This article relates two categories of music-theoretical graphs, in which points represent notes and chords, respectively. It unifies previous work by Brower, Callender, Cohn, Douthett, Gollin, O’Connell, Quinn, Steinbach, and myself, while also introducing new models of voice-leading structure—including a three-note octahedral Tonnetz and tetrahedral models of four-note diatonic and chromatic chords.

Thanks to Richard Cohn and Gilles Baroin for helpful comments.

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Author notes

Dmitri Tymoczko is a composer and music theorist who teaches at Princeton University. His book A Geometry of Music was published in 2011, and his album Beat Therapy has recently been released on the Bridge label. A second CD, featuring chamber music for string quartet and other instruments, will be released by Bridge in 2012.