EP01

EP01: Chord Space and the Torus

Tonnetz, PLR变换与轨形

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EP08 Shepard Tones and the Tritone Paradox EP10 Negative Harmony and the Dihedral Group D₁₂ EP14 Tonnetz Hodge Duality

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Overview

If I told you that moving from a C major chord to an A minor chord takes exactly one step in a certain geometric space — and that this space has the shape of a donut — would you believe it?

This episode develops the mathematical backbone of Neo-Riemannian theory: three involutory chord transformations (P, L, R) that generate a dihedral group acting on 24 major/minor triads, a torus-shaped visualization called the Tonnetz, and Dmitri Tymoczko’s general theory of voice-leading orbifolds.

中文: “从数学角度看，这三种变换生成了一个群。具体来说，是二面体群D12，一共有24个元素。这个群作用在24个大小三和弦上，而且是simply transitive的。”

Prerequisites

Basic group theory: groups, generators, homomorphisms
Topology: quotient spaces, torus as product of circles
Pitch-class topology (EP08) — $\mathbb{Z}_{12}$ as the circle of pitch classes
Motif networks (EP17) — graph models of musical structure

Definitions

Definition 1.1 (Major and Minor Triad)

A triad is an ordered triple of pitch classes $(r, m, \ell) \in \mathbb{Z}_{12}^3$ satisfying the spacing constraint:

Major triad rooted at $r$ : pitch classes $\{r,\; r+4,\; r+7\} \pmod{12}$
Minor triad rooted at $r$ : pitch classes $\{r,\; r+3,\; r+7\} \pmod{12}$

There are 12 major triads and 12 minor triads, giving a set $\mathcal{T}$ of 24 triads. We write $[r,+]$ for the major triad rooted at $r$ and $[r,-]$ for the minor triad rooted at $r$ .

Definition 1.2 (PLR Transformations)

Three functions $P, L, R : \mathcal{T} \to \mathcal{T}$ are defined by the rule: each transformation preserves two common tones and moves the remaining tone by the minimal semitone distance.

P (Parallel): changes mode while keeping root fixed.
$P([r,+]) = [r,-], \qquad P([r,-]) = [r,+]$ Concretely: in $[r,+] = \{r, r{+}4, r{+}7\}$ , the note $r{+}4$ moves by one semitone to $r{+}3$ .
L (Leading-tone exchange): in a major triad, the root moves down by semitone; in minor, the fifth moves up by semitone.
$L([r,+]) = [r+4,-], \qquad L([r,-]) = [r+8,+]$ (indices mod 12)
R (Relative): relates a major triad to its relative minor (sharing the same key signature).
$R([r,+]) = [r+9,-], \qquad R([r,-]) = [r+3,+]$

Each of P, L, R is an involution: $P^2 = L^2 = R^2 = \mathrm{id}$ .

Visual: C major $\{C, E, G\}$ under P becomes C minor $\{C, E\flat, G\}$ — only E moves, dropping one semitone to E $\flat$ . Under R it becomes A minor $\{A, C, E\}$ — only G moves, ascending a whole tone to A.

Definition 1.3 (Neo-Riemannian Group)

The Neo-Riemannian group $\mathcal{R}$ is the subgroup of $\mathrm{Sym}(\mathcal{T})$ generated by $\{P, L, R\}$ :

\mathcal{R} = \langle P, L, R \rangle \leq \mathrm{Sym}(\mathcal{T})

Since $P = (LR)^3 \cdot L$ (a provable relation), $\mathcal{R}$ is already generated by $L$ and $R$ alone. As a group, $\mathcal{R} \cong D_{24}$ , the dihedral group of order 24.

Definition 1.4 (Tonnetz)

The Tonnetz (German: “tone network”) is a graph $G = (V, E)$ where:

Vertices $V = \mathbb{Z}_{12}$ : the 12 pitch classes.
Edges $E$ : pairs $\{a, b\}$ with $|a - b| \equiv 7 \pmod{12}$ (perfect fifth), $|a - b| \equiv 4 \pmod{12}$ (major third), or $|a - b| \equiv 3 \pmod{12}$ (minor third).

