Module 3 · Weeks 6–7 · Graduate

Group Theory of Musical Transformations

The twelve pitch classes form a cyclic group. Transposition, inversion, and neo-Riemannian operations reveal the algebraic skeleton hidden inside every tonal chord progression.

1. Musical Transformations as Group Actions

We model the twelve pitch classes as \( \mathbb{Z}_{12} = \{0,1,\ldots,11\} \). The two fundamental operations are:

Transposition by n semitones:

\( T_n(p) = p + n \pmod{12} \)

Inversion about pitch class n:

\( I_n(p) = n - p \pmod{12} \)

Together, \( \{T_0, T_1, \ldots, T_{11}, I_0, I_1, \ldots, I_{11}\} \) form the dihedral group \( D_{12} \) of order 24. The 12 transpositions form a normal cyclic subgroup \( \mathbb{Z}_{12} \trianglelefteq D_{12} \), while adding any single inversion completes the semidirect product:

\( D_{12} \cong \mathbb{Z}_{12} \rtimes \mathbb{Z}_2 \)

This group acts faithfully on \( \mathbb{Z}_{12} \): each non-identity element moves at least one pitch class. Composition is \( T_m \circ T_n = T_{m+n} \) and \( I_m \circ I_n = T_{m-n} \).

2. The 24 Triads and D₁₂

A major triad is the set \( \{p, p+4, p+7\} \subseteq \mathbb{Z}_{12} \). Transposing gives 12 major triads; inverting a major triad yields a minor triad. Thus \( D_{12} \) acts transitively on the 24 consonant triads (12 major + 12 minor).

Orbit-stabilizer theorem:

\( |D_{12}| = |\text{Orb}(X)| \cdot |\text{Stab}(X)| = 24 \cdot 1 = 24 \)

The stabilizer of any triad is trivial—no non-identity element of \( D_{12} \) fixes a consonant triad setwise.

In serial (twelve-tone) music the four row forms are: Prime (T), Inversion (I), Retrograde (R), Retrograde-Inversion (RI). These too live inside \( D_{12} \) when we view a tone row as an ordered 12-tuple.

3. Neo-Riemannian Theory: P, L, R

Hugo Riemann's harmonic dualism, revived by Lewin, Hyer, and Cohn, defines three parsimonious voice-leading operations on triads. Each moves exactly one note by one or two semitones:

P (Parallel)

Keeps root and fifth, moves the third:

\( \{0,4,7\} \xrightarrow{P} \{0,3,7\} \)

C major ↔ C minor

L (Leading-tone)

Keeps minor third and fifth, moves root:

\( \{0,4,7\} \xrightarrow{L} \{11,4,7\} = \text{E minor} \)

C major ↔ E minor

R (Relative)

Keeps root and third, moves the fifth:

\( \{0,4,7\} \xrightarrow{R} \{9,0,4\} = \text{A minor} \)

C major ↔ A minor

Each operation is an involution (\( P^2 = L^2 = R^2 = \text{id} \)). Together they generate a group isomorphic to \( D_{12} \) acting on the 24 triads. The composition \( L \circ P \) has order 12, and \( P \circ R \) has order 12, giving the full dihedral structure.

Python Simulation: Group Actions on Pitch Classes

Visualising the dihedral group D₁₂ acting on pitch classes: transpositions vs inversions of C major, the PLR graph on the 24 consonant triads, the Z₁₂ Cayley table, and Messiaen modes of limited transposition.

Python

script.py178 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
from matplotlib.patches import FancyArrowPatch
from math import gcd

fig, axes = plt.subplots(2, 2, figsize=(14, 12))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes.flat:
    ax.set_facecolor('#0d0d24')

NOTE_NAMES = ['C','C#','D','Eb','E','F','F#','G','Ab','A','Bb','B']

# --- Plot 1: Clock-face showing T_n and I_n images of C major triad ---
ax1 = axes[0, 0]
ax1.set_aspect('equal')
ax1.set_xlim(-1.5, 1.5)
ax1.set_ylim(-1.5, 1.5)
ax1.axis('off')
ax1.set_title('T_n and I_n orbits of C major', color='#a0c4ff', fontsize=11, pad=8)

R = 1.1
for i in range(12):
    angle = np.pi/2 - 2*np.pi*i/12
    x, y = R*np.cos(angle), R*np.sin(angle)
    ax1.add_patch(plt.Circle((x, y), 0.12, color='#1e293b', zorder=2))
    ax1.text(x, y, NOTE_NAMES[i], ha='center', va='center', color='#94a3b8',
             fontsize=7, fontweight='bold', zorder=3)

def triad_angles(root, quality):
    if quality == 'maj':
        pcs = [(root)%12, (root+4)%12, (root+7)%12]
    else:
        pcs = [(root)%12, (root+3)%12, (root+7)%12]
    return pcs

