The geometry of music

The deep connection between music and mathematics was recognized at least as early as the time of Pythagoras. Now, Ian Quinn, an assistant professor in Yale's music department and its cognitive science program, and his colleagues have devised a new mathematical means of understanding music. This "geometrical music theory" can translate the language of music theory into that of contemporary geometry and create visual representations of music's underlying mathematical structure.

In the April 18 issue of Science, they describe five ways ("symmetries") of categorizing groups of notes that are similar but not identical: the same note in different octaves, or the same group of notes in a different order. Then they show how these symmetries can be combined to map musical works in coordinate space where, for instance, two-note chords take the shape of a Mobius strip, three-note chord types take the shape of a three-dimensional cone, and four-note chord types somewhat resemble a pyramid.

"We can put any music into the model," Quinn says, "and visualize the structure behind similarities and differences among musical styles—why Chopin, for instance, sounds different from Mozart." Or Lennon from McCartney.

The translation of music theoretical terms into precise geometrical language provides a framework for investigating contemporary music-theoretical topics, Quinn says. It can also be useful in analysis, composition, pedagogy, and even the design of new kinds of instruments. Adds Quinn, "My students have used the models to write in the styles of various composers. Somewhat to my surprise these complex topics are fairly easily taught."