Each triangular face of this planar graph corresponds to a triad: upward-pointing triangles are major triads, downward-pointing triangles are minor triads.

A PLR transformation corresponds to a flip across a shared edge: P flips across the minor-third edge, L flips across the major-third edge, R flips across the perfect-fifth edge.

Definition 1.5 (Voice-Leading Distance)

The voice-leading distance between two chords $A = \{a_1, \ldots, a_n\}$ and $B = \{b_1, \ldots, b_n\}$ (as multisets in $\mathbb{R}/12\mathbb{Z}$ ) is

d_{\mathrm{VL}}(A, B) = \min_{\sigma \in S_n} \sum_{i=1}^n \min_{k \in \mathbb{Z}} |a_i - b_{\sigma(i)} + 12k|

where the outer minimum is over all bijections $\sigma$ pairing voices, and the inner minimum realizes the shortest semitone displacement respecting octave equivalence.

Main Theorems

Theorem 1.1 (PLR Are Involutions)

Each of the three transformations $P$ , $L$ , $R$ is an involution on $\mathcal{T}$ :

P^2 = L^2 = R^2 = \mathrm{id}_{\mathcal{T}}

Proof.

We verify for $P$ ; the argument for $L$ and $R$ is analogous.

By Definition 1.2, $P([r,+]) = [r,-]$ and $P([r,-]) = [r,+]$ . Thus $P^2([r,+]) = P(P([r,+])) = P([r,-]) = [r,+]$ , and similarly $P^2([r,-]) = [r,-]$ . Since $P^2$ fixes every element of $\mathcal{T}$ , we have $P^2 = \mathrm{id}$ . $\square$

Theorem 1.2 (Simple Transitivity of D₁₂ on Triads)

The Neo-Riemannian group

\mathcal{R} \cong D_{24}

acts simply transitively on

\mathcal{T}

: for any two triads

s, t \in \mathcal{T}

, there exists a unique group element

g \in \mathcal{R}

such that

g \cdot s = t

Proof.

Transitivity (any triad reachable from any other): The 24 triads form a single orbit under $\mathcal{R}$ . Starting from $[C,+]$ , repeated application of $L$ and $R$ generates a connected path through all 24 triads — this can be verified by explicitly tracing the “Chicken-wire” cycle $L, R, L, R, \ldots$ (the 24-cycle that returns to the start), which visits all elements of $\mathcal{T}$ .

Free action (no nonidentity element fixes any triad): $|\mathcal{R}| = 24 = |\mathcal{T}|$ . By the orbit-stabilizer theorem,

|\mathcal{R}| = |\mathrm{Orb}(s)| \cdot |\mathrm{Stab}(s)|

Since

|\mathrm{Orb}(s)| = 24

(transitivity) and

|\mathcal{R}| = 24

, we get

|\mathrm{Stab}(s)| = 1

, meaning only the identity fixes any element.

\square

Visual: Place all 24 triads as vertices of a 24-gon. Draw edges for each generator: blue edges for $L$ , red edges for $R$ , green edges for $P$ . The result is a Cayley graph of $D_{24}$ — every vertex has exactly three colored edges emanating from it.

Theorem 1.3 (Tonnetz Torus Identification)

Under enharmonic equivalence (identifying $G\sharp = A\flat$ , etc.), the Tonnetz wraps into a torus $T^2 = S^1 \times S^1$ .

Specifically:

Ascending by a perfect fifth (7 semitones) twelve times returns to the starting pitch class: $0 \xrightarrow{+7} 7 \xrightarrow{+7} 2 \xrightarrow{+7} \cdots \xrightarrow{+7} 0 \pmod{12}$ . This identifies opposite vertical edges.
Ascending by a major third (4 semitones) three times traverses the augmented triad and returns: $0 \xrightarrow{+4} 4 \xrightarrow{+4} 8 \xrightarrow{+4} 0 \pmod{12}$ . Combined with the $\mathbb{Z}_{12}$ structure, this identifies opposite horizontal edges.

