EP10

EP10: Negative Harmony and the Dihedral Group D₁₂

Reflection in ℤ₁₂, D₁₂ Group Action, Voice-Leading Symmetry

▶ 7:38 Abstract AlgebraGroup Theory

前置知识

EP04 All-Interval Rows and ℤ₁₂ EP01 Chord Space and the Torus

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Overview

The perfect authentic cadence — dominant seventh to tonic — has been the gravitational engine of Western tonal music for four centuries. It pulls inward: the leading tone $B$ rises to $C$ , the tritone $B$ – $F$ contracts, the bass falls a fifth.

中文: “但是，如果我告诉你，它有一个’镜像版本'——同样平滑，同样到达，但引力方向完全相反——你会相信吗？”

This mirror version is negative harmony: a reflection of the chromatic circle that preserves every voice-leading distance while reversing every harmonic direction. The operation is not a curiosity — it is one of the 12 reflections in the dihedral group $D_{12}$ , the full symmetry group of the regular 12-gon. Western tonal harmony has systematically explored the 12 rotations (transpositions) of $D_{12}$ while largely leaving the 12 reflections unexplored.

This episode is the first in a four-episode arc on Jacob Collier’s mathematics. The framework connects to EP04 ( $\mathbb{Z}_{12}$ and group actions) and forward to EP17 (Neo-Riemannian theory).

Prerequisites

The All-Interval Row and $\mathbb{Z}_{12}$ (EP04) — pitch classes, group actions, stabilizers
Overtone Series and Intervals (EP01) — just intervals, harmonic series
Basic group theory: groups, generators, relations, group homomorphisms

Definitions

Definition 10.1 (Pitch Class Group ℤ₁₂)

The pitch class group is the cyclic group $\mathbb{Z}_{12} = \{0, 1, 2, \ldots, 11\}$ under addition modulo 12. Each element represents one of the 12 chromatic pitch classes (C=0, C♯=1, …, B=11), with octave equivalence: pitch classes differing by a multiple of 12 are identified.

The transposition by $k$ semitones is the map $T_k: x \mapsto x + k \pmod{12}$ . Transpositions form the cyclic subgroup $\{T_0, T_1, \ldots, T_{11}\} \cong \mathbb{Z}_{12}$ of the group of pitch-class transformations.

Definition 10.2 (Pitch-Class Inversion)

The inversion (or reflection) through axis $n/2$ is the map

I_n : \mathbb{Z}_{12} \to \mathbb{Z}_{12}, \qquad I_n(x) = n - x \pmod{12}

Geometrically, $I_n$ reflects the 12-gon through the axis passing through the midpoint of pitch classes $n/2$ and $n/2 + 6$ (mod 12).

For negative harmony in C major, the axis passes through the midpoint between E (pitch class 4) and E♭ (pitch class 3), equivalently between G (7) and C (0) — giving $n = 7$ and the formula $I_7(x) = 7 - x \pmod{12}$ :

Pitch	PC	$I_7$ image	Image name
C	0	7	G
D	2	5	F
E	4	3	E♭
F	5	2	D
G	7	0	C
A	9	10	B♭
B	11	8	A♭

Note that C and G exchange, and E and E♭ exchange — as stated in the narration: “主音 C 和属音 G 互相交换；降E 和 E 也互换.”

Definition 10.3 (Dihedral Group D₁₂)

The dihedral group $D_{12}$ is the group of symmetries of a regular 12-gon. It has order 24 and is generated by two elements:

$r$ : rotation by $2\pi/12 = 30°$ (transposition by 1 semitone, $T_1$ )
$s$ : a fixed reflection (e.g., $I_0$ )

Subject to the relations:

r^{12} = e, \qquad s^2 = e, \qquad srs = r^{-1}

The 24 elements decompose as:

12 rotations: $\{e, r, r^2, \ldots, r^{11}\}$ — transpositions $T_0, \ldots, T_{11}$
12 reflections: $\{s, sr, sr^2, \ldots, sr^{11}\}$ — inversions $I_0, I_1, \ldots, I_{11}$

The relation $srs = r^{-1}$ encodes: a reflection composed with a rotation composed with the same reflection gives the inverse rotation — “reflection reverses the direction of rotation.”

Definition 10.4 (Fixed Points of a Reflection)

A pitch class $x \in \mathbb{Z}_{12}$ is a fixed point of $I_n$ if $I_n(x) = x$ , i.e., $n - x \equiv x \pmod{12}$ , i.e., $2x \equiv n \pmod{12}$ .

If $n$ is even, this has two solutions: $x = n/2$ and $x = n/2 + 6$ (mod 12). These are the two pitch classes on the reflection axis.
If $n$ is odd, the equation $2x \equiv n \pmod{12}$ has no solutions in $\mathbb{Z}_{12}$ (since 2 is not invertible mod 12). The axis passes between pitch classes, and there are no fixed pitch classes — all 12 form six exchange pairs.

