BAIN MUSC 525
Post-Tonal Theory

Mathematical Terms & Concepts

Return to: MUSC 525

Post-tonal theory (Straus 2016) invokes a 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. What is more, the emerging field of mathematical music theory is constantly expanding the repertoire of mathematical approaches that are used to understand musical objects, entities, operations, relations, and spaces (e.g., see the Journal of Music and Mathematics). For a historical overview of the development of pitch-class set theory, I recommend Babbitt 2003 and Schuijer 2008. For a historical overview of twelve-tone theory, I recommend Morris 2007.

Below you will find links to online definitions of the 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 perform calculations. Most of the calculations we need to perform (e.g, a mod n, n factorial, n choose k, etc.) may be quickly performed using Google Calculator. For a more powerful online calculator, see Desmos.

Keywords:
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}
   • Integers {MW}
     • Z = {..., –3, –2, –1, 0, 1, 2, 3, …}
     • Z⁺ = {1, 2, 3, ...}
   • Modular arithmetic, or "clock arithmetic" {MW; WP}
   • Clock face {Ex., Theory Shop}
   • mod 12
     
   • a mod n
     Try various values for a and n using Wolfram Alpha
     
   • Modulus {MW}
     
   • Congruence {MW}
   • Residue {MW}
2. Computational Discrete Mathematics
• Computation {MW}
  
• Algorithm {MW | WP}
• Array {MW | WP}
• List {MW | WP}
  
• Matrix {WP}
• String {MW | WP}
• Tuples {MW | WP}
3. Set Theory {MW}
   • Naive set theory {WP} vs. axiomatic set theory {WP}
     
   • Set {MW; WP}
     • Finite sets {MW} vs. infinite sets {MW}
     • Set notation {WP}
       • Property notation {WP}
     • Cardinality {MW}
     • Universal set {MW}
     • Ex. Integers mod 12: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
       
     • Empty set {MW}
     • Singleton set {MW}
   • Set Operations and Relations
     • Union {MW}
     • Intersection {MW}
     • Complement {MW}
     • Venn Diagrams
       • Union {WP} | Intersection {WP} | Subset {WP}
         Create Venn diagrams using Wolfram Alpha
   • Subsets {MW; WP}
     • Inclusion
     • Proper subset {MW}
     • Superset {MW}
     • Power set {MW}
       Ex.: P {1, 2, 3} = {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
       Compute power sets using Wolfram Alpha
     • Hasse Diagram {MW | WDP}
     • Set partition {MW}
   • Other operations
     • Set difference {MW; Venn diagram: Wolfram Alpha}
     • Symmetric difference {MW; Venn diagram: Wolfram Alpha}
     • Cartesian product {MW | WP}
4. Combinatorics {MW | WP}
   • Does order matter? Is repetition allowed?
     1. Set {MW}
     2. Permutation, or arrangement {MW | WP}
     3. Multiset {MW}
     4. Sequence {MW}
   • Counting sets
     • Combination {MW}
       • Binomial coefficient {MW}
         • n choose k {MW}
           Compute n choose k using Google's built in calculator or Wolfram Alpha
     • Pascal's Triangle {MW | WP}
   • Permutations
     Compute permutations using Wolfram Alpha
     • Permutation {MW}
       
     • Factorial {MW}
   • Necklace {MW | WP}
     
   • Combinatorial necklaces and bracelets (Davis 2012)
     Compute necklaces and bracelets with Davis 2012
   • e.g., n=12, k=2
   • Partition {MW}
     Compute integer partitions using Wolfram Alpha
     
     • Integer partitions of 12
       
   • Advanced: (Hook 2007)
     • Enumeration {MW | WP}
     • Pólya enumeration theorem {MW | WP}
5. Graph Theory
   
• Graph {MW}
• Nodes (or vertices) {MW}
• Edges {MW}
• Directed {MW} vs. undirected {MW} graphs
• Arrows (directed graph edge)
• Isographic networks
6. Algebra {MW}
   • Equivalence relation {MW}
     • Reflexive
       
     • Symmetric
     • Transitive
   • Equivalence class {MW}
   • Binary operation {MW}
   • Binary operator {MW}
   • Function {MW}
     
   • Function composition {MW}
   • Involution {WP}
   • Isomorphism {WP}
     
7. Abstract Algebra {MW; Socratica, Abstract Algebra)
   • Group Theory (See also: Lewin 1987)
     • Group {MW; WP}
     • Cyclic group {MW | WP}
     • Ex. Integers mod 12 under addition (Z₁₂)
       
     • Dihedral group {MW | WP} (Crans et al. 2009)
       
     • Symmetric group {MW | WP}
   • Ring {MW}
8. Geometry  {MW} (See also: Tymoczko 2011; Toussaint 2019)
   • Euclidean Plane {MW}
   • Circle {MW}
   • Polygon {MW}
   • Symmetry {WP}
     
9. Topology {MW}
• Space {MW; WP} (See also: Lerdahl 2004, Tymoczko 2011, Hook 2022)
  
• Möbius strip {MW; WP}
  
• Torus {MW; WP}
• Quotient space {WP}
• Orbifold {WP}
  
10. Other
    • Category theory {WP} (Mazzola 2002)
      
    • Fourier transform {WP; 3Blue1Brown: YouTube} (Yust 2016)
    • Knot theory {WP} (Jedrejewski 2006)

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 {MusMat.org}

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)

x

Fiore, Music and Mathematics {Author's website}

Shilito, Introduction to Higher Mathematics (2013-20), video series {YouTube Playlist}

• 5. Set Theory {YouTube}; 8. Relations {YouTube}; 9. Functions {YouTube}
• 10. Number Theory {YouTube}; 11. Modular Arithmetic {YouTube}
• 14. Topology {YouTube}
• 16. Group Theory {YouTube}

Socratica, Abstract Algebra, video series. Available online at: <https://www.socratica.com/subject/abstract-algebra>.

Wikipedia, Available online at: <https://www.wikipedia.org>

Wolfram MathWorld (MW). Available online at: <https://mathworld.wolfram.com>.

Wolfram Research, Wolfram Alpha: A Computational Knowledge Engine. Available online at: <https://www.wolframalpha.com>.

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. {MMA.org}

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}