Post-Tonal Theory

Mathematical Terms & Concepts

Return to: MUSC 525

Post-tonal music theory (Straus 2016) invokes a wide variety of basic mathematical concepts – especially concepts from discrete mathematics (e.g., number theory, set theory, combinatorics, and graph theory). Concepts from algebra, abstract algebra (especially group theory) and geometry have also played a significant role in its development. For a historical overview of the development of pitch-class set theory, see Schuijer 2008. Recently, the emerging field of mathematical music theory (e.g., see the Journal of Music and Mathematics) has invoked an even wider array of mathematical concepts to model musical objects, entities, operations, relations, and spaces.

Below you will find links to online definitions of mathematical concepts we will encounter this semester.
Wolfram's MathWorld {MW} provides authoritative definitions of mathematical terms. Wikipedia {WP} provides highly visual starting points for additional learning and discovery. You will also find links to Wolfram Alpha (and other Web-based apps) that allow you to interactively explore selected calculations. Many of these calculations (e.g, a mod n, n factorial, and n choose k) may be performed using Google Calculator. For a more powerful calculator, see Desmos.

Geometrical music theory (Hall 2008),
Mathematical music theory (
Mazzola et al. 2016), Neo-Riemannian theory (Cohn 2012), Pitch-class set theory (Babbitt 2003, Forte 1973, Rahn 1980 & Morris 1987), Scale theory (Tymoczko 2011), Transformational theory (Lewin 1987), Twelve-tone theory (Morris 2007),

  1. Number Theory {MW} Zahlen
  2. Computational Discrete Mathematics
  3. Venn diagram: intersectionSet Theory {MW}
  4. Pascal's triangleCombinatorics {MW | WP}
  5. Cyclic groupGraph Theory
  6. Algebra {MW}
  7. Abstract Algebra {MW; Socratica, Abstract Algebra)
  8. Mobius stripGeometry  {MW} (See also: Tymoczko 2011; Toussaint 2019)
  9. Topology {MW}
  10. Other

Click on an image to see the image credit.

Journals & Conference Proceedings

Journal of Mathematics and Music {Taylor & Francis; SMCM}

This journal features emerging research in mathematical and computational approaches to music theory, analysis, composition and performance. UofSC maintains an electronic subscription to this journal.

MusMat – Brazilian Journal of Music and Mathematics {}

Society for Mathematics and Computation in Music {SMCM}

Geometrical Music Theory

The following introductory articles appeared in the journal Science between 2006 and 2008:

Rachel Wells Hall, Geometrical Music Theory {JSTOR; Science} (Hall 2008)

Dmitri Tymoczko, The Geometry of Musical Chords {JSTOR; Author's Website} (Tymoczko 2006)

Julian Hook, Exploring Musical Spaces {JSTOR} (Hook 2006; See also: Hook 2022)

Clifton Callender, Ian Quinn, and Dmitri Tymoczko, Generalized Voice-Leading Spaces {JSTOR} (Callender et al. 2008)

Fiore, Music and Mathematics {Author's website}

Shilito, Introduction to Higher Mathematics (2013-20), video series {YouTube Playlist}
Socratica, Abstract Algebra, video series. Available online at: <>.

Wikipedia, Available online at: <>

Wolfram MathWorld (MW). Available online at: <>.

Wolfram Research, Wolfram Alpha: A Computational Knowledge Engine. Available online at: <>.


See also: BAIN MUSC 525 Music and Mathematics Bibliography

Babbitt, Milton. 2003. The Collected Essays of Milton Babbitt, edited by Stephen Peles, with Stephen Dembski, Andrew Mead and Joseph N. Straus. Princeton: Princeton University Press. {Full text: JSTOR}
Callender, Clifton, Ian Quinn, and Dmitri Tymoczko. 2008. "Generalized Voice-Leading Spaces." Science 320/5874 (April 18, 2008): 346–348. {JSTOR}
Cohn, Richard. 2012. Audacious Euphony: Chromatic Harmony and the Triad's Second Nature. New York: Oxford University Press. {GB}

Crans, A.; Fiore, T.; and Satyendra, R. 2009. "Musical Actions of Dihedral Groups." The American Mathematical Monthly, 116/6 (2009): 479–495. {}

Forte, Allen. 1973. The Structure of Atonal Music. New Haven: Yale University Press
. {Full text: JSTOR}

Hall, Rachel Wells. 2008. "Geometrical Music Theory." Science 320/5874 (April 18, 2008): 328–329.
{JSTOR; Science}

Hook, Julian. 2022. Exploring Musical Spaces: A Synthesis of Mathematical Approaches. New York: Oxford. {GB}

__________. 2007. "Why Are There Twenty-Nine Tetrachords? A Tutorial on Combinatorics and Enumeration in Music Theory? Music Theory Online 13/4 (December 2007). {MTO}

Jedrejewski, Frank. 2006. Mathematical Theory of Music. Paris: Ircam-Centre Pompidou. {Delatour}

Johnson, Timothy A. 2008. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. New York: Scarecrow Press. {GB}

Lerdahl, Fred. 2004. Tonal Pitch Space. New York: Oxford University Press. {GB}

Lewin, David. 1987. Generalized Musical Intervals and Transformations (GMIT). New Haven: Yale University Press. {GB}

Mazzola, Guerino. 2002. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance, Volume 1. Basel: Birkhäuser Verlag. {GB}

Mazzola, Guerino, Maria Mannone, and Yan Pang. 2016. Cool Math for Hot Music: A First Introduction to Mathematics for Music Theorists. New York: Spring. {GB}

Morris, Robert. 2007. "Mathematics and the Twelve-Tone System: Past, Present, and Future." Perspectives of New Music 45/2 (Summer, 2007), pp. 76-107. {JSTOR}

____________. 1991a. Class Notes for Atonal Theory. Lebanon, NH: Frog Peak {GBd}

____________. 1991b. Class Notes for Advanced Atonal Theory. Lebanon, NH: Frog Peak. {GBd}

____________. 1987. Composition with Pitch Classes: A Theory of Compositional Design. New Haven: Yale University Press. {GBd; Full text: JSTOR}

Rahn, John. 1980. Basic Atonal Theory. New York: Schirmer. {GBd}

Schuijer, Michiel. 2008. Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. Rochester: University of Rochester Press. {Full text: JSTOR; GB}

Straus, Joseph N. 2016. Introduction to Post-Tonal Theory, 4th ed. New York: Norton. {GB}

Toussaint, Godfried T. 2019.The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good?, 2nd ed. Boca Raton, FL: CRC Press. {GB; Full text: Ebook Central}

Tymoczko, Dmitri. 2011. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford. {GB; Full-text: Ebook Central}

__________________. 2006. "The Geometry of Musical Chords." Science 313 (2006): 72–74.
{JSTOR; Author's Website}

Yust, Jason. "Special Collections: Renewing Set Theory." Journal of Music Theory 60/2 (October 2016): 213–262. {JSTOR}

Updated: July 20, 2023

Reginald Bain | University of SouthCarolina | School of Music