EP14

EP14: Tonnetz Hodge Duality

Simplicial Complexes, Hodge Star, and the Decomposition of Harmony

▶ 16:27 TopologyLinear AlgebraHarmonic Analysis

前置知识

EP01 Chord Space and the Torus EP02 String Vibration and the Wave Equation EP04 All-Interval Rows and ℤ₁₂

Overview

The Tonnetz — Euler’s 1739 interval lattice, rediscovered by neo-Riemannian theorists — is not merely a picture of chord relationships. It is a simplicial complex triangulating a torus, and every chord progression on it is a cochain that decomposes, uniquely and orthogonally, into three mathematically distinct components.

This episode applies discrete Hodge theory (Eckmann 1944; Lim 2020) to the Tonnetz. The central result:

Any 1-cochain on the Tonnetz torus decomposes as:
$\omega = \omega_{\text{exact}} \;+\; \omega_{\text{coexact}} \;+\; \omega_{\text{harmonic}}$ where the three summands are pairwise orthogonal and uniquely determined.

The three components correspond — as testable hypotheses — to functional cadences, chromatic PLR cycles, and remote modulations along the torus topology.

Prerequisites

Chord Space as Torus (EP01) — Tonnetz and PLR transformations
Wave equation and Fourier series (EP02) — analogy with heat diffusion on manifolds
All-Interval Rows and ℤ₁₂ (EP04) — cyclic group structure underlying pitch classes

Preamble: What Is a “Hole”? Building Homology from Scratch

Who this is for: You have watched the video and understand the setup (boundary matrices, chain groups, Betti numbers), but the definition $H_k = \ker(\partial_k)/\operatorname{im}(\partial_{k+1})$ feels like a formula dropped from the sky. This preamble builds it from a single question: what actually is a hole?

P.1 — The Naive Question

A doughnut has one hole. A pretzel has two. A solid ball has none.

First attempt: “a hole is a region with nothing in it." That doesn’t work — the inside of a sphere is also “nothing,” but a sphere doesn’t have the same kind of hole as a torus.

Second attempt: “a hole is something you can stick a finger through." That’s dimension-specific and requires 3D embedding.

The algebraic topology approach sidesteps all of this. Instead of defining what a hole is, we define what a hole does: it obstructs you from filling in a closed curve.

P.2 — The Loop Test

Draw a closed loop on a surface. Try to “fill it in” — can you find a 2-dimensional patch whose boundary is exactly that loop?

On the sphere: every closed loop bounds a disk — no obstruction.

On the torus: the loop going through the hole cannot bound any patch. The hole is that obstruction.

The algebraic challenge: detect this without geometry or deformation. The answer is the boundary operator $\partial$ .

Loop test: which loops can be filled in?

Loops that cannot shrink to a point on the torus

P.3 — Chains: The Algebraic Skeleton

Replace the geometric surface with an oriented combinatorial skeleton. Work with formal linear combinations over $\mathbb{R}$ :

A 0-chain: $3A - 2B + C$ — weighted vertices.
A 1-chain: $[A,B] + 2[B,C] - [A,C]$ — weighted oriented edges.
A 2-chain: $[A,B,C] - [D,E,F]$ — weighted oriented triangles.

Orientation: $[B,A] = -[A,B]$ . These form real vector spaces $C_0, C_1, C_2, \ldots$

P.4 — The Boundary Operator $\partial$

$\partial_k$ takes a $k$ -chain and returns its $(k-1)$ -dimensional boundary:

\partial_0(A) = 0, \qquad \partial_1([A,B]) = B - A

\partial_2([A,B,C]) = [B,C] - [A,C] + [A,B]

Sign rule: delete each vertex in turn, alternating signs — the result is the counterclockwise perimeter.

Boundary operator: triangle → its three edges

P.5 — $\partial^2 = 0$ : The Fundamental Identity

Theorem P.1 (Boundary of a Boundary Vanishes)

For any simplicial complex,

\partial_k \circ \partial_{k+1} = 0

Proof.

Compute $\partial_1(\partial_2([A,B,C]))$ :

\partial_1\bigl([B,C] - [A,C] + [A,B]\bigr) = (C-B) - (C-A) + (B-A) = 0 \;\checkmark

In general, applying $\partial_{k-1}$ to $\partial_k[v_0,\ldots,v_k] = \sum_i (-1)^i [v_0,\ldots,\hat{v}_i,\ldots,v_k]$ and collecting terms by pairs $(i,j)$ with $i < j$ shows every pair cancels with opposite sign. $\square$

Geometric meaning: The boundary of a solid region is a closed surface — which has no boundary itself. Algebraic signs enforce this cancellation. This one identity is what makes all of homology work.