The resulting quotient space is homeomorphic to the torus $T^2$ .

Proof.

The Tonnetz can be given coordinates in

\mathbb{Z}_{12} \times \mathbb{Z}_{12}

(or more precisely, in

\mathbb{Z}_{12} \times \mathbb{Z}_3

with the diagonal fifth-direction). The key point is that the lattice of pitch classes generated by the intervals

\{3, 4, 7\}

modulo 12 is a quotient

\mathbb{Z}^2 / \Lambda

for some sublattice

\Lambda

. The identifications imposed by

\mathbb{Z}_{12}

(the group of pitch classes) cause both pairs of opposite sides of the fundamental domain to be glued with the same orientation, producing

T^2

rather than

S^2

K

(the Klein bottle). The formal identification is: the fifth-direction generator

e_1

satisfies

12 e_1 = 0

, and the third-direction generator

e_2

satisfies

3 e_2 = 0

within the fundamental domain, with both identifications orientation-preserving.

\square

Theorem 1.4 (Tymoczko Orbifold)

The space of $n$ -voice chords (unordered multisets of $n$ pitch classes in $\mathbb{R}/12\mathbb{Z}$ ) is the orbifold

\mathcal{O}_n = T^n / S_n

where $T^n = (\mathbb{R}/12\mathbb{Z})^n$ is the $n$ -torus of ordered $n$ -tuples, and $S_n$ acts by permuting coordinates.

Special cases:

$n = 2$ : $T^2 / S_2 \cong$ Möbius band (with boundary = unison/octave dyads)
$n = 3$ : $T^3 / S_3 \cong$ a twisted triangular prism with three singular edges

Voice-leading distance $d_{\mathrm{VL}}$ equals the path length in $\mathcal{O}_n$ under the quotient metric.

Proof.

(Sketch) $T^n$ carries the flat metric from $(\mathbb{R}/12\mathbb{Z})^n$ . The action of $S_n$ by coordinate permutation is an isometric group action, so the quotient $T^n/S_n$ inherits a well-defined metric (on the regular part) and becomes an orbifold at fixed points of the $S_n$ -action (i.e., at chords with repeated notes). The geodesic distance in $\mathcal{O}_n$ between two points $[A]$ and $[B]$ is

d_{\mathcal{O}_n}([A],[B]) = \min_{\sigma \in S_n} d_{T^n}(A, \sigma(B))

which is exactly the formula for $d_{\mathrm{VL}}$ in Definition 1.5. The homeomorphism type of $\mathcal{O}_2$ follows from the standard identification of the quotient of $[0,12)^2$ by $(a,b) \sim (b,a)$ as a Möbius band when one accounts for the periodic boundary. $\square$

Prop 1.1 (Parsimony of PLR)

Each PLR transformation moves exactly one pitch class by one or two semitones and fixes the other two. More precisely:

$P$ moves one note by exactly 1 semitone (the third of the triad).
$L$ moves one note by exactly 1 semitone (the root of a major triad, or the fifth of a minor triad).
$R$ moves one note by exactly 2 semitones (the fifth of a major triad, or the root of a minor triad).

Consequently, $d_{\mathrm{VL}}(t, P(t)) \leq 1$ , $d_{\mathrm{VL}}(t, L(t)) \leq 1$ , $d_{\mathrm{VL}}(t, R(t)) \leq 2$ for all $t \in \mathcal{T}$ .

Proof.

Direct computation. For

P

: in

[C,+] = \{0,4,7\}

, we have

P([C,+]) = [C,-] = \{0,3,7\}

. The note 4 (E) moves to 3 (E

\flat

), a displacement of 1 semitone; notes 0 (C) and 7 (G) are fixed. For

L

L([C,+]) = [E,-] = \{4,7,11\}

. Note 0 (C) moves to 11 (B), a displacement of 1 semitone downward; notes 4 and 7 are fixed. For

R

R([C,+]) = [A,-] = \{9,0,4\}

. Note 7 (G) moves to 9 (A), a displacement of 2 semitones; notes 0 and 4 are fixed.

