, including articles by Grisey, Murail, and major analysts. The best single source for understanding the spectral school.

Group Theory and Set Theory

Forte, Allen. The Structure of Atonal Music. Yale University Press, 1973. The founding text of pitch-class set theory. Dense but remarkably clear. The appendix of set-class tables is an invaluable reference.

Straus, Joseph N. Introduction to Post-Tonal Theory. 4th ed. Norton, 2016. The standard undergraduate textbook on post-tonal music theory, including pitch-class set theory, twelve-tone theory, and their applications. Clear and well-organized.

Mazzola, Guerino. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, 2002. An extraordinary and extraordinarily demanding application of advanced mathematics (category theory, algebraic geometry) to music theory. Not for beginners, but for those wanting to see how far the mathematical analysis of music can go.

Algorithmic Composition

Cope, David. Virtual Music: Computer Synthesis of Musical Style. MIT Press, 2001. Cope's account of his Experiments in Musical Intelligence (EMI) system, which generated new "Bach chorales," "Mozart sonatas," and "Beethoven symphonies" using statistical analysis of existing works. Includes the controversial blind-test experiments in which experts were fooled.

Nierhaus, Gerhard. Algorithmic Composition: Paradigms of Automated Music Generation. Springer, 2009. A comprehensive technical survey of algorithmic composition methods from Baroque dice games through modern machine learning approaches. Balanced between technical detail and musical context.

Briot, Jean-Pierre, Gaëtan Hadjeres, and François-David Pachet. Deep Learning Techniques for Music Generation. Springer, 2019. A current survey of neural network-based music generation, including historical context and evaluation methodologies. Appropriate for those interested in where algorithmic composition has arrived with modern AI tools.