Boundary-squared = 0: opposite boundary edges cancel

P.6 — Cycles, Boundaries, and the Definition of a Hole

Definition P.1 (Cycles and Boundaries)

A $k$ -cycle is a $k$ -chain with zero boundary: $Z_k = \ker(\partial_k)$ .

A $k$ -boundary is a $k$ -chain in $\operatorname{im}(\partial_{k+1})$ .

Since $\partial^2 = 0$ : every boundary is automatically a cycle:

B_k \;\subseteq\; Z_k \;\subseteq\; C_k

Chain	Cycle?	Boundary?
Open path $[A,B]$	No	—
Triangle perimeter $\partial_2[A,B,C]$	Yes	Yes
Loop through torus hole	Yes	No

Cycles vs boundaries: which loops can be filled?

A $k$ -dimensional hole is a $k$ -cycle that is NOT the boundary of any $(k+1)$ -chain.

Holes as obstructions to filling: the algebraic picture

Two cycles are homologous if they differ by a boundary: $c_1 \sim c_2 \iff c_1 - c_2 \in B_k$ .

$B_k \subseteq Z_k \subseteq C_k$ ">

Definition P.2 (Homology Groups and Betti Numbers)

H_k = Z_k / B_k = \ker(\partial_k) \,/\, \operatorname{im}(\partial_{k+1})

The $k$ -th Betti number $\beta_k = \dim(H_k)$ counts independent $k$ -dimensional holes.

P.7 — Connecting to the Tonnetz Numbers

The video computes $\dim(\ker \partial_1) = 25$ and $\operatorname{rank}(\partial_2) = 23$ . Therefore:

\beta_1 = 25 - 23 = 2

Of 25 closed loops, 23 can be filled by triangles (fake holes). The remaining 2 equivalence classes are genuine: one along the circle of fifths, one along the major third cycle. These two holes are why the harmonic component of the Hodge decomposition is non-trivial.

Water-flow analogy: Boundaries $B_k$ are currents driven by a potential — remove the pump and they stop. Non-boundary cycles are currents with no source, no sink, no pump — they persist because the topology allows it. $H_k = Z_k/B_k$ is the space of “eternal currents.”

$H_k = Z_k / B_k$ , homologous cycles identified">

Two homologous cycles on the torus: same hole, different paths

The Tonnetz as a Simplicial Complex

Definition 0.1 (Tonnetz Simplicial Complex)

The Tonnetz is a simplicial complex $K$ triangulating the torus $T^2 = S^1 \times S^1$ :

Dimension	Cells	Count	Musical meaning
0 (vertices)	Pitch classes $\{C, C\sharp, \ldots, B\}$	12	Individual notes
1 (edges)	Minor 3rd + Major 3rd + Perfect 5th	36	Consonant intervals
2 (faces)	Major triads + Minor triads	24	Triads

Euler characteristic: $\chi(K) = 12 - 36 + 24 = 0$ . Since $\chi = 2 - 2g$ for orientable closed surfaces, $g = 1$ — confirming the torus.

音乐联系

中文: “Tonnetz 的顶点是十二个音高类，边是音程，三角形是三和弦。整个结构是一个三角剖分的环面。”

The 36 edges are exactly the consonant intervals of Western voice leading: perfect fifths (3:2), major thirds (5:4), minor thirds (6:5). Going up 12 perfect fifths returns to the starting pitch class mod 12; going up 4 major thirds does the same ( $4 \times 3 = 12$ semitones). Both directions “close up,” forcing the torus topology.

Betti numbers (from $\operatorname{rank}(\partial_1) = 11$ , $\operatorname{rank}(\partial_2) = 23$ ):

$\beta_k$	Value	Meaning
$\beta_0 = 12 - 11$	1	Connected: all pitch classes in one piece
$\beta_1 = 25 - 23$	2	Two non-contractible loops: circle of fifths + major third cycle
$\beta_2 = 24 - 23$	1	One enclosed cavity (the torus encloses a void)

中文: “Betti 数 1, 2, 1 是环面 $T^2$ 的拓扑签名。”

The Tonnetz torus: two non-contractible loops, Betti numbers (1,2,1)

The dual complex $K^*$ (chicken-wire torus): Reversing the roles of vertices and faces — each triangle of $K$ becomes a vertex, each shared edge becomes an edge, each vertex becomes a hexagonal face — gives the Poincaré dual $K^*$ :

In $K$	In $K^*$
12 vertices (pitch classes)	12 hexagonal faces
36 edges (intervals)	36 edges (PLR voice leadings)
24 triangular faces (triads)	24 vertices (triads)

$\chi(K^*) = 0$ — still a torus. This is the chicken-wire torus of Douthett & Steinbach (1998), where P, L, R transformations correspond to edges of $K^*$ .

