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 $p_k = k \cdot 120^\circ \bmod 1200$ (Wu, 2016). The circle of fifths is generated by the addition of $+7$ modulo 10, producing a decagonal cycle: $$0\rightarrow7\rightarrow4\rightarrow1\rightarrow8\rightarrow5\rightarrow2\rightarrow9\rightarrow6\rightarrow3\rightarrow0,$$ which serves as the fundamental cyclic structure for tonal relativity and modal construction.

2. Intervallic Logic and the Structural Parameter $\Delta$

Central to harmonic theory in 10-TET is the fixing of the "perfect fifth" as a seven-step interval ($q=7$ in $\mathbb{Z}_{10}$). The major third ($t$) and minor third ($s$) are then defined by $t+s\equiv q\pmod{10}$ and distinguished by the structural parameter $\Delta = t-s \pmod{10}$. Only odd values of $\Delta$ yield viable triadic systems, resulting in three principal harmonic regimes:

System Label	$\Delta$	Major Third $t$	Minor Third $s$
Acoustic	$1$	$4$	$3$
Tritone	$3$	$5$	$2$
Wide	$5$	$6$	$1$

Major and minor triads at root $r$ are constructed as: $D_r = (r, r+t, r+q), \quad M_r = (r, r+s, r+q)$ (Nurowski, 5 Jan 2026). The combinatorial properties and symmetries of these triad families are deeply dependent on the choice of $\Delta$.

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 $\Delta=1$ (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 $\mathrm{GP}(10,1)$ (decagonal prism), a 3-regular bipartite graph on 20 vertices.
• For $\Delta=3$ (Tritone system), no degeneracy occurs. Major and minor triads are distinct sets, with the three moves generating $\mathrm{GP}(10,1)$—the same decagonal prism topology.
• For $\Delta=5$ (Wide system), harmonic connectivity is maximized. The resultant Levi graph is the Desargues graph, isomorphic to the symmetric $(10_3)$ 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
$\Delta=1$	Prism $\mathrm{GP}(10,1)$	Disjoint cycles (restored: prism)
$\Delta=3$	Prism $\mathrm{GP}(10,1)$	Decagonal Prism
$\Delta=5$	Desargues graph	$(10_3)_1$ 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 $10$, using the full shift on $\{0,\dots,9\}$ and tools from symbolic dynamics (Aíza, 2020). The number of $k$-note scales (orbital dimension) and their modal families (transversal dimension) is given by: $$C_{10,k} = \binom{9}{k-1}, \qquad W_{10,k} = \frac{1}{k} \sum_{d|\gcd(10,k)} \phi(d) \binom{(10/d)-1}{(k/d)-1}$$ where $\phi(d)$ is Euler's totient function. For example, for $k=5$, there are $126$ compositions (scales) and $26$ wheels (cyclic classes).

Explicit examples include:

• Heptatonic scale: $S_{10,7} = \{0,1,2,4,5,7,8\}$
• Pentatonic scale: $P_{10,5} = \{0,2,4,6,8\}$

Modes correspond to cyclic permutations of the interval sequence, and every scale possesses a defined set of degrees by algebraic constructions in $\mathbb{Z}_{10}$ (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 $L = \{\log_2(i)\}$ by multiplication and rounding, yielding step sizes of $1/10$ octaves. Each ET step $k$ corresponds to frequency $f_0 \cdot 2^{k/10}$, preserving product-compatibility.

Approximation of just intervals is quantified:

• Perfect fifth ($3:2$), closest ET step $k=6$, error $+18.07$ cents
• Perfect fourth ($4:3$), $k=4$, error $-18.24$ cents
• Major third ($5:4$), $k=3$, error $-26.31$ cents

Furthermore, for $m=10$, 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 $\mathrm{GP}(10,1)$ 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.