Univesity of South Carolina
School of Music

Music and Mathematics

Spring 2015

Blackboard | Google Books | Grove | JSTOR | MathWorld | Naxos | WorldCat

C O U R S E   M O D U L E S

Introduction, Background &

II. Tuning Theory
III. Musical Spaces

Mathematical Music Theory

Composing with Numbers

Electronic Reserves

Grove Music Online
Wolfram: Alpha | MathWorld



There is geometry in the humming of the strings...there is music in the spacing of the spheres.

-- Pythagoras

Course Documents & Resources

  1. Syllabus (pdf)
  2. Daily Schedule
  3. Homework Assignments
  4. Online Listening
  5. Software
  6. Online Media
  7. Final Presentation
  8. Mathematical Music Theory Resources
  9. Related courses


Fauvel, John, Raymond Flood, and Robin J. Wilson, ed. 2003/2006. Music and Mathematics: From Pythagoras to Fractals.
      New York: Oxford University Press. {GB | OUP}

Online Introductions

  1. Mathematics
  2. Tuning Theory
  3. Acoustics
  4. Music Theory
  5. Music and Mathematics
  6. Combinatorics

Introduction | Tuning Theory | Musical Spaces | Mathematical Music Theory | Composing with Numbers | Bibliography

Daily Schedule

  1. Introduction, Background & Overview (Music and Mathematics Resources)
    1. Mon., Jan. 12: Course Introduction
    2. Wed., Jan. 14: Music & Mathematics: An Overview (Fauvel Preface)
      • Materials:
    3. Fri., Jan. 16: The Music of the Spheres (Fauvel Ch. 2)
  2. Tuning Theory (Tuning Theory Resources)
    1. Wed., Jan. 21: Pythagorean Mathematics
    2. Fri., Jan. 23: The Pythagorean Scale (Fauvel Ch. 1 & 4)
    3. Mon., Jan. 26: Just Intonation & The Harmonic Series
    4. Wed. Jan. 28: cont.
    5. Fri., Jan. 30: Temperaments: 1/4-comma mean-tone & 12tet
    6. Mon., Feb. 2: The Mathematics of Musical Sound (Fauvel Ch. 3 & 5)
    7. Wed., Feb. 4: Consonance and Dissonance
    8. Fri, Feb. 6: cont.
      Exam 1 (2/6-2/13)

  3. Musical Spaces
    1. Mon., Feb. 9: Spaces, Objects/Entities, Relations & Operations
      Definition: Binomial coefficient
      • Materials:
        • Dmitri Tymoczko, A Geometry of Music (GOM), Ch. 2 {GB}
    2. Wed., Feb. 11:
      • Group Presentations:
        • Dmitri Tymoczko, A Geometry of Music (GOM), Ch. 1 {GB}
    3. Fri., Feb. 13: Tn & TnI {pdf}
      Definition: Modular Arithmetic {MW | WP}
    4. Mon., Feb. 16: Musical Set Theory
      Definition: See Set Theory
    5. Wed., Feb. 18: Interval Content
      Definition: Binomial coefficient
      • Materials:
        • Polygon Notation {pdf}
    6. Fri., Feb. 20: Normal Form, Prime Form & Set Class
    7. Mon., Feb. 23: cont.
    8. Wed., Feb. 25: The Geometry of Rhythm
      • Materials:
        • Godfried Toussaint, The Geometry of Musical Rhythm {GB}
        • Rachel W. Hall, The Mathematics of Rhythm {Website}
        • Rachel W. Hall and Paul Klingsberg, Asymmetric Rhythms and Tiling Canons
    9. Fri., Feb. 27 cont.
      • Group Presentations:
        • Godfried T. Toussaint, The Rhythm that Conquered the World: What Makes a 'Good' Rhythm Good {Author's website}
    10. Mon., March 2: cont.
    11. Wed., March 4: Symmetry and Transformations in the Musical Plane (Fauvel Ch. 6; see Ch. 6 Examples)
      • Definitions:
        • Euclidean space
        • Isometry (WP)
          • Translation, Rotation, Reflection and Glide reflection
      • Materials:
        • Vi Hart, Symmetry and Transformations in the Musical Plane {Bridges 2009}
    12. Fri., March 6: One-one-one meetings