Definition 0.2 (k-Cochain)

A $k$ -cochain $\phi \in C^k(K)$ is a real-valued function on oriented $k$ -simplices, satisfying $\phi(-\sigma) = -\phi(\sigma)$ .

0-cochain $f \in C^0$ : a “tonal weight” on pitch classes. Example: $f(C) = 1$ , $f(G) = 0.8$ , …
1-cochain $\omega \in C^1$ : a signed flow on directed intervals. Example: $\omega([C,G]) = +7$ (up a fifth), $\omega([G,C]) = -7$ .
2-cochain $g \in C^2$ : a signed weight on oriented triads.

The coboundary $\delta_k = \partial_{k+1}^T : C^k \to C^{k+1}$ satisfies $\delta_k \circ \delta_{k-1} = 0$ — the cochain analogue of $\partial^2 = 0$ .

A 1-form (cochain) assigns signed values to oriented edges of the Tonnetz

Section 1: The Hodge Star Operator ★

中文: “离散 Hodge 星算子，记作 ★，是连接原始复形 $K$ 和对偶复形 $K^*$ 的代数算子。”

Definition 1.1 (Discrete Hodge Star)

Let $K$ triangulate an orientable closed $n$ -manifold with dual $K^*$ . The discrete Hodge star is the linear map

\star_k : C^k(K) \to C^{n-k}(K^*)

sending each basis

k

-cochain (dual to

\sigma \in K

) to the basis

(n-k)

-cochain (dual to

\sigma^* \in K^*

For the Tonnetz torus ( $n = 2$ ):

Map	Dimensions	Musical translation
$\star_0 : C^0(K) \to C^2(K^*)$	$12 \to 12$	Pitch-class weights → hexagonal face weights
$\star_1 : C^1(K) \to C^1(K^*)$	$36 \to 36$	Interval flows ↔ PLR voice-leading flows
$\star_2 : C^2(K) \to C^0(K^*)$	$24 \to 24$	Triad weights → dual vertex weights

中文: “★ 把 1-上链映射到 1-上链。边上的流映射到对偶边上的流。36 对 36，一维在二维流形上是自对偶的。”

Worked examples:

$\star_0$ : $f \in C^0$ with $f(C)=1$ , all others $0$ → $\star_0 f$ assigns $1$ to the hexagonal face of $K^*$ surrounding $C$ , bounded by the six triads containing $C$ . Translates “note C” into “hexagonal region of triadic space centered on C.”
$\star_1$ (self-dual): Flow $+1$ on edge $[C,E]$ (major third) → the dual edge connects the two triads sharing $C$ and $E$ : C major and A minor. Translates “interval flow” into “PLR voice-leading motion between the two triads sharing that interval.”

Theorem 1.1 (Poincaré Duality — Cell Count)

For an orientable closed

n

-manifold, the number of

k

-cells in

K

equals the number of

(n-k)

-cells in

K^*

. Consequently

\star_k

is a square matrix.

Proof.

The construction of

K^*

gives a bijection

\sigma \mapsto \sigma^*

between

k

-simplices of

K

and

(n-k)

-cells of

K^*

. Therefore

\dim C^k(K) = \dim C^{n-k}(K^*)

\square

中文: “Poincaré 对偶保证原始 $k$ -胞腔数等于对偶 $n-k$ 胞腔数，所以 ★ 总是方阵。”

Self-duality at the middle dimension: When $k = n/2$ , $\star_k$ maps a space to itself. For the Tonnetz torus ( $n=2$ , $k=1$ ): 1-cochains are self-dual. Same phenomenon: the electromagnetic field tensor $F$ is a 2-form on Minkowski spacetime ( $n=4$ , $k=2$ ), and Maxwell’s equations are $dF = 0$ , $d{\star}F = J$ .

中文: “其实 Maxwell 方程组可以写成两行： $dF = 0$ ， $d\star F = J$ 。整个电磁学就是外微分 $d$ 和 Hodge 星的组合。”

Section 2: Inner Product and Adjoint Operators

Definition 2.1 (Standard Inner Product on Cochain Spaces)

Equip $C^k$ with the inner product declaring all basis $k$ -simplices orthonormal:

\langle e_i, e_j \rangle = \delta_{ij}

中文: “我们给链空间 $C_k$ 装一个标准内积，声明基底正交归一。”

The inner product identifies chains with cochains canonically, and gives us the adjoint of $\partial$ — the key ingredient for the Hodge Laplacian.

