Toroidal models in tonal theory and pitch- class analysis
Hendrik Purwins, Benjamin Blankertz and Klaus Obermayer
In: Tonal Theory for the Digital Age, Computing in Musicology (2008) MIT Press , Menlo Park , pp. 74-103.

Abstract

The doubly circular relations of the major and minor keys based on all twelve pitchclasses can be depicted in toroidal models. We demonstrate a convergence of derivations from the different bases of conventional harmonic theory and recent experiments in music psychology. We present a formalization of the music-theoretical derivation from Gottfried Weber’s 1817 chart of tone-centers by using a topographic ordering map. We find the results to be consistent with Krumhansl and Kessler’s 1982 visualization of perceptual ratings.

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