  4. Mathematical Music Theory
    1. Mon., March 16: Change Ringing (Fauvel Ch. 7; Ch. 7 Examples)
    2. Wed., March 18: Mathematics and the Twelve-Tone System (Fauvel Ch. 8, pp. 130-139)
      • Materials:
        • Milton Babbitt, Twelve-Tone Invariants as Compositional Determinants {JSTOR}
        • Robert Morris, Mathematics and the Twelve-Tone System: Past, Present, and Future {JSTOR}
        • Harald Fripertinger and Peter Lackner, “Tone Rows and Tropes” {JM&M}
        • Robert Morris, Review of “Tone Rows and Tropes” by Harald Fripertinger and Peter Lackner {JM&M}
    3. Fri., March 20: cont.
      Exam 2 (3/20 - 3/27)
    4. Mon., March 23: cont.
    5. Wed., March 25: Geometrical Music Theory
    6. Fri., March 27: Generalized Voice-Leading Spaces (OPTIC)
      • Definitions: Multiset | R | R^n | Torus {WP}
      • Reading:
        • Clifton Callender, Ian Quinn, and Dmitri Tymoczko, Generalized Voice-Leading Spaces {JSTOR}
      • Materials:
        • Supporting Online Material {Science}
          • See Figures S1, S2 & S3 and Table S1
    7. Mon., March 30: cont.
    8. Wed., April 1: The Geometry of Musical Chords
      • Reading:
        • Dmitri Tymoczko, The Geometry of Musical Chords {JSTOR}
      • Materials:
        • Supporting Online Material {Science}
          • See Fig. 1, Fig. 2, Movies S1-S4, and Table 1
        • See also: Dmitri Tymoczko, GOM | Science Articles
    9. Fri., April 3: cont.
    10. Mon., April 6: Rings, Intervals, Transformations, and Tonal Analysis
      • Group Presentations:
        • Steven Rings, Tonality and Transformation: Ch. 1 Intervals, Transformations, and Tonal Analysis {GB}
      • Materials:
        • David Lewin, Generalized Musical Intervals and Transformations. New York: Oxford University Press. {GB}
        • See also: Rings 2006 {JSTOR}; Hall 2009 {JSTOR}; Tymoczko 2009 {JSTOR}
    11. Wed., April 8: cont. - Group Presentations (cont.)
    12. Fri., April 10: Guest speaker: Dmitri Tymoczko, Princeton University {4/10 Talk Schedule}
    13. Mon., April 13: Audacious Euphony
      • Reading:
        • Julian Hook, Exploring Musical Space {JSTOR}
        • Guy Capuzzo, Neo-Riemannian Theory and the Analysis of Pop-Rock Music (JSTOR}
      • Materials:
        • Richard Cohn
          • An Introduction to Neo-Riemannian Theory: A Survey and Historical Perspective {JSTOR}
          • Neo-Riemannian Operations, Parsimonious Trichords, and Their 'Tonnetz' Representations {JSTOR}
    14. Wed., April 15: The PLR Group
      • Materials:
        • Crans, Fiore & Satyendra, Musical Actions on Dihedral Groups {CiteSeer]
        • Robert Morris, Voice-Leading Spaces {JSTOR}
        • Dmitri Tymoczko, The Generalized Tonnetz {Author's website}
    15. Fri., April 17: Final Presentation Day 1
    16. Mon., April 20: Final Presentation Day 2
    17. Wed., April 22: Final Presentation Day 3
    18. Fri., April 24: Final Presentation Day 4
      Exam 3 (4/24 - 5/1)
    19. Mon., April 27: Final Presentation Day 5
  5. Composing with Numbers

Selected Bibliography (See also: Math Resources; BAIN MUSC 525)

Aceff-Sánchez, et al. An Introduction to Group Theory: Applications to Mathematical Music Theory. Available online at: <https://bookboon.com/en/an-introduction-to-group-theory-ebook>.