Definition 2.2 (Coboundary (Adjoint) Operator)

The coboundary $\delta_k := \partial_{k+1}^T : C^k \to C^{k+1}$ is the transpose of the boundary operator. It runs in the opposite direction:

C_2 \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0 \qquad \text{(boundary, downward)}

C^0 \xrightarrow{\delta_0 = \partial_1^T} C^1 \xrightarrow{\delta_1 = \partial_2^T} C^2 \qquad \text{(coboundary, upward)}

The adjoint property:

\langle \partial_{k+1}\beta, \alpha \rangle = \langle \beta, \delta_k \alpha \rangle

中文: “伴随算子的方向和 $\partial$ 相反。 $\partial$ 从高维到低维，它的转置从低维到高维。”

Worked example ( $\delta_0$ — discrete gradient): Let $f \in C^0$ have $f(C) = 1$ , $f = 0$ elsewhere. Then $(\delta_0 f)([C,E]) = f(E) - f(C) = -1$ and $(\delta_0 f)([G,C]) = f(C) - f(G) = +1$ . So $\delta_0 f$ assigns $-1$ to every edge leaving $C$ and $+1$ to every edge arriving at $C$ — a discrete gradient converging toward the high-potential note.

Prop 2.1 (Coboundary Squared = 0)

\delta_k \circ \delta_{k-1} = 0

Proof.

\delta_k \circ \delta_{k-1} = \partial_{k+1}^T \circ \partial_k^T = (\partial_k \circ \partial_{k+1})^T = 0^T = 0

\square

Section 3: The Hodge Laplacian

中文: “ $L_k$ 的两个矩阵项分别检测两类’非调和’分量。 $\partial_k^T \partial_k$ 检测非零梯度， $\partial_{k+1} \partial_{k+1}^T$ 检测非零旋度。”

Definition 3.1 (Hodge Laplacian)

The $k$ -th Hodge Laplacian is the self-adjoint operator $L_k : C^k \to C^k$ :

L_k = \partial_k^T \partial_k + \partial_{k+1} \partial_{k+1}^T

$\partial_k^T \partial_k$ (down-up): penalizes non-zero divergence (sources/sinks at vertices).
$\partial_{k+1} \partial_{k+1}^T$ (up-down): penalizes non-zero curl (circulation around faces).

$L_1 \omega = 0$ means the 1-cochain $\omega$ has neither sources/sinks nor local whirlpools.

Heat diffusion analogy: Imagine edge flows as temperatures on a donut-shaped pipe network. $L_1 \omega$ measures how far $\omega$ is from thermal equilibrium. The equilibrium states — where no gradient wants to push heat downhill and no eddy stirs heat in circles — are the harmonic forms.

Prop 3.1 (Hodge Laplacian is PSD)

L_k

is symmetric and positive semi-definite:

\langle L_k \omega, \omega \rangle = \|\partial_k \omega\|^2 + \|\partial_{k+1}^T \omega\|^2 \geq 0

Proof.

\langle L_k \omega, \omega \rangle = \langle \partial_k^T \partial_k \omega, \omega \rangle + \langle \partial_{k+1} \partial_{k+1}^T \omega, \omega \rangle = \|\partial_k \omega\|^2 + \|\partial_{k+1}^T \omega\|^2 \geq 0

\square

Theorem 3.1 (Discrete Hodge Theorem)

For any finite simplicial complex $K$ with the standard inner product:

\ker(L_k) \cong H^k(K;\mathbb{R}), \qquad \dim(\ker L_k) = \beta_k

Elements of

\ker(L_k)

are called **harmonic forms** (调和形式).

Proof.

Since $L_k$ is PSD:

\omega \in \ker(L_k) \iff \langle L_k \omega, \omega \rangle = 0 \iff \|\partial_k \omega\|^2 + \|\partial_{k+1}^T \omega\|^2 = 0

\iff \partial_k \omega = 0 \;\;\text{AND}\;\; \partial_{k+1}^T \omega = 0

So $\ker(L_k) = \ker(\partial_k) \cap \ker(\partial_{k+1}^T)$ — the set of $k$ -cycles that are orthogonal to all boundaries.

