The Geometry of Harmony

What if I told you that the reason a C major chord resolving to G major sounds right has nothing to do with aesthetics, cultural conditioning, or divine proportion — and everything to do with statistics?

Over the past five days, I've been chasing a connection between two fields that seem to have no business talking to each other: neo-Riemannian music theory (a branch of mathematics that models chord relationships as geometric transformations) and information geometry (the study of probability distributions as points on curved manifolds). The results have been, frankly, a little shocking. The structures that music theorists have studied for over a century — the Tonnetz, the circle of fifths, the tension between consonance and dissonance — all of them emerge naturally from treating chords as probability distributions and measuring the distance between them using the Fisher-Rao metric.

No tuning. No aesthetic judgment. Just information theory.

The first move is the crucial one, and it's deceptively simple: every chord defines a probability distribution over the twelve pitch classes.

Take a C major triad — the notes C, E, and G played together. When a physical string vibrates at the frequency of C, it doesn't produce a pure tone. It produces a whole tower of harmonics: the octave (2×), the fifth above the octave (3×), two octaves up (4×), the major third above that (5×), and so on, decaying in amplitude roughly as $1/n$ for the $n$-th harmonic. Each harmonic, when reduced modulo the octave, lands on some pitch class in $\mathbb{Z}_{12}$.

So for every note in the chord, we can compute how much "spectral weight" it contributes to each of the 12 pitch classes via its harmonic series. Sum the contributions from all chord tones. Normalize. What you get is a probability distribution $p_C$ on $\mathbb{Z}_{12}$ — a spectral fingerprint of the chord.

This embedding maps all 24 major and minor triads to 24 points in $\Delta_{11}$, the 11-dimensional probability simplex. Every chord becomes a point in a well-understood mathematical space — and that space comes equipped with a natural, unique, information-theoretically motivated metric.

A few things to notice:

• Every pitch class gets nonzero weight. The higher harmonics spray spectral energy across all 12 pitch classes, so the distributions are fully supported.
• The embedding is physically grounded. The harmonic series is a fact about vibrating strings and air columns — no arbitrary choices.
• It's transposition-invariant. D major's spectral distribution is just a cyclic shift of C major's.

The natural question: does this embedding know anything about music theory?

The Tonnetz (German for "tone network") is one of the most important structures in music theory. Originally sketched by Euler in 1739 and formalized by neo-Riemannian theorists like Richard Cohn in the 1990s, it's a graph where the 24 major and minor triads are vertices, connected by three types of edges — the PLR operations:

• P (Parallel): C major ↔ C minor. Same root, flip the quality.
• R (Relative): C major ↔ A minor. The "relative minor" relationship.
• L (Leading tone exchange): C major ↔ E minor. Root drops by a semitone.

Each operation changes exactly one note by one or two semitones — the most parsimonious possible voice leadings between triads. The PLR group is isomorphic to $D_{12}$, and the Tonnetz itself is topologically a torus.

When I computed the Fisher-Rao distance between the spectral distributions of all 24 triads:

and checked whether the nearest neighbors in this metric correspond to PLR neighbors on the Tonnetz:

The Tonnetz is the nearest-neighbor graph of the spectral statistical manifold.

The ordering $R < L < P$ is itself fascinating. You might expect P to be closest — parallel major-minor pairs share a root and fifth. But P changes the third, which is the interval that defines quality. That single semitone shift changes the chord's spectral character more than the root-movements in R and L, because the third appears at the 5th harmonic and carries more perceptual weight.

I also ran multidimensional scaling on the $24 \times 24$ Fisher distance matrix. The eigenvalue spectrum:

Degenerate pairs. This is the eigenvalue signature of $T^2 = S^1 \times S^1$. Each circle contributes a sine-cosine pair. The Tonnetz isn't just combinatorially a torus — the Fisher metric knows it's a torus.

And this isn't fragile. I tested five different spectral models — $1/n$, $1/n^2$, $1/\sqrt{n}$, 8 harmonics, 32 harmonics. All five give 100% PLR recovery, identical ordering, identical torus signature. The Tonnetz is an intrinsic property of pitch-class overlap under harmonic alignment.

The Tonnetz gives us a triangulated torus — 12 vertices, 24 triangular faces, 36 edges. With Fisher-metric edge lengths, we can compute the Regge curvature at each vertex:

For the equal-temperament Tonnetz: the curvature is identically zero at all 12 vertices.

Every triangle sums to exactly 180°. Gauss-Bonnet is satisfied: $\sum \delta(v) = 2\pi \chi(T^2) = 0$. The Fisher-Tonnetz is perfectly flat.

Why? Because equal temperament has perfect $\mathbb{Z}_{12}$ symmetry. Every pitch class is related to every other by the same transposition operation. With three edge types, all identical within type, on a vertex-transitive triangulation, the geometry is forced flat.

This seems like a dead end. But actually, it's the most important result of the entire project.

When a listener hears music in C major, not all pitch classes are equal. C is the tonic — the center of gravity. G is the dominant. The chromatic notes fade into the background. This is a deformation of the flat, symmetric picture.

I modeled this with key-weighted spectral distributions: each pitch class's spectral envelope boosted or dampened by its role in the key. A strength parameter $s \in [0, 1]$ blends between unweighted ($s = 0$) and full key context ($s = 1$).

