The Geometry of Music

Three note chords

The Geometry of Musical Chords

Dmitri Tymoczko, Science 313: 72-74 (2006).

Provides a geometrical model of musical structure, and uses this model to explain how harmony and counterpoint can be combined.

The paper comes in two parts:

  1. The three-page summary, as it actually appeared in the magazine. (A link to a PDF is in the upper left.)
  2. The supplementary materials. These include a 28-page paper containing the substance of the argument. (Click here for the supplement.) There are also four movies and a soundfile here illustrating points made in the article.

Four note chords

Generalized Voice-Leading Spaces

Clifton Callender, Ian Quinn, and Dmitri Tymoczko, Science 320: 346-348 (2008).

Generalizes and extends the ideas in the previous paper to provide a comprehensive translation manual between music theory and geometry.

Also in two parts:

  1. The three-page summary.
  2. The supplementary materials.


You may also find it useful to download ChordGeometries, a computer program that allows readers to interact with the geometrical spaces described in these papers.