Since $Z^k = B^k \oplus (Z^k \cap (B^k)^\perp)$ (orthogonal decomposition), we have $H^k = Z^k/B^k \cong Z^k \cap (B^k)^\perp = \ker(L_k)$ . Therefore $\dim \ker(L_k) = \beta_k$ . $\square$

中文: “Hodge 定理说， $L_k$ 的 kernel 同构于第 $k$ 个上同调群。kernel 里的元素叫调和形式。”

Concrete numbers for the Tonnetz:

Laplacian	Size	$\ker$ dimension	Musical interpretation
$L_0$	$12\times 12$	$\beta_0 = 1$	One constant harmonic function (connected)
$L_1$	$36\times 36$	$\beta_1 = 2$	Two independent harmonic interval flows
$L_2$	$24\times 24$	$\beta_2 = 1$	One harmonic 2-cochain (volume form)

The two harmonic 1-cochains in $\ker(L_1)$ correspond to:

The circle of fifths direction: C→G→D→A→E→B→F♯→C♯→A♭→E♭→B♭→F→C
The major third direction: C→E→G♯→C

These are the only non-zero flows on the torus that are simultaneously divergence-free and curl-free. A sphere ( $\beta_1 = 0$ ) would have no such flows; a double torus ( $\beta_1 = 4$ ) would have four.

中文: “对 Tonnetz 上的 1-上链来说，调和空间的维度等于 $\beta_1 = 2$ ，对应五度圈方向和大三度方向的两个独立全局循环。”

Section 4: The Hodge Decomposition Theorem

中文: “任意 $k$ 维上链都可以唯一正交分解为三个分量。”

Theorem 4.1 (Discrete Hodge Decomposition)

For any finite simplicial complex $K$ with standard inner product, the cochain space $C^k$ admits an orthogonal direct sum decomposition:

C^k = \operatorname{im}(\partial_k^T) \;\oplus\; \operatorname{im}(\partial_{k+1}) \;\oplus\; \ker(L_k)

Every $\omega \in C^k$ has a unique decomposition:

\omega = \underbrace{\omega_{\text{exact}}}_{\in\,\operatorname{im}(\partial_k^T)} \;+\; \underbrace{\omega_{\text{coexact}}}_{\in\,\operatorname{im}(\partial_{k+1})} \;+\; \underbrace{\omega_{\text{harmonic}}}_{\in\,\ker(L_k)}

Moreover: $\|\omega\|^2 = \|\omega_\text{exact}\|^2 + \|\omega_\text{coexact}\|^2 + \|\omega_\text{harmonic}\|^2$ (Pythagorean theorem for orthogonal components).

Proof.

Step 1: Pairwise orthogonality.

(a) $\operatorname{im}(\partial_k^T) \perp \operatorname{im}(\partial_{k+1})$ : For $u = \partial_k^T \alpha$ and $v = \partial_{k+1}\beta$ :

\langle u, v \rangle = \langle \partial_k^T \alpha, \partial_{k+1}\beta \rangle = \langle \alpha, \partial_k \partial_{k+1}\beta \rangle = \langle \alpha, 0 \rangle = 0 \;\checkmark

(b) $\operatorname{im}(\partial_k^T) \perp \ker(L_k)$ : For $u = \partial_k^T \alpha$ and $\omega \in \ker(L_k)$ (which gives $\partial_k \omega = 0$ ):

\langle u, \omega \rangle = \langle \partial_k^T \alpha, \omega \rangle = \langle \alpha, \partial_k \omega \rangle = 0 \;\checkmark

(c) $\operatorname{im}(\partial_{k+1}) \perp \ker(L_k)$ : For $v = \partial_{k+1}\beta$ and $\omega \in \ker(L_k)$ (which gives $\partial_{k+1}^T \omega = 0$ ):

\langle v, \omega \rangle = \langle \partial_{k+1}\beta, \omega \rangle = \langle \beta, \partial_{k+1}^T \omega \rangle = 0 \;\checkmark

Step 2: The three subspaces span $C^k$ .

We show $\dim(\operatorname{im}(\partial_k^T)) + \dim(\operatorname{im}(\partial_{k+1})) + \dim(\ker(L_k)) = \dim(C^k)$ .