Agmon, Eyton 2013. The Languages of Western Tonality. New York: Springer. {GB}

Arom. S. 2004. African Polyphony and Polyrhythm. Cambridge: Cambridge University Press.

Babbitt, Milton. The Collected Essays of Milton Babbitt, ed. Stephen Peles, Stephen Dembski, Andrew Mead, and Joseph N. Straus. Princeton, NJ: Princeton University Press: 2003. {GB}

Bamberger, Jeanne. 2000. Developing Musical Intuitions: A Project-based Introduction to Making and Understanding Music. New York: Oxford University Press. {GBd}

Barbour, J. Murray. 1951/2004. Tuning and Temperament: A Historical Survey. Mineola, NH: Dover. {GB}

Benson, David. 2007. Music: A Mathematical Offering. Cambridge: Cambridge University Press. {GB | Website}

Bregman, Albert. 1990. Auditory Scene Analysis: The Perceptual Organization of Sound. Cambridge, MA: MIT Press. {GB}

Brindle, Reginald Smith. 1987. The New Music: The Avant-Garde Since 1945, 2nd ed. New York: Oxford University Press. {GBd}

Calter, Paul A. 2008. Squaring the Circle: Geometry in Art and Architecture. New York: John Wiley & Sons Inc. {GBd | Website}

Campbell, Murray and Clive Greated. 1987/2001. The Musician's Guide to Acoustics. New York: Oxford University Press. {GB}

Castine, Peter. 1991. Set Theory Objects: Abstractions for Computer-Aided Analysis and Composition of Serial and Atonal Music. Berlin: Peter Lang. {GBd}

Chew, Elaine. 2013. Mathematical and Computational Modeling of Tonality: Theory and Applications. New York: Springer. {GB}

Christensen, Thomas. 2002. The Cambridge History of Western Music Theory. Cambridge: Cambridge University Press. {GB}

Cohn, Richard. 2012. Audacious Euphony. New York: Oxford. {GB}

Cowell, Henry. 1930/1996. New Musical Resources. Cambridge, MA: Havard University Press. {GB}

Deutsch, Diana. ed. 1982/2012. The Psychology of Music, 3rd ed. New York: Academic Press {GB}

Devlin, Keith. 1994. Mathematics: The Science of Patterns. New York: Scientific American Library. {GBd}

Doty, David B. 1994/2002. The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 3rd ed.
      San Francisco, CA: Just Intonation Network. {Website}

Douthett, Jack M. Martha M. Hyde, Charles J. Smith, and John Clough, 2008. Music Theory and Mathematics:
     Chords, Collections, and Transformations. Rochester, NY: University of Rochester Press. {GB}

Dummit, Steven and Richard Foote. 2004. Abstract Algebra, 3rd ed. New York: Wiley. {GB}

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

Gann, Kyle. 2019. The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician. Champaign, IL: University of Illinois Press {U. of IL Press; Companion Website}

Gollin, Michael and Alexander Rehding, eds. 2011. The Oxford Handbook of Neo-Riemannian Music Theories. New York: Cambridge University Press. {GB}

Hardy, G.H. and E.M. Wright. 1938/2008. An Introduction to the Theory of Numbers. New York: Oxford University Press. {GBd}

Hofstadter, Douglas. 1979. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books. {WP}

Jedrzejewski, F. 2006. Mathematical Theory of Music. Paris: Ircam-Centre Pompidou.

Johnson, Tim. 2003/2008. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals.
      Lanham, MD: Scarecrow Press. {GB}

Keith, Michael. 1991. From Polychords to Polya: Adventures in Musical Combinatorics. Princeton: Vinculum Press.