\square

Musical Connection

音乐联系

The Tristan Chord and the Tonnetz

Wagner’s opera Tristan und Isolde (1865) opens with the famous “Tristan chord” $\{F, B, D\sharp, G\sharp\}$ , which resisted definitive functional-harmonic analysis for over a century. Neo-Riemannian theory offers an alternative reading.

The chord progression in the opening bars — Tristan chord → E major → (continuation) — can be tracked as a path on the Tonnetz. The Tristan chord is an instance of a “French augmented sixth” spelling, but on the Tonnetz it occupies a position adjacent to multiple triads. The initial resolution traces a short geodesic on $T^2$ .

中文: “在Tonnetz上，我们可以清晰地追踪它的路径。这些和弦在环面上形成了一个优美的轨迹，绕着Torus表面滑行。”

Romantic-era chromatic harmony exploited exactly the smooth voice-leading that the Tonnetz geometry quantifies. Schubert’s “Wanderer” Fantasy and Liszt’s harmonic progressions follow paths through the Tonnetz that minimize total voice-leading displacement — instinctively approaching the geodesics that Tymoczko’s orbifold formalism later made rigorous.

Practical use in analysis: music theorist Brian Hyer (1995) and Richard Cohn (1996–1998) revived 19th-century Riemannian ideas precisely because PLR transformations capture how chromatic composers modulated through unrelated keys with minimal voice-leading effort. The mathematical structure predicts which progressions will sound “smooth” even when harmonically distant in the traditional functional sense.

Limits and Open Questions

Extension to seventh chords: PLR theory is cleanest for triads. Extending to dominant seventh chords, half-diminished sevenths, etc. requires defining additional generators and working in a larger group. The resulting orbifold $T^4/S_4$ is considerably more complex.
Metric vs. combinatorial voice-leading: Definition 1.5 gives a metric, but actual voice-leading in score involves register (which octave each note is in) and voice-crossing constraints. A full model requires lifting from $\mathbb{R}/12\mathbb{Z}$ to $\mathbb{R}$ and imposing ordering constraints.
Perception vs. geometry: It remains an open empirical question whether listeners perceive voice-leading distance as the geodesic metric in $\mathcal{O}_n$ . Psychoacoustic experiments (e.g., Callender, Quinn, Tymoczko 2008) support the orbifold model, but individual variation and musical context complicate direct verification.
Higher-dimensional orbifold topology: The orbifold $T^n/S_n$ has non-trivial orbifold fundamental group and orbifold Euler characteristic. For $n \geq 4$ , the topology of these spaces is not yet fully classified in music-theoretically useful terms.

Conjecture (Optimality of PLR in Voice-Leading)

Among all involutory chord transformations that preserve two common tones in a triad, PLR uniquely minimizes the maximum single-voice displacement over all 24 triads. No other set of three involutions achieves the same parsimony bound simultaneously for all elements of

\mathcal{T}

Academic References

Cohn, R. (1997). Neo-Riemannian operations, parsimonious trichords, and their Tonnetz representations. Journal of Music Theory, 41(1), 1–66.
Tymoczko, D. (2006). The geometry of musical chords. Science, 313(5783), 72–74.
Tymoczko, D. (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford University Press.
Hyer, B. (1995). Reimag(in)ing Riemann. Journal of Music Theory, 39(1), 101–138.
Callender, C., Quinn, I., & Tymoczko, D. (2008). Generalized voice-leading spaces. Science, 320(5874), 346–348.
Cohn, R. (2012). Audacious Euphony: Chromaticism and the Triad’s Second Nature. Oxford University Press.
Riemann, H. (1880). Skizze einer neuen Methode der Harmonielehre. Leipzig: Breitkopf und Härtel.