For $I_7$ (negative harmony in C major), $n = 7$ is odd, so there are no fixed pitch classes. Every note maps to a different note.

Main Theorems

Theorem 10.1 (Distance Preservation Under Reflection)

For any two pitch classes $x, y \in \mathbb{Z}_{12}$ and any inversion $I_n$ ,

|I_n(x) - I_n(y)| \equiv |x - y| \pmod{12}

That is, the interval between two pitch classes is preserved by $I_n$ .

Proof.

Compute directly:

I_n(x) - I_n(y) = (n - x) - (n - y) = y - x \pmod{12}

Therefore $|I_n(x) - I_n(y)| = |y - x| = |x - y|$ (mod 12). $\square$

中文: “反射变换 $n$ 映射到 $7$ 减 $n$ 。两个音之间的距离： $n$ 减 $m$ 的绝对值。反射后的距离：$7$ 减 $n$ ，减去 $7$ 减 $m$ 的绝对值，等于 $m$ 减 $n$ 的绝对值。完全相等。”

Corollary: Voice leading — the pattern of semitone distances each voice moves through — is identical in a progression and its negative-harmony mirror. If V→I moves each voice by $d_1, d_2, \ldots, d_k$ semitones, then the mirrored progression moves each voice by $d_1, d_2, \ldots, d_k$ semitones as well.

Theorem 10.2 (Negative Harmony of G7 in C Major)

Under $I_7: x \mapsto 7 - x \pmod{12}$ , the G dominant seventh chord $\{7, 11, 2, 5\}$ (G–B–D–F) maps to $\{0, 8, 5, 2\}$ (C–A♭–F–D), which is an F minor sixth chord (Fm6) in first inversion.

The cadential motion G7→C (pitch classes $\{7,11,2,5\} \to \{0,4,7\}$ ) maps to Fm6→Cm ( $\{0,8,5,2\} \to \{0,3,7\}$ ).

Proof.

Apply $I_7$ componentwise to G7 = $\{7, 11, 2, 5\}$ :

I_7(7) = 0 \text{ (C)}, \quad I_7(11) = -4 \equiv 8 \text{ (A♭)}, \quad I_7(2) = 5 \text{ (F)}, \quad I_7(5) = 2 \text{ (D)}

The image set $\{0, 2, 5, 8\}$ = $\{C, D, F, A♭\}$ = Fm6 (F minor with added sixth: F–A♭–C–D).

For the tonic C major $\{0, 4, 7\}$ (C–E–G):

I_7(0) = 7 \text{ (G)}, \quad I_7(4) = 3 \text{ (E♭)}, \quad I_7(7) = 0 \text{ (C)}

Image: $\{0, 3, 7\}$ = C minor (Cm).

So G7→Cmaj becomes Fm6→Cm under negative harmony. The dominant-to-tonic authentic cadence becomes a subdominant-to-tonic plagal motion. $\square$

Theorem 10.3 (D₁₂ Group Structure)

The 24 elements of $D_{12}$ act on $\mathbb{Z}_{12}$ as follows:

Rotations $T_k(x) = x + k \pmod{12}$ : form the normal subgroup $\mathbb{Z}_{12} \trianglelefteq D_{12}$
Reflections $I_n(x) = n - x \pmod{12}$ : form a coset of $\mathbb{Z}_{12}$ in $D_{12}$ , not closed under composition

The composition law is:

T_k \circ T_j = T_{k+j}, \qquad I_m \circ I_n = T_{m-n}, \qquad T_k \circ I_n = I_{n+k}, \qquad I_m \circ T_k = I_{m-k}

Proof.

For rotations: $(T_k \circ T_j)(x) = T_k(x+j) = x+j+k = T_{k+j}(x)$ . Clear.

For two reflections: $(I_m \circ I_n)(x) = I_m(n-x) = m-(n-x) = m-n+x = T_{m-n}(x)$ . The composition of two reflections is a rotation by $m-n$ .

For rotation then reflection: $(T_k \circ I_n)(x) = T_k(n-x) = n-x+k = (n+k)-x = I_{n+k}(x)$ .

For reflection then rotation: $(I_m \circ T_k)(x) = I_m(x+k) = m-(x+k) = (m-k)-x = I_{m-k}(x)$ .

These are precisely the multiplication rules of $D_{12}$ with generators $r = T_1$ and $s = I_0$ , verifying $D_{12}$ structure. $\square$

Prop 10.1 (Neo-Riemannian Connection)

The neo-Riemannian group generated by the PLR transformations (Parallel, Leading-tone exchange, Relative) on major and minor triads is isomorphic to

D_{12}

. The

P

L

R

operations are inversions

I_n

for specific

n

depending on the triad root, composed with triad-type switching. This establishes negative harmony as a special case of neo-Riemannian theory.