Since $\partial_{k+1}^T \partial_k^T = (\partial_k \partial_{k+1})^T = 0$ , the cross-term in $L_k$ vanishes when $L_k$ acts on $\operatorname{im}(\partial_k^T)$ : for $u = \partial_k^T \alpha$ , $L_k u = \partial_k^T \partial_k \partial_k^T \alpha$ . One verifies $L_k$ acts injectively (and thus surjectively) on each of $\operatorname{im}(\partial_k^T)$ and $\operatorname{im}(\partial_{k+1})$ separately. Therefore $\operatorname{im}(L_k) = \operatorname{im}(\partial_k^T) \oplus \operatorname{im}(\partial_{k+1})$ , and by rank-nullity:

\dim(\operatorname{im}(\partial_k^T)) + \dim(\operatorname{im}(\partial_{k+1})) + \dim(\ker(L_k)) = \dim(\operatorname{im}(L_k)) + \dim(\ker(L_k)) = \dim(C^k) \;\square

中文: “这不是近似，是精确的数学定理。”

Dimension count for $k=1$ on the Tonnetz:

Component	Subspace	Dimension
Exact $\omega_\text{exact} = \partial_1^T f$	$\operatorname{im}(\partial_1^T)$	$11 = \operatorname{rank}(\partial_1)$
Coexact $\omega_\text{coexact} = \partial_2 g$	$\operatorname{im}(\partial_2)$	$23 = \operatorname{rank}(\partial_2)$
Harmonic	$\ker(L_1)$	$2 = \beta_1$
Total	$C^1$	36 ✓

Meaning of each component:

Component	Formula	Vector calculus analogy	Characterization
Exact	$\partial_1^T f$	Gradient $\nabla f$	Acyclic; flow on $[u,v]$ is $f(v)-f(u)$
Coexact	$\partial_2 g$	Curl $\nabla \times \mathbf{A}$	Circulates locally around triangular faces
Harmonic	$L_1 \omega = 0$	Harmonic function	Divergence-free AND curl-free simultaneously

中文: “Exact 分量——梯度。Coexact 分量——旋度。Harmonic 分量——沿着环面的拓扑洞做全局循环。”

Water-on-a-donut analogy:

Type	Water analogy	Musical analogy
Exact	Flows downhill from peak to valley	V→I cadence; directed tension-resolution
Coexact	Whirlpools spinning in tight circles	PLR cycles, chromatic voice leading
Harmonic	A river circling the donut — no hill drives it, no eddy spins it	Tonal center migrating along circle of fifths

The decomposition is orthogonal — the three types never interfere.

Computing the decomposition (for implementation): The orthogonal projections use Moore–Penrose pseudoinverses:

\omega_\text{exact} = \partial_1^T(\partial_1 \partial_1^T)^\dagger \partial_1 \omega, \quad \omega_\text{coexact} = \partial_2(\partial_2^T \partial_2)^\dagger \partial_2^T \omega, \quad \omega_\text{harmonic} = \omega - \omega_\text{exact} - \omega_\text{coexact}

Section 5: Three Musical Hypotheses

中文: “Hodge 分解是定理。接下来我们提出三个可检验的音乐假设。”

Encoding a chord progression: Assign weights to Tonnetz edges traversed (e.g., proportional to interval activation duration). Apply the decomposition. The energy ratios $E_\text{exact}/\|w\|^2$ , $E_\text{coexact}/\|w\|^2$ , $E_\text{harmonic}/\|w\|^2$ give the harmonic profile.

Conjecture (H1: Exact Energy ≈ Cadential Harmony)

A passage with high $E_\text{exact}/\|w\|^2$ tends to exhibit strong cadences and functional harmony.

Reasoning: $\omega_\text{exact} = \partial_1^T f$ means flow on edge $[u,v]$ equals $f(v) - f(u)$ : the progression flows from high-potential to low-potential pitch classes. Dominant (high potential) → tonic (low potential) is exactly this gradient structure. The exact component is automatically acyclic — consistent with goal-directed harmonic motion.

Example: Final bars of Bach chorale BWV 269. The cadence ii⁶–V–V⁷–I is almost pure gradient flow: $f(D) > f(G) > f(C)$ , monotonically descending to the tonic.

Falsification: Find high $E_\text{exact}$ with no cadential structure.

Conjecture (H2: Coexact Energy ≈ Chromatic Cycles (PLR))

A passage with high $E_\text{coexact}/\|w\|^2$ tends to exhibit PLR-type short cycles and chromatic voice leading.

Reasoning: $\omega_\text{coexact} = \partial_2 g$ is a sum of triangle boundaries — local circulation around triads. Neo-Riemannian P, L, R transformations create short cycles wrapping around adjacent triangles (each step moves one voice by a semitone or whole tone).

Example: Opening of Brahms Intermezzo Op. 119 No. 1 — chains of third-related triads (B min → D maj → F♯ min → A maj → …), each a P or R step, locally circular with no global modulation. The hexatonic cycle (C maj → C min → A♭ maj → A♭ min → E maj → E min → C maj) lies entirely in $\operatorname{im}(\partial_2)$ .

