The application of mathematics in the theory of music in the West is generally thought to start with Pythagoras and progress though the works of Euler, Mersenne, Helmholtz, and others. Recently, mathematical music theory has gained impetus through, among others, the Neo-Riemannian theory of harmony, David Lewin’s research on transformational theory, and Guerino Mazzola’s application of topos theory.
As well, the application of science and math to the theory of human perception of sound? is known as psychoacoustics and has had a major impact in today’s musically digital world. All of these viewpoints have greatly enhanced the variety of mathematical concepts employed in the study of music that today include tools from set, group, and category theory.
John Baez: This Week’s Finds (Week 234), June 12, 2006. (link)
n-Café: The Mathematics of Music at Chicago, May 31, 2009. (link)
For applications of mathematics to music, see Thomas M. Fiore's page.
For discussions on the interaction of category theory and music, see the MaMux seminar at IRCAM, Paris.
A comprehensive anthology of Riemannian and Neo-Riemannian music theory is:
An excellent textbook on the mathematics of music that ranges from Fourier analysis over digital music to compositional symmetries is
Comparable in range though presumably more digestible for the non mathematical reader is
Some canonical music theory texts that rely on mathematics:
“The Structure of Atonal Music ranks as one of the most important contributions to music theory in the twentieth century. His use of the computer as well as traditional means of analysis has led to fuller knowledge of musical structure and thereby enhanced both the understanding and the enjoyment of music.”
“David Lewin’s Generalized Musical Intervals and Transformations is recognized as the seminal work paving the way for current studies in mathematical and systematic approaches to music analysis.”
“The third edition stays abreast of recent theoretical developments by including discussions of transformational networks and graphs, contour theory, atonal voice leading, triadic post-tonality (including neotonality), inversional symmetry, and interval cycles.”
A nice first introduction to music theory and what groups can do for it is
More mathematically inclined are the following
Emilio Lluis-Puebla, Ocatavio A. Augustín-Aquino (eds.), Memoirs of the Fourth International Seminar on Mathematical Music Theory, Sociedad Matemática Mexicana 2010. (pdf)
Guerino Mazzola, Geometrie der Töne, Birkhäuser Basel 1990.
Guerino Mazzola, The Topos of Music, Birkhäuser Basel 2002.
Guerino Mazzola, Mathematical Music Theory: Status Quo 2010, pp.11-42 in Lluis-Puebla, Augustín-Aquino (2010).
Dimitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, Oxford University Press 2011.
Dmitri Tymoczko, Generalizing Musical Intervals, Journal of Music Theory 53 no.2 (2009) pp.227-254. (draft)
An introduction to group theory with applications to neo-Riemannian theory is
Besides the monumental Mazzola (2002), the following employ category theory:
Thomas Noll, The topos of triads, pp.1-26 in Fripertinger, Reich (eds.), Colloquium on Mathematical Music Theory, Grazer Math. Ber. 347 (2005). (citeseer)
Alexandre Popoff, Moreno Andreatta, Andree Ehresmann, Groupoids and wreath products of musical transformations: A categorical approach from poly-Klumpenhouwer networks, pp. 33-45 in Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science, vol 11502. Springer. (pdf)
Alexandre Popoff, Jason Yust, Meter networks: a categorical framework for metrical analysis, Journal of Mathematic and Music. (pdf)
Methods of physics are employed to study tonality in
Hegel’s philosophy of music can be found in his lectures on aesthetics though these reflect to a large extent the view of his pupil Heinrich Gustav Hotho who compiled the posthumeous publication:
A development of the basic concepts of music theory along his ideas on dialectics can be found in
The following introductory texts on psychoacoustics help provide a foundation to better understand how sound becomes emotion becomes music:
Arthur H. Benade, Fundamentals of Musical Acoustics Dover New York 1976.
Hugo Fastl, Eberhard Zwicker, Psychoaccoustics: facts and models, Springer, Heidelberg 1990.
Juan G. Roederer, The Physics and Psychophysics of Music, Springer Heidelberg 2008$^4$.
A theory of emotions which applies beyond music in narrative arts is in
An introduction to Haskell programming systematically employing examples from music is
Other “mathemusical” topics include:
Milton Babbit, The Function of Set Structure in the Twelve-Tone System, Ph.D. dissertation, Princeton University 1946.
E. Chew, Towards a Mathematical Model of Tonality, Ph.D. dissertation, MIT Cambridge 2000.
David Clampitt, Thomas Noll, Modes, the Height-Width Duality, and Handschin’s Tone Character, Music Theory Online 17 no.1 (2011).
Thomas M. Fiore, Ramon Satyendra, Generalized Contextual Groups, Music Theory Online 11 no.3 (2005).
J. Hook, Uniform triadic transformations, Journal of Music Theory 46 no.1-2 (2002) pp.57–126.
T. A. Johnson, Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals, Scarecrow Press 2008.
Fred Lerdahl, Tonal Pitch Space, Oxford University Press 2004.
Miguel F. M. Lima, J. A. Tenreiro Machado, António C. Costa, A Multidimensional Scaling Analysis of Musical Sounds Based on Pseudo Phase Plane, Abstract and Applied Analysis Vol. 2012. (Article ID 436108)
Thomas Noll, Sturmian sequences and morphisms: a music theoretical application, Journée annuelle (2008) pp.79–102.
R. N. Shepard, Geometrical approximations to the structure of musical pitch, Psychological Review, 89 no.4 (1982) 305.
J. Tennenbaum, The Foundations of Scientific Musical Tuning, in Staff, Sigerson (eds.), A Manual On the rudiments of Tuning and Registration, Schiller Institute Washington 1992.
J. S. Walker, D. W. Gary, Mathematics and Music: Composition, Perception, and Performance, Taylor&Francis Boca Raton 2013.
Last revised on November 10, 2022 at 17:09:35. See the history of this page for a list of all contributions to it.