Kung, David. 2013. How Music and Mathematics Relate. DVD. Chantilly, VA: The Great Courses. {Website}

Lerdahl, Fred. 2001. Tonal Pitch Space. New York: Oxford University Press. {GB}

Lerdahl, Fred and Ray Jackendoff. 1983/2010. A Generative Theory of Tonal Music. Cambridge: MIT Press. {GB}

Lewin, David. 1987/2007. Generalized Musical Intervals and Transformations. New York: Oxford University Press. {GB}

Loy, D. G. 2006. Musimathics: The Mathematical Foundations of Music, Vol. 1-2. Cambridge, Mass: MIT Press. {Website | Vol. 2: GB}

Mazzola, Guerino. 2002. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. New York: Springer. {GB | Springer}

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

Morris, Robert. 1987. Composition with Pitch Classes. New Haven: Yale University Press. {GBd}

___________. 2001. Class Notes for Atonal Theory. Lebanon, NH: Frog Peak. {GBd}

___________. 2001. Class Notes for Advanced Atonal Theory. Lebanon, NH: Frog Peak. {GBd}

Pierce, John. 1992. The Science of Musical Sound. W H Freeman. {GBd}

Pinter, Charles. 1982/1990. A Book of Abstract Algebra: Second Edition. Mineola, NY: Dover {GB}

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

Rings, Steven. 2011. Tonality and Transformation. New York: Oxford University Press. {GB}

Roberts, Gareth E. 2016. From Music to Mathematics: Exploring the Connections. Baltimore, MD: John Hopkins University Press. {GB}

Roederer, Juan. 1973/2008. The Physics and Psychophysics of Music: An Introduction, 4th ed. New York: Springer. {GB}

Rothstein, Edward. 1995/2006. Emblems of Mind: The Inner Life of Music and Mathematics. Chicago: University of Chicago Press. {GBd}

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

Schillinger, Joseph. The Mathematical Basis of the Arts. 1943. New York: Philophical Library. {Archive.org}

_______________. The Schillinger System of Musical Composition. 1946. New York: Carl Fischer. {Archive.org}

Sethares, William A. 2005. Tuning, Timbre, Spectrum, Scale, 2nd ed. New York: Springer. {GB | Website}

Straus, Joseph N. 2005. Introduction to Post-Tonal Theory, 3rd ed. Englewood Cliffs, NJ: Prentice Hall. {GBd}

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

Tatlow, Ruth. 2015. Bach's Numbers: Compositional Proportion and Significance. Cambridge: Cambridge University Press. {GB}

Temperley, David. 2010. Music and Probability. Cambridge: MIT Press. {GBd}

Toussaint, Godfried T. 2013. The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? Boca Raton, FL: CRC Press. {GB}

Tymoczko, Dmitri. 2011. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice.
      New York: Oxford University Press. {GB | OUP | Companion Website | Author's Website}

Walker, James S. Mathematics and Music. CRC Press. Boca Raton, FL: CRC Press. {GB}

Xenakis, Iannis. 1992. Formalized Music: Thought and Mathematics in Composition. Pendragon Revised Edition. Hillsdale, NY: Pendragon. {GB}


Reginald Bain, MUSC 726 Music and Mathematics {WorldCat} *

Computer Music Journal, Computer Music {MIT Press}

David Huron, Music 829B: Bibliography of Consonance and Dissonance {OSU}

Huygens-Fokker Foundation, Tuning and Temperament {HFF}

IRCAM, Pitch Class-Set Theory, Diatonic Theory and Neo-Riemannian Theory {IRCAM}

- Book citations refer to this bibliography

GB - Google Books; GBd - Google Books description; MTO - Music Theory Online; JSTOR - USC JSTOR subscription

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Updated: April 20, 2019

Reginald Bain | University of South Carolina | School of Music