Falsification: Find high $E_\text{coexact}$ with no chromatic activity.

Conjecture (H3: Harmonic Energy ≈ Remote Modulation)

A passage with high $E_\text{harmonic}/\|w\|^2$ corresponds to modulation along the torus topology — the tonal center migrates globally.

Reasoning: $\omega_\text{harmonic} \in \ker(L_1)$ means simultaneously divergence-free ( $\partial_1 \omega = 0$ ) and curl-free ( $\partial_2^T \omega = 0$ ). The only non-zero flows on a torus satisfying both are those along the two non-contractible loops: circle of fifths or major third axis. The tonal center itself migrates — no local potential drop or eddy explains it; the flow is topological.

Examples:

Development of Schubert’s String Quintet D. 956, mvt. 2: E maj → F min → chain of major-third related keys — traverses the major-third non-contractible loop.
Development of Beethoven’s “Waldstein” Sonata Op. 53: C → E → A♭ → C, a complete circle of major thirds — one generator of $H_1$ .

Falsification: Find high $E_\text{harmonic}$ with only nearby key relations.

Caveats:

中文: “三个分量是结构性分解，不是价值判断。Exact 不等于’好'，harmonic 不等于’高级'。”

No value judgment: “Exact” does not mean better; “harmonic” does not mean more sophisticated.
Model sensitivity: Results depend on which Tonnetz variant is used, how durations are weighted, and how musical passages are segmented.
Validation required: Statistical testing on corpora (Bach chorales, common-practice sonatas, jazz standards) against baseline tonal-distance features is needed before these hypotheses can be accepted.

中文: “验证需要在真实音乐语料库上做统计检验。”

Section 6: Langlands Duality and the Tonnetz

中文: “数学中的 Langlands 对偶，交换根与余根，恰好对应音乐中的大小调对偶。这是一个严格的结构性对应，不是松散类比。”

The standard Tonnetz triangulates the torus $T^2 = \mathbb{R}^2 / \Lambda$ , where $\Lambda$ is the $A_2$ root lattice — the lattice generated by the simple roots of the Lie algebra $\mathfrak{sl}(3)$ .

Definition 6.1 (Langlands Dual (Root System Level))

For a semisimple Lie algebra $\mathfrak{g}$ with root system $\Phi$ , the coroot of $\alpha \in \Phi$ is $\alpha^\vee = 2\alpha / \langle \alpha, \alpha \rangle$ . The Langlands dual $\mathfrak{g}^\vee$ is obtained by $\Phi \leftrightarrow \Phi^\vee$ .

For the $A_2$ root system (which is simply laced — all roots have the same length), roots and coroots differ only by uniform scaling, so $A_2$ is self-dual.

Theorem 6.1 (Langlands Duality = Major/Minor Duality)

(Rietsch, 2024) On the

A_2

Tonnetz, the Langlands involution — interchange of roots and coroots — structurally corresponds to major/minor duality: every major triad maps to its parallel minor.

Proof.

(Sketch — see Rietsch 2024 §3.) The two simple root directions in the

A_2

lattice correspond to the major third axis (

+4

semitones) and the minor third axis (

+3

semitones) of the Tonnetz. Swapping roots with coroots exchanges these two axes. In the Tonnetz, edge

C \to E

(major third,

+4

) maps to

C \to E\flat

(minor third,

+3

), and edge

E \to G

(minor third,

+3

) maps to

E\flat \to G

(major third,

+4

). The triangle

\{C, E, G\}

(C major) maps to

\{C, E\flat, G\}

(C minor). This involution acts globally on the entire

A_2

lattice simultaneously, yielding the parallel-minor operation across all 24 triads.

\square

音乐联系

Euler 1739 → Rietsch 2024: A 285-year arc

中文: “Euler 在 1739 年画了一张音程网格。285 年后，我们才认出它里面藏着的代数拓扑结构。”

Euler drew his interval lattice as a tuning tool with no concept of simplicial homology, differential forms, or root systems. Yet the structure he wrote down is recognizable, 285 years later, as:

A simplicial triangulation of a torus with Betti numbers $(1,2,1)$
A space where the Hodge decomposition gives a tripartite analysis of harmony: gradients (cadences) + curls (PLR cycles) + topological flows (modulations)
A quotient of the $A_2$ root lattice — making its major/minor duality an instance of Langlands duality

The Hodge decomposition yields testable predictions: Bach chorales should have high $E_\text{exact}$ (strong cadences); Brahms intermezzos high $E_\text{coexact}$ (chromatic PLR); late Schubert high $E_\text{harmonic}$ (third-relation modulations). Whether these energy ratios outperform simpler tonal-distance metrics is an open empirical question — but the mathematical structure is exact.