At $s = 0$: flat. At $s > 0$: curvature appears and grows monotonically.

Negative curvature = gravitational wells. The tonic, dominant, and subdominant — the three pillars of functional harmony — are literally the deepest points on the curved manifold. Resolution means rolling downhill.

Positive curvature = repulsion. The supertonic and mediant are "hills" — and indeed, these are the least stable scale degrees. Tension, geometrically, is being on top of a hill.

The deep insight: the Tonnetz is flat in abstract pitch space; tonality curves it toward a center. Music is geometry being deformed by context.

If being in a key means curving the Tonnetz around a tonal center, then modulating to a new key means moving the curvature. The gravitational wells shift. Pitch classes that were stable become unstable, and vice versa.

I computed the curvature landscape for all 12 major keys and measured the $L^2$ distance between them. Every step along the circle of fifths has exactly the same curvature distance: 0.393. Standard deviation less than $10^{-6}$.

This perfectly recapitulates the classical hierarchy of closely related keys. G and F are nearest — one sharp or flat away. F♯ is farthest — the tritone. The geometry of curvature flow encodes what music students learn in their first year of theory.

For modulation from C to G, the most stable pitch class is E (curvature change: 0.018) — exactly the note a composer would use as a pivot. The geometry predicts what composers have done intuitively for centuries.

And a surprise: parallel keys are closer than relative keys in curvature space. C major ↔ C minor (0.501) vs. C major ↔ A minor (0.517). Sharing the same tonic matters more than sharing pitch content. The gravitational center stays fixed; only the landscape's shape changes.

Everything above is in equal temperament, where all pitch classes are symmetric under $\mathbb{Z}_{12}$. But for most of Western music history, instruments were tuned differently. Information geometry explains exactly what those tradeoffs are.

In just intonation, intervals are simple frequency ratios: a perfect fifth is exactly $3/2$, a major third is $5/4$. But twelve perfect fifths $(3/2)^{12} \neq 2^7$. The Pythagorean comma means the circle of fifths doesn't close. Something has to give.

In JI, the hardest fifth-modulation is 3 times harder than the easiest. The hardest is always F♯ → C♯ — the wolf fifth, where accumulated error piles into a single, painfully wide interval.

Werckmeister III, the temperament Bach likely used for The Well-Tempered Clavier, has the smallest irregularity among historical tunings and the smallest wolf-fifth excess. It distributes the comma so all keys are playable while keeping commonly-used keys pure.

Each tuning system defines a different Riemannian metric on the Tonnetz. JI: near-key heaven, far-key hell. Pythagorean: pure fifths, harsh thirds. Meantone: beautiful in C, terrifying in F♯. Werckmeister: the careful compromise. ET: perfect democracy, all intervals equally impure, all modulations equally easy.

The wolf fifth is the cost of having pure nearby intervals. Equal temperament pays a small, uniform tax on every interval to eliminate that cost entirely. The history of tuning is the history of navigating this geometric tradeoff.

Here's where things get delightfully strange. An L-system is a formal grammar where every symbol is rewritten simultaneously at each step — invented to model plant growth, but capable of generating beautiful self-similar structures.

What happens if you use L-system symbols as PLR operations and walk the Tonnetz?

A simple Fibonacci-like L-system $A \to AB$, $B \to A$, interpreting $A$ as P and $B$ as R, generates:

The resulting path on the Fisher-Rao manifold has measurable properties: total path length, curvature accumulation, spectral entropy. Different L-systems produce different geometric signatures — fractal-like walks on a statistical manifold.

This connects three seemingly unrelated domains: the L-system doesn't know about music theory, the Fisher metric doesn't know about grammars. But when connected, chord progressions are simultaneously outputs of a formal language (computation), paths on a Riemannian manifold (geometry), and sequences of harmonic events (music). Three perspectives on the same object.

Music theory has always had a dual character. On one hand, a system of rules and conventions. On the other, a persistent intuition that these rules reflect something deeper — some structure in the physics of sound or the geometry of perception.

What this research suggests is that the deeper structure is information-geometric. The chain:

1. Chords are probability distributions (spectral envelopes on $\mathbb{Z}_{12}$).
2. The Fisher-Rao metric is the unique invariant metric on probability distributions (Čencov's theorem).
3. The Tonnetz emerges as the nearest-neighbor graph.
4. Equal temperament implies flatness (by $\mathbb{Z}_{12}$ symmetry).
5. Tonality is curvature — choosing a key breaks symmetry, creating gravitational wells.
6. Modulation is curvature flow — the circle of fifths is a regular geodesic.
7. Temperament shapes the metric — each system defines a different Riemannian geometry.

The music theory that emerged from centuries of compositional practice turns out to be information geometry in disguise. The PLR group is the symmetry group of the spectral statistical manifold. The circle of fifths is a geodesic. Functional harmony is curvature dynamics.

From cellular automata to category theory, from fractals to type theory, the same pattern keeps appearing: simple structures, viewed through the right lens, reveal infinite depth. The spectral distribution is simple — just harmonics with $1/n$ decay. The Fisher metric is simple — just the Riemannian metric on the probability simplex. But together they reconstruct the entire architecture of Western harmony.

Music theory IS information geometry.