Chapter 16 Further Reading: Symmetry in Music and Physics

Primary Texts and Foundational Works

Weyl, Hermann. Symmetry. Princeton University Press, 1952. The foundational humanistic treatment of symmetry by one of the twentieth century's greatest mathematicians. Weyl covers visual, physical, and aesthetic symmetry in prose accessible to non-mathematicians. Still the best single introduction to the concept of symmetry across domains.

Noether, Emmy. "Invariante Variationsprobleme." Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1918. The original paper presenting Noether's theorem. In German and technically demanding, but important to know exists. An accessible English discussion can be found in many physics history sources.

Bach, Johann Sebastian. Musikalisches Opfer (Musical Offering), BWV 1079. 1747. The score is freely available through the IMSLP (International Music Score Library Project). Recordings by Musica Antiqua Köln (Reinhard Goebel) and the Academy of Ancient Music are recommended. Listen with the score if possible.

Music Theory and Group Theory

Mazzola, Guerino. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, 2002. The comprehensive mathematical treatment of music theory using category theory and algebraic geometry. Technically demanding but philosophically profound. Chapter 1 provides a useful overview accessible to determined non-mathematicians.

Babbitt, Milton. "Set Structure as a Compositional Determinant." Journal of Music Theory 5, no. 1 (1961). The foundational paper applying group theory explicitly to twelve-tone composition. Babbitt introduces the concepts of combinatoriality and set-theoretical analysis of tone rows.

Forte, Allen. The Structure of Atonal Music. Yale University Press, 1973. The standard reference for pitch-class set theory — the application of combinatorics and group theory to post-tonal music analysis. Technical but accessible to music students.

Morris, Robert. Composition with Pitch-Classes. Yale University Press, 1987. A more compositionally oriented treatment of pitch-class set theory and symmetry operations, with many analytical examples.

Physics and Symmetry

Lederman, Leon, and Christopher Hill. Symmetry and the Beautiful Universe. Prometheus Books, 2004. An accessible treatment of Noether's theorem and its implications for physics. Written for general readers with no mathematics background beyond arithmetic. Highly recommended for students who want to understand the physics side of this chapter without advanced mathematics.

Greene, Brian. The Elegant Universe. W. W. Norton, 1999. A popular treatment of string theory and symmetry, including accessible discussions of the Standard Model's symmetry groups and spontaneous symmetry breaking. Chapters 1-4 are directly relevant to this chapter.

Feynman, Richard P. The Character of Physical Law. MIT Press, 1965. Based on Feynman's Messenger Lectures at Cornell, this slim book contains the clearest accessible discussion of conservation laws, symmetry, and the nature of physical law ever written. Chapter 4 on symmetry is essential reading for anyone interested in the physics side of this chapter.

Kane, Gordon. Modern Elementary Particle Physics. Perseus Books, 1993. More technical treatment of the Standard Model and its symmetry groups. Chapters 1-3 give a good introduction to what SU(3) × SU(2) × U(1) means in practice.

History and Biography

Osen, Lynn M. Women in Mathematics. MIT Press, 1974. Contains a chapter on Emmy Noether's life and work. A useful supplement to the technical material.

Srinivasan, Bhama, and Judith Sally, eds. Emmy Noether in Bryn Mawr. Springer, 1983. A collection of essays on Noether's life and mathematics, including her time in the United States after fleeing Nazi Germany.

Gould, James. "The Discovery of the 230 Space Groups." Historical Studies in the Physical Sciences 19 (1989). An accessible history of the crystallographic classification problem and its solution by Fedorov and Schoenflies.

Cultural and Cross-Cultural Studies

Lerdahl, Fred, and Ray Jackendoff. A Generative Theory of Tonal Music. MIT Press, 1983. The standard reference for a Chomsky-inspired transformational grammar of Western tonal music. The symmetry operations of the grammar are related to the musical symmetries discussed in this chapter.

Toussaint, Godfried. The Geometry of Musical Rhythm. CRC Press, 2013. A mathematical treatment of rhythm using geometry and combinatorics, with extensive coverage of African and other non-Western rhythmic patterns. Chapters on clave rhythms and their rotational equivalences are directly relevant.

Tenzer, Michael, ed. Analytical Studies in World Music. Oxford University Press, 2006. Contains detailed analyses of gamelan music, Indian classical music, and other non-Western traditions using modern music-theoretical tools. Useful for understanding how symmetry concepts translate across cultural contexts.

Online Resources

IMSLP (International Music Score Library Project): imslp.org Free access to scores of all works discussed in this chapter, including the Musical Offering and Schoenberg's twelve-tone works.

The MacTutor History of Mathematics Archive (St Andrews University): mathshistory.st-andrews.ac.uk Accessible biographies of Noether, Galois, Abel, and other mathematicians mentioned in this chapter, with discussion of their mathematical contributions.

Guerino Mazzola's website: includes downloadable papers and a summary of his Topos of Music project accessible to non-specialists.