Harmony in 10-TET: From Parallel Universes to the Desargues Configuration

Abstract: The 10-tone equal temperament (10-TET) constitutes a distinct harmonic environment with step sizes exceeding the standard semitone, yet it has historically remained on the fringes of music theory. This paper proposes a radical shift in perspective, treating $\mathbb{Z}_{10}$ not as an imperfect imitation of 12-TET, but as an independent harmonic universe with its own rigorous logic. We classify all possible harmonic systems in 10-TET using a single structural parameter $Δ$, representing the difference between the harmonic mediants (the major and minor thirds'') which sum to the generator (thefifth''). Our analysis reveals a rich landscape of possibilities. The system defined by $Δ=1$, which is naturally the first to consider, initially appears to suffer from a fundamental structural flaw: its harmonic space fractures into two disjoint universes. Conversely, the Wide'' system ($Δ=5$) emerges as a geometrically superior structure, isomorphic to the \textbf{Desargues Configuration} $(10_3)_1$ and forming a fully connected, vertex-transitive graph. Between these extremes lies theTritone'' system ($Δ=3$), which forms a connected Decagonal Prism. Our main result is the discovery of a topological isomorphism between the ``restored'' Acoustic system ($Δ=1$) and the Tritone system ($Δ=3$): despite disjoint intervallic definitions, both generate the Generalized Petersen graph $GP(10,1)$. These results suggest that in 10-TET, the path to harmonic symmetry lies in the geometry of the projective plane and prism graphs rather than pure acoustics.