def draw_triangle(ax, pcs, color, alpha, lw=1.5, r=1.1):
    pts = []
    for pc in pcs:
        ang = np.pi/2 - 2*np.pi*pc/12
        pts.append((r*np.cos(ang), r*np.sin(ang)))
    tri = plt.Polygon(pts, fill=False, edgecolor=color, linewidth=lw, alpha=alpha, zorder=4)
    ax.add_patch(tri)

c_maj = [0, 4, 7]
# T_n images (blue faded)
for n in range(12):
    pcs = [(p+n)%12 for p in c_maj]
    draw_triangle(ax1, pcs, '#3b82f6', 0.18 if n != 0 else 0.0)
# I_n images (green faded)
for n in range(12):
    pcs = [(n-p)%12 for p in c_maj]
    draw_triangle(ax1, pcs, '#22c55e', 0.18)
# C major itself bold blue
draw_triangle(ax1, c_maj, '#60a5fa', 1.0, lw=2.5)
ax1.text(0, -1.38, 'Blue = T_n  |  Green = I_n  |  Bold = C major',
         ha='center', color='#64748b', fontsize=7)

# --- Plot 2: PLR graph on 24 triads ---
ax2 = axes[0, 1]
ax2.set_facecolor('#0d0d24')
ax2.set_aspect('equal')
ax2.axis('off')
ax2.set_title('PLR graph: orbit of C major under neo-Riemannian group', color='#a0c4ff', fontsize=10, pad=8)

# 24 triads: 12 major (outer ring) + 12 minor (inner ring)
n_triads = 24
node_pos = {}
for i in range(12):
    angle = np.pi/2 - 2*np.pi*i/12
    node_pos[('maj', i)] = (1.2*np.cos(angle), 1.2*np.sin(angle))
for i in range(12):
    angle = np.pi/2 - 2*np.pi*i/12 + np.pi/12
    node_pos[('min', i)] = (0.65*np.cos(angle), 0.65*np.sin(angle))

def plr_edges():
    edges = []
    for root in range(12):
        # P: major <-> minor same root
        edges.append((('maj', root), ('min', root), 'P', '#ef4444'))
        # L: C major <-> E minor  (maj root) <-> (min (root+4)%12)
        edges.append((('maj', root), ('min', (root+4)%12), 'L', '#f59e0b'))
        # R: C major <-> A minor  (maj root) <-> (min (root+9)%12)
        edges.append((('maj', root), ('min', (root+9)%12), 'R', '#22c55e'))
    return edges

edges = plr_edges()
for (n1, n2, label, color) in edges:
    x1, y1 = node_pos[n1]
    x2, y2 = node_pos[n2]
    ax2.plot([x1, x2], [y1, y2], color=color, alpha=0.25, linewidth=0.7, zorder=1)

for (q, root), (x, y) in node_pos.items():
    color = '#3b82f6' if q == 'maj' else '#22c55e'
    ax2.add_patch(plt.Circle((x, y), 0.085, color=color, zorder=3))
    ax2.text(x, y, NOTE_NAMES[root]+('m' if q == 'min' else ''),
             ha='center', va='center', color='white', fontsize=5, fontweight='bold', zorder=4)

legend_elements = [
    mpatches.Patch(color='#3b82f6', label='Major'),
    mpatches.Patch(color='#22c55e', label='Minor'),
    mpatches.Patch(color='#ef4444', label='P edges'),
    mpatches.Patch(color='#f59e0b', label='L edges'),
    mpatches.Patch(color='#22c55e', label='R edges'),
]
ax2.legend(handles=legend_elements, loc='lower right', fontsize=6,
           facecolor='#1e293b', edgecolor='#334155', labelcolor='white')
ax2.set_xlim(-1.5, 1.5)
ax2.set_ylim(-1.5, 1.5)

# --- Plot 3: Cayley table heatmap for Z_12 ---
ax3 = axes[1, 0]
Z12 = np.array([[(i+j)%12 for j in range(12)] for i in range(12)])
im = ax3.imshow(Z12, cmap='plasma', aspect='auto', vmin=0, vmax=11)
ax3.set_xticks(range(12))
ax3.set_yticks(range(12))
ax3.set_xticklabels(range(12), color='#94a3b8', fontsize=8)
ax3.set_yticklabels(range(12), color='#94a3b8', fontsize=8)
ax3.set_title('Cayley table for Z_12 (addition mod 12)', color='#a0c4ff', fontsize=11, pad=8)
ax3.set_xlabel('j', color='#64748b', fontsize=9)
ax3.set_ylabel('i', color='#64748b', fontsize=9)
for i in range(12):
    for j in range(12):
        ax3.text(j, i, str((i+j)%12), ha='center', va='center',
                 color='white', fontsize=6, fontweight='bold')
cbar = fig.colorbar(im, ax=ax3, shrink=0.85)
cbar.ax.yaxis.set_tick_params(color='#64748b')
plt.setp(cbar.ax.yaxis.get_ticklabels(), color='#94a3b8', fontsize=7)

