10-TET: A Unique Equal Temperament System

• 10-TET is a tuning system that divides the octave into ten equal steps using modular arithmetic and precise pitch-class labeling.
• Its harmonic structure is defined through innovative algebraic, geometric, and graph-theoretic frameworks, including decagonal cycles and the Desargues configuration.
• The system offers alternative compositional tools and theoretical insights by integrating symbolic dynamics and tempered monoid theory for just intonation approximations.

The 10-Tone Equal Temperament (10-TET) is a musical system in which the octave is divided into ten equal steps, with each pitch-class forming an element of the additive cyclic group $\mathbb{Z}_{10}$. This structure is not to be regarded merely as a diminished approximation to 12-TET, but as a fully autonomous harmonic universe possessing its own combinatorial, algebraic, and geometric properties (Nurowski, 5 Jan 2026). The foundation of 10-TET scales, intervals, tonal relations, and harmonic graphs draws on abstract group theory, projective geometry, and symbolic dynamics—yielding both novel compositional tools and mathematical frameworks.

1. Algebraic Foundations and Pitch-Class Structure

At the core of any $N$-TET system is the identification of pitch classes with the cyclic group $\mathbb{Z}/N\mathbb{Z}$, where addition operates modulo $N$ and enharmonic equivalence follows directly from identifying integers differing by multiples of $N$. In 10-TET, the pitch classes are labeled $0$ through $9$, each corresponding to a step size of $2^{1/10}$ in frequency, forming the scale: $f_k = f_0 \times 2^{k/10}, \quad k=0,1,\dots,9.$ One semitone in 10-TET measures exactly $120$ cents, rendering the k-th pitch class as $N$0 (Wu, 2016). The circle of fifths is generated by the addition of $N$1 modulo 10, producing a decagonal cycle: $N$2 which serves as the fundamental cyclic structure for tonal relativity and modal construction.

2. Intervallic Logic and the Structural Parameter $N$3

Central to harmonic theory in 10-TET is the fixing of the "perfect fifth" as a seven-step interval ($N$4 in $N$5). The major third ($N$6) and minor third ($N$7) are then defined by $N$8 and distinguished by the structural parameter $N$9. Only odd values of $\mathbb{Z}/N\mathbb{Z}$0 yield viable triadic systems, resulting in three principal harmonic regimes:

System Label	$\mathbb{Z}/N\mathbb{Z}$1	Major Third $\mathbb{Z}/N\mathbb{Z}$2	Minor Third $\mathbb{Z}/N\mathbb{Z}$3
Acoustic	$\mathbb{Z}/N\mathbb{Z}$4	$\mathbb{Z}/N\mathbb{Z}$5	$\mathbb{Z}/N\mathbb{Z}$6
Tritone	$\mathbb{Z}/N\mathbb{Z}$7	$\mathbb{Z}/N\mathbb{Z}$8	$\mathbb{Z}/N\mathbb{Z}$9
Wide	$N$0	$N$1	$N$2

Major and minor triads at root $N$3 are constructed as: $N$4 (Nurowski, 5 Jan 2026). The combinatorial properties and symmetries of these triad families are deeply dependent on the choice of $N$5.

3. Graph-Theoretic Classification: Tonnetz and Harmonic Connectivity

The interaction of triads under the Riemannian moves—Parallel (P), Relative (R), and Leading-tone (L)—determines the topology of the Tonnetz graph in each system:

• For $N$6 (Acoustic system), modal degeneracy arises: each major triad coincides set-wise with a minor triad seven steps away, resulting in two disjoint 10-cycles if chord sets are identified solely pitch-class-wise. However, by distinguishing root-functionality, this degenerate graph is restored to the Generalized Petersen graph $N$7 (decagonal prism), a 3-regular bipartite graph on 20 vertices.
• For $N$8 (Tritone system), no degeneracy occurs. Major and minor triads are distinct sets, with the three moves generating $N$9—the same decagonal prism topology.
• For $N$0 (Wide system), harmonic connectivity is maximized. The resultant Levi graph is the Desargues graph, isomorphic to the symmetric $N$1 Desargues configuration from projective geometry:
  • Vertex-transitive under cyclic action,
  • Girth = 6,
  • Unique symmetric configuration.

The following table recapitulates the principal Tonnetz topologies:

System	Tonnetz Graph	Geometric Configuration
$N$2	Prism $N$3	Disjoint cycles (restored: prism)
$N$4	Prism $N$5	Decagonal Prism
$N$6	Desargues graph	$N$7 configuration

These findings establish that harmonic symmetry and progression in 10-TET are geometric rather than acoustic in essence (Nurowski, 5 Jan 2026).

4. Scales, Modes, and Symbolic Dynamical Classification

Scales in 10-TET are classified by compositions and cyclic compositions ("wheels") of $N$8, using the full shift on $N$9 and tools from symbolic dynamics (Aíza, 2020). The number of $0$0-note scales (orbital dimension) and their modal families (transversal dimension) is given by: $0$1 where $0$2 is Euler's totient function. For example, for $0$3, there are $0$4 compositions (scales) and $0$5 wheels (cyclic classes).

Explicit examples include:

• Heptatonic scale: $0$6
• Pentatonic scale: $0$7

Modes correspond to cyclic permutations of the interval sequence, and every scale possesses a defined set of degrees by algebraic constructions in $0$8 (Wu, 2016).

5. Tempered Monoids, Fractality, and Just Intonation Approximation

From the perspective of tempered monoids (Bras-Amorós, 2017), 10-TET arises via discretizing the logarithmic monoid $0$9 by multiplication and rounding, yielding step sizes of $9$0 octaves. Each ET step $9$1 corresponds to frequency $9$2, preserving product-compatibility.

Approximation of just intervals is quantified:

• Perfect fifth ($9$3), closest ET step $9$4, error $9$5 cents
• Perfect fourth ($9$6), $9$7, error $9$8 cents
• Major third ($9$9), $2^{1/10}$0, error $2^{1/10}$1 cents

Furthermore, for $2^{1/10}$2, the discretizations of the logarithmic monoid and the golden fractal monoid coincide, and 10-TET is shown to be "odd-filterable": the odd-indexed subset of steps remains algebraically closed under addition, in analogy with 12-TET, though 12 is the maximal division with this property (Bras-Amorós, 2017).

6. Geometric Paradigms and Conclusions

The emergence of harmonic symmetry in 10-TET is fundamentally geometric. While 12-TET derives its intervallic logic from acoustic ratios and associated integer frequency divisions, 10-TET achieves maximal symmetry through combinatorial and projective geometric constructs, notably the Desargues configuration. The restored Acoustic and Tritone systems, though intervallically distinct, both realize the decagonal prism $2^{1/10}$3 via suitable mappings, whereas the Wide system uniquely achieves a vertex-transitive, high-girth Levi graph structure (Nurowski, 5 Jan 2026).

A plausible implication is that future explorations of equal-temperament systems should prioritize projective and graph-theoretic universality rather than mere rational approximation. The 10-TET case demonstrates a coherent theoretical model, supported by symbolic dynamics, algebraic topology, and tempered monoid theory, for treating alternative equal divisions as independently valid harmonic worlds.