Research-grade graduate course
Music &
Mathematics
From the Pythagorean comma to the topology of chord spaces β music as a laboratory for mathematical ideas, from eigenvalue theory to orbifolds.
The Harmonic Series: Where Music Meets Mathematics
A vibrating string produces frequencies that are integer multiples of the fundamental
Six Mathematical Domains, One Musical Thread
The Tonnetz: Harmony as Geometry
Horizontal = perfect fifth (+7), diagonal = major third (+4) β triads are triangles
About This Course
Music and mathematics have been intertwined since antiquity. The Pythagoreans discovered that consonant intervals correspond to simple integer ratios; Fourier showed that any sound can be decomposed into sinusoidal components; group theory explains the symmetries of musical transformations; and topology reveals the geometry of chord progressions.
This course explores these connections rigorously, starting from the wave equation and Fourier series, through Diophantine approximation (why 12 tones?), group actions on the Tonnetz, Euclidean rhythms, information-theoretic models of musical expectation, orbifold voice-leading geometry, algorithmic composition, and 12 open research problems at the frontier.
Every module includes interactive components: a Fourier synthesizer, a Tonnetz explorer, a tuning comparator, a Euclidean rhythm generator, a Markov harmony engine, and an orbifold chord-space visualizer. Key equations are rendered with MathJax: from\(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\)(wave equation) to \(\mathcal{O}_n = (\mathbb{R}/12\mathbb{Z})^n / S_n\) (chord orbifold).
Module Outline
Integer ratios, consonance, the harmonic series, the comma
Wave equation, eigenmodes, LΒ² theory, STFT, psychoacoustics
Just intonation, continued fractions, why 12 tones? Xenharmonic theory
Dihedral group Dββ, Tonnetz, neo-Riemannian theory, Lewinβs GIS
Euclidean rhythms, necklaces, Vuza canons, DFT on β€/nβ€
Shannon entropy, Markov harmony, Bayesian expectation, neural coding
Orbifolds (β/β€)βΏ/Sβ, voice-leading geodesics, persistent homology
L-systems, cellular automata, Xenakis stochastic, deep generative models
12 unsolved questions at the musicβmathematics frontier
Interactive Components
Fourier Synthesizer
Build any timbre from harmonics, hear the result in real time
Tonnetz Explorer
Navigate P/L/R neo-Riemannian transformations on the torus
Tuning Comparator
Play the same chord in 6 tuning systems simultaneously
Euclidean Rhythm Generator
Generate (k,n) rhythms, visualise necklace equivalence
Markov Harmony Engine
Probabilistic chord sequences trained on Bach chorales
Orbifold Chord Space
3D visualization of 2- and 3-voice chord orbifolds
Prerequisites
Key References
MIT 21M.383: Computational Music Theory
47 video lectures covering music representation, pitch theory, corpus analysis, and computational approaches to music theory.
0. Overview for OCW Learners
1. How Do Computers Represent Music?
2. Notes, Pitches & Durations
3a. Pitch Representation
3b. Pitch: Pros, Cons & Stakeholders
4. Scores & Music Representation
5. Music Representation (II)
6a. Representation (III): Unlocking Pitch
7. Representation (IV) & Hierarchies
8. Hierarchies (II): Streams & Recursions
9a. Music Information Retrieval
10. Equivalence & Intervals (I)
11. Equivalence & Intervals (II)
12a. Painting Emotions in Music
12b. Chorales as a Corpus
13. Corpus Studies & Statistics
βMusic is the pleasure the human mind experiences from counting without being aware that it is counting.β
β Leibniz
Begin Module 0 β