# --- Plot 4: Modes of limited transposition ---
ax4 = axes[1, 1]
ax4.set_facecolor('#0d0d24')
ax4.axis('off')
ax4.set_title('Modes of limited transposition (orbit sizes)', color='#a0c4ff', fontsize=11, pad=8)

modes = [
    ('Whole-tone', 2, '#3b82f6'),
    ('Octatonic', 3, '#8b5cf6'),
    ('Chromatic', 1, '#ef4444'),
]
pie_axes_coords = [
    [0.05, 0.15, 0.27, 0.65],
    [0.37, 0.15, 0.27, 0.65],
    [0.69, 0.15, 0.27, 0.65],
]
for idx, (name, n_orbits, color) in enumerate(modes):
    sub_ax = fig.add_axes(
        [axes[1,1].get_position().x0 + pie_axes_coords[idx][0]*axes[1,1].get_position().width,
         axes[1,1].get_position().y0 + pie_axes_coords[idx][1]*axes[1,1].get_position().height,
         pie_axes_coords[idx][2]*axes[1,1].get_position().width,
         pie_axes_coords[idx][3]*axes[1,1].get_position().height]
    )
    sub_ax.set_facecolor('#0d0d24')
    sizes = [1]*n_orbits
    colors_pie = [color] + ['#1e293b']*(n_orbits-1) if n_orbits > 1 else [color]
    sub_ax.pie(sizes, colors=colors_pie, wedgeprops={'edgecolor': '#0a0a1a', 'linewidth': 1.5})
    sub_ax.set_title(name + chr(10) + str(n_orbits) + ' orbit' + ('s' if n_orbits != 1 else ''),
                     color='#94a3b8', fontsize=8, pad=4)

plt.suptitle('Group Theory of Musical Transformations', color='#e2e8f0',
             fontsize=14, fontweight='bold', y=1.01)
plt.tight_layout(pad=1.5)
plt.savefig('output.png', dpi=150, bbox_inches='tight',
            facecolor='#0a0a1a', edgecolor='none')
plt.close()

print('Group Theory Simulation Results:')
print('  D_12 order: 24 (12 transpositions + 12 inversions)')
print('  Orbit of C major under D_12: all 24 consonant triads')
print('  Stabilizer of any consonant triad: trivial (order 1)')
print('  Orbit-stabilizer: 24 = 24 x 1 confirmed')
print('  Whole-tone scale orbits under Z_12: 2')
print('  Octatonic scale orbits under Z_12: 3')
print('  Z_12 Cayley table: addition mod 12, cyclic structure visible')
print('  PLR graph: 24 nodes, edges P(red) L(amber) R(green)')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

4. Interactive Tonnetz Explorer

The Tonnetz is a planar lattice where the horizontal axis moves by fifths (7 semitones) and the vertical axis by major thirds (4 semitones). Triads appear as triangles. Click any hexagon to set a new root, or use P/L/R to navigate.

Current: C[C, E, G]

Transformation history:

C major

5. Lewin's Generalized Interval Systems

David Lewin unified musical interval theory in his 1987 monograph by defining a Generalized Interval System (GIS) as a triple\( (S, G, \text{int}) \) where:

\( S \) is a set of musical objects (pitch classes, time points, etc.)
\( G \) is a group (the “interval group”)
\( \text{int}: S \times S \to G \) satisfies\( \text{int}(r,t) = \text{int}(r,s) \cdot \text{int}(s,t) \)

Torsor interpretation:

A GIS is precisely a principal homogeneous space (torsor) for \( G \). The group \( G \) acts simply transitively on \( S \): for any\( s, t \in S \) there exists a unique \( g \in G \) with \( g \cdot s = t \). This is the algebraic meaning of “interval.”

The power of Lewin's framework: the same formalism describes pitch-class intervals (\( G = \mathbb{Z}_{12} \)), duration ratios (\( G = \mathbb{Q}^+ \)), and even timbral distances when \( G \) is a continuous group.

6. Messiaen's Modes of Limited Transposition

Olivier Messiaen catalogued pitch-class sets that are invariant under some non-trivial transposition \( T_k \) with \( 0 < k < 12 \). The stabilizer of such a set under the transposition group \( \mathbb{Z}_{12} \) is a non-trivial subgroup.

Example — the whole-tone scale:

\( W = \{0,2,4,6,8,10\} \). Its stabilizer is\( \text{Stab}(W) = \{T_0, T_2, T_4, T_6, T_8, T_{10}\} \cong \mathbb{Z}_6 \).

Orbit-stabilizer theorem:

\( |\text{Orb}(W)| = \frac{|\mathbb{Z}_{12}|}{|\text{Stab}(W)|} = \frac{12}{6} = 2 \)

There are only 2 distinct whole-tone scales—the minimum possible number of transpositions.

Messiaen's seven modes correspond to subgroups of \( \mathbb{Z}_{12} \) of orders 2, 3, 4, and 6. The octatonic scale (diminished) has stabilizer \( \mathbb{Z}_3 \) giving 3 transpositions, while the augmented scale has \( \mathbb{Z}_4 \) giving 3 transpositions as well. No scale with a stabilizer \( \mathbb{Z}_1 \) (trivial) qualifies—such scales have the full 12 transpositions.