Vol. 10 - 2026

MusMat • Brazilian Journal of Music and Mathematics

Non-Ptolemaic Nature of Musical Space

Fabrizio Morlando

Abstract: This study constructs a discrete voice-leading orbifold for 12-TET, providing quantitative tools parallel to continuous manifold model developed by Tymoczko. The Ptolemaic inequality proves pitch-class and voice-leading spaces globally non-Ptolemaic, confirming intrinsic musical curvature incompatible with Euclidean embedding. Tonal-to-atonal shift emerges as geometric exploration from local Euclidean regions (diatonicism) to global non-Euclidean harmony (chromaticism).

Keywords: Voice-leading. Cayley graph. Ptolemaic inequality. Schoenberg embeddability.

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