Classical musical harmonies like the C major triad map perfectly onto rigid geometric shapes used in advanced finite mathematics.
April 23, 2026
Original Paper
Tonnetz Theory, Classical Harmony, and the Combinatorial Geometry of Abstract Musical Resources
arXiv · 2604.19960
The Takeaway
Combinatorial geometry explains why certain musical chords feel harmonious to the human ear. Diatonic and pentatonic scales are not just cultural preferences, they are reflections of the Fano and Desargues configurations. This discovery links the emotional feeling of music to the absolute logic of finite geometry. The field of music theory has long lacked a unified mathematical grounding of this depth. It suggests that musical beauty is a structural property of the universe rather than a subjective human experience.
From the abstract
In a previous submission, we established a fundamental relation between tone networks and configurations. It was shown that the Eulerian tonnetz can be represented by a $\{12_3\}$ of Daublebsky von Sterneck type D222. We also constructed a tonnetz for Tristan-genus chords (dominant sevenths and half-diminished sevenths) and we showed that this tonnetz can be represented by a $\{12_3\}$ of type D228. In both of these constructions the associated Levi graphs play an important role. Here we look at