Additional results (Rietsch 2024): A Tonnetz on a sphere encoding all major ninth chords; the transformation group of the seventh-chord Tonnetz is $S_5 \times \mathbb{Z}_{12}^4$ .

Scope note: The “Langlands duality” here operates at the level of root systems of $A_2$ — a very special case. It does not claim connections to the full Langlands program in number theory. The correspondence is precise but limited: a structural coincidence at the $A_2$ lattice level that turns out to have musical meaning.

Historical Timeline

Year	Development	Key figure(s)
1739	Euler’s interval lattice (Tentamen novae theoriae musicae)	Leonhard Euler
1941	Hodge theory on Riemannian manifolds	W.V.D. Hodge
1944	Discrete Hodge theory on simplicial complexes	Beno Eckmann
1998	Chicken-wire torus from parsimonious voice leading	Douthett & Steinbach
2011	HodgeRank: Hodge decomposition for statistical ranking	Jiang, Lim, Yao & Ye
2013	Tonnetz as simplicial complex; computational homology	Bigo & Andreatta
2020	Hodge Laplacians on Graphs (key reference, free PDF)	Lek-Heng Lim
2020	Generalized Tonnetze topology and homology	Jason Yust
2024	Langlands duality = major/minor duality on $A_2$ Tonnetz	Konstanze Rietsch

Academic References

Euler, L. (1739). Tentamen novae theoriae musicae. St. Petersburg.
Hodge, W.V.D. (1941). The Theory and Applications of Harmonic Integrals. Cambridge University Press.
Eckmann, B. (1944). “Harmonische Funktionen und Randwertaufgaben in einem Komplex.” Commentarii Mathematici Helvetici 17, 240–255.
Douthett, J. & Steinbach, P. (1998). “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition.” Journal of Music Theory 42(2), 241–263.
Friedman, J. (1998). “Computing Betti Numbers via Combinatorial Laplacians.” Algorithmica 21(4), 331–346.
Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. Free PDF: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html
Edelsbrunner, H. & Harer, J. (2010). Computational Topology: An Introduction. AMS.
Jiang, X., Lim, L.-H., Yao, Y. & Ye, Y. (2011). “Statistical Ranking and Combinatorial Hodge Theory.” Mathematical Programming 127, 203–244.
Bigo, L. & Andreatta, M. (2013). “Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes.” MCM 2013.
Frankel, T. (2012). The Geometry of Physics, 3rd ed. Cambridge University Press. Ch. 14.
Cannas, S. & Andreatta, M. (2018). “A Generalized Dual of the Tonnetz for Seventh Chords.” Bridges 2018, 301–308.
Lim, L.-H. (2020). “Hodge Laplacians on Graphs.” SIAM Review 62(3), 685–715. DOI: 10.1137/18M1223101. Free PDF: https://www.stat.uchicago.edu/~lekheng/work/hodge-graph.pdf — Primary reference for all discrete Hodge theory in EP14.
Yust, J. (2020). “Generalized Tonnetze and Zeitnetze, and the Topology of Music Concepts.” Journal of Mathematics and Music 14(2), 170–203.
Humphreys, J. (1972). Introduction to Lie Algebras and Representation Theory. Springer GTM 9.
Rietsch, K. (2024). “Generalisations of Euler’s Tonnetz on Triangulated Surfaces.” Journal of Mathematics and Music. DOI: 10.1080/17459737.2024.2362132

前置知识

Overview

Prerequisites

Preamble: What Is a “Hole”? Building Homology from Scratch

P.1 — The Naive Question

P.2 — The Loop Test

P.3 — Chains: The Algebraic Skeleton

P.4 — The Boundary Operator

P.5 — : The Fundamental Identity

P.6 — Cycles, Boundaries, and the Definition of a Hole

P.7 — Connecting to the Tonnetz Numbers

The Tonnetz as a Simplicial Complex

Section 1: The Hodge Star Operator ★

Section 2: Inner Product and Adjoint Operators

Section 3: The Hodge Laplacian

Section 4: The Hodge Decomposition Theorem

Section 5: Three Musical Hypotheses

Section 6: Langlands Duality and the Tonnetz

Historical Timeline

Academic References

P.4 — The Boundary Operator $\partial$

P.5 — $\partial^2 = 0$ : The Fundamental Identity