Musical Connection

音乐联系

The two halves of $D_{12}$

Western tonal harmony from 1600–1900 systematically exploits the rotation subgroup of $D_{12}$ : modulation is $T_k$ , the circle of fifths is traversal of $T_7^n$ , all 24 major and minor keys are orbits under rotation. The reflection half remained largely implicit — present in the mirror counterpoint of Bach’s Art of Fugue, in Hindemith’s symmetrical constructions, in Bartók’s axis system — but was never given a unified theoretical name until Ernst Levy’s A Theory of Harmony (1985).

中文: “几百年来，西方和声学主要系统探索了旋转——大小调、转调、移调，都是旋转操作。反射操作相对较少被系统使用。负和声不是’另一种功能'——它是’另一种引力方向'。对称性的另一半。”

Why the reflection “works” musically

Theorem 10.1 guarantees that every voice-leading motion in the original progression has an equally smooth counterpart in the reflection. The perception of smoothness (Tymoczko 2011) depends only on the sum of voice-leading distances, not on direction. Thus negative harmony is, in a precise sense, the maximally voice-leading-equivalent reharmonization.

However, direction matters for harmonic function: the leading tone $B$ resolves upward to $C$ in G7→C because $B$ is a half-step below $C$ . Its mirror $A♭$ resolves downward to $G$ in $Fm6→Cm$ because $A♭$ is a half-step above $G$ . The tension is real and comparable, but the functional grammar (dominant→tonic vs. subdominant→tonic) is inverted.

中文: “两条路，长度相同，方向相反。这就是负和声的本质：距离不变，功能镜像。”

Historical trajectory: Levy → Coleman → Collier

Ernst Levy formulated the theory algebraically in the 1940s–1980s. Jazz saxophonist Steve Coleman brought it into improvisational practice in the 1980s. Jacob Collier demonstrated it publicly in 2017, causing Levy’s out-of-print book to sell 800 copies overnight. The viral dissemination of a mathematical idea about group symmetry through a YouTube video is itself remarkable.

中文: “1940年代，瑞士理论家Ernst Levy在极性理论中首先探索了这种对称性……2017年，Jacob Collier在一次访谈中演示了它。Levy那本只卖了32本的书，一夜之间卖出了800多本。”

Limits and Open Questions

Axis choice is culturally determined. The axis $n = 7$ (between C and G, axis at E♭/E) is natural for C major because it maps tonic to tonic and dominant to dominant. But the choice of axis depends entirely on the tonal center of the music. For atonal music, there is no canonical axis, and the operation becomes purely a structural transformation without harmonic-functional content.
Seventh chords and extensions. Theorem 10.2 shows G7 maps to Fm6. But jazz chords with ninths, elevenths, and thirteenths have more pitch classes; the reflected chord may not correspond to any standard chord label. The operation is always well-defined algebraically, but the resulting chord may be “unnameable” within standard nomenclature.
Higher-dimensional analogies. $D_{12}$ acts on the one-dimensional “chromatic circle.” Can one define an analogous dihedral action on the two-dimensional torus of keys (EP12), or on the three-dimensional tonnetz? Cohn (2012) and others have explored Riemannian extensions that suggest the answer is yes, but the resulting groups are larger and less tractable.
Empirical perceptual studies. The claim that negative-harmony cadences are perceived as “equally smooth but harmonically reversed” (Theorem 10.1 + musical analogy) has not been tested in controlled psychoacoustic experiments. Whether listeners unfamiliar with the concept perceive the symmetry, or only those trained in it, is an open empirical question.

Conjecture (Auditory Symmetry Detection)

Naive listeners (without music theory training) hearing a V–I cadence followed by its

I_7

-reflection will not spontaneously identify the mirror relationship, but will rate both cadences as approximately equally “complete” or “resolved.” This would confirm that voice-leading smoothness (distance-preserved) rather than harmonic function (direction-sensitive) drives the perception of cadential closure.

Academic References

Levy, E. (1985). A Theory of Harmony. SUNY Press.
Cohn, R. (1998). Introduction to Neo-Riemannian theory: A survey and a historical perspective. Journal of Music Theory, 42(2), 167–180.
Tymoczko, D. (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford University Press.
Hook, J. (2002). Uniform triadic transformations. Journal of Music Theory, 46(1–2), 57–126.
Straus, J. N. (2005). Introduction to Post-Tonal Theory (3rd ed.). Prentice Hall. Ch. 2 (Pitch-class sets and operations).
Collier, J. (2017). Negative harmony demonstration. YouTube (various interviews). [Publicly available.]
Fiore, T. M., & Satyendra, R. (2005). Generalized contextual groups. Music Theory Spectrum, 27(1), 99–120.