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

Paweł Nurowski

TL;DR

This work derives a self-contained harmonic theory for 10-TET by classifying harmonic universes in $\mathbb{Z}_{10}$ through the single parameter $\Delta=t-s$, with $t+s\equiv q$. It shows three viable geometries: the Narrow $(t,s)=(4,3)$ with modal degeneracy into two disjoint Tonnetz cycles, the Wide $(t,s)=(6,1)$ realizing the Desargues configuration as a vertex-transitive Levi graph, and the Tritone $(t,s)=(5,2)$ forming a Decagonal Prism isomorphic to $GP(10,1)$ once connectivity is restored. A key result is an isomorphism between the Acoustic (4,3) and Tritone (5,2) systems, demonstrating that distinct intervallic rules can share the same topological backbone. The paper connects musical harmony in 10-TET to projective geometry and graph theory, showing that harmonic symmetry can arise from prism- and Desargues-like geometries rather than pure acoustics. These insights offer a rigorous, geometry-driven framework for exploring non-12-TET tunings and their rich Tonnetz/topological structures with potential compositional applications in decaphonic tuning.”

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 (the ``fifth''). 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 the ``Tritone'' 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.

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

TL;DR

This work derives a self-contained harmonic theory for 10-TET by classifying harmonic universes in through the single parameter , with . It shows three viable geometries: the Narrow with modal degeneracy into two disjoint Tonnetz cycles, the Wide realizing the Desargues configuration as a vertex-transitive Levi graph, and the Tritone forming a Decagonal Prism isomorphic to once connectivity is restored. A key result is an isomorphism between the Acoustic (4,3) and Tritone (5,2) systems, demonstrating that distinct intervallic rules can share the same topological backbone. The paper connects musical harmony in 10-TET to projective geometry and graph theory, showing that harmonic symmetry can arise from prism- and Desargues-like geometries rather than pure acoustics. These insights offer a rigorous, geometry-driven framework for exploring non-12-TET tunings and their rich Tonnetz/topological structures with potential compositional applications in decaphonic tuning.”

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 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 , representing the difference between the harmonic mediants (the major and minor ``thirds'') which sum to the generator (the ``fifth''). Our analysis reveals a rich landscape of possibilities. The system defined by , 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 () emerges as a geometrically superior structure, isomorphic to the \textbf{Desargues Configuration} and forming a fully connected, vertex-transitive graph. Between these extremes lies the ``Tritone'' system (), which forms a connected Decagonal Prism. Our main result is the discovery of a topological isomorphism between the ``restored'' Acoustic system () and the Tritone system (): despite disjoint intervallic definitions, both generate the Generalized Petersen graph . 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.
Paper Structure (62 sections, 8 theorems, 35 equations, 20 figures, 9 tables)

This paper contains 62 sections, 8 theorems, 35 equations, 20 figures, 9 tables.

Table of Contents

  1. Introduction
  2. From Pythagorean tuning to its $n$-tone generalization
  3. General n-TET Pythagorean Systems
  4. Why n=10 Is Distinguished
  5. Labelling of Intervals and the Generator
  6. Generalization of Major and Minor Chords
  7. Mathematical Definition of Chords
  8. The Reference Model: The Classical 12-TET System
  9. Solutions for Thirds in 12-TET
  10. 12-TET Chords and the Tonnetz
  11. 12-TET Chords and the Tonnetz
  12. 12-TET Chords and Their Tonnetz Assignments
  13. The 10-TET System and the Choice of Thirds
  14. Hypothesis 1: The Diatonic Difference ($\Delta=1$)
  15. Hypothesis 2: The Whole-Tone Difference ($\Delta=2$)
  16. ...and 47 more sections

Key Result

Theorem 1

In the 10-TET system with $(t=4, s=3)$, the set of pitch classes forming a major chord $D_r$ is identical to the set forming the minor chord $M_{r+7}$.

Figures (20)

  • Figure 1: Complete enumeration of 12-TET chords.
  • Figure 2: The connectivity of the 12-TET Tonnetz in integer notation.
  • Figure 3: The 12-TET Tonnetz.
  • Figure 4: The 12-TET Tonnetz as a Levi bipartite graph of 12 points and 12 lines, sharing the incidence relation where each line contains 3 points, and every three lines meet at precisely one point lane.
  • Figure 5: Explicit pitch sets of 10-TET chords. Note the highlighted entries.
  • ...and 15 more figures

Theorems & Definitions (15)

  • Theorem 1: Modal Identity
  • proof
  • Theorem 2: Topological Decomposition
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 5 more