The theoretically minded of you out there should be aware of the work of Dmitri Tymoczko. Tymoczko is a composer and teaches at Princeton. An active music theorist, his recent work develops geometric models for the mapping of musical space. His paper “The Geometry of Musical Chords” was published last fall in Science magazine; it was the first music theory paper the publication has accepted in its over one hundred years of existence.
In collaboration with colleagues in math and science, Tymoczko demonstrates in the paper the efficacy of orbifolds for mapping musical space. Orbifolds are multi-dimensional non-Euclidean shapes whose properties more or less resemble a metastasizing mobius strip. As a very simple example of geometric musical modeling – and the only one I’m really comfortable imparting – take an octave: the endpoints of an octave are both the same pitch class, but they really aren’t the same pitch; one “C,” say, is twice (or one-half) the frequency of the other. Talking a walk up or down an octave, you end up in the same – yet different – place. Geometrically, this is just like tracing a line around/inside/outside a mobius strip: you end up in the same place, but on the other side.
The headline of Tymoczko’s orbifold work is this: consonant sonorities tend to cluster around the center of an orbifold, whereas dissonant sonorities tend to occupy disparate points around the periphery. Such being this case, Tymoczko’s mappings offer a precise way of articulating musical impressions that often only find realization in nebulous emoting. His mappings also give renewed interest to the centuries-old discussion about music’s relationship to mathematics, and refresh conceptualization of the interplay between harmony and counterpoint.
But here’s the really fun part: Tymoczko has created a free computer program called ChordGeometries 1.1 that lets you futz around with different modes of geometrical modeling – including orbifolds. You can enter chords via a MIDI keyboard, or simply poke them out on the keyboard in the program!
Could this be the next internet craze? It’s certainly more interesting than “fling the cow.”