Module 6 · Weeks 12–13 · Research

Topology of Musical Space

An \(n\)-voice chord lives in the orbifold \(\mathcal{O}_n = (\mathbb{R}/12\mathbb{Z})^n / S_n\), where \(S_n\) permutes voices and \(\mathbb{R}/12\mathbb{Z}\) wraps pitch classes mod the octave. Voice leading between two chords is a path in this space; the shortest path — the geodesic — is the most efficient way to move every voice from one chord to the next.

Interactive Orbifold Explorer

Root

Type

Mathematical Framework

Following Tymoczko (2006, 2011), the space of \(n\)-note chords modulo octave equivalence and voice permutation is the orbifold

\[\mathcal{O}_n \;=\; \bigl(\mathbb{R}/12\mathbb{Z}\bigr)^n \,/\, S_n\]

This is a singular space: the fixed points of the \(S_n\) action (chords with repeated pitch classes) form singular strata of lower dimension. For \(n = 2\), the orbifold is a Möbius strip; for \(n = 3\), it is a cone over the 2-simplex with identifications.

Voice-Leading Norm

The \(L^1\) voice-leading distance between chords \(X = (x_1, \ldots, x_n)\) and \(Y = (y_1, \ldots, y_n)\) is

\[d_{VL}(X,Y) \;=\; \min_{\sigma \in S_n} \sum_{i=1}^{n} \bigl| x_i - y_{\sigma(i)} \bigr|_{12}\]

where \(|a|_{12} = \min(|a|, 12 - |a|)\) is distance on the circle \(\mathbb{R}/12\mathbb{Z}\).

Key Results

Efficient voice leading corresponds to short geodesics in \(\mathcal{O}_n\) (Tymoczko, A Geometry of Music, 2011).
Completely even chords (e.g. augmented triads, diminished sevenths) sit at the singular points of the orbifold.
The Callender–Quinn–Tymoczko framework (Science, 2008) extends to continuous pitch spaces and arbitrary equivalence relations (transposition, inversion, cardinality).
Persistent homology applied to the orbifold reveals clustering of common-practice harmony in tonal music.

References

Tymoczko, D. (2006). “The Geometry of Musical Chords.” Science 313(5783), 72–74.
Tymoczko, D. (2011). A Geometry of Music. Oxford University Press.
Callender, C., Quinn, I., & Tymoczko, D. (2008). “Generalized Voice-Leading Spaces.” Science 320(5874), 346–348.

Simulation: Voice-Leading Distances in Chord Orbifolds

Compute and visualise the L1 voice-leading distance matrix across all major and minor triads, trace smooth paths through the orbifold, and project the 24 triads onto sum/difference coordinates.

Python

script.py134 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from itertools import permutations

# ---------- helpers ----------
NOTE_NAMES = ['C','C#','D','D#','E','F','F#','G','G#','A','A#','B']

def triad(root, kind):
    if kind == 'major':
        return [root, (root+4)%12, (root+7)%12]
    else:
        return [root, (root+3)%12, (root+7)%12]

def vl_dist(a, b):
    best = float('inf')
    for perm in permutations(b):
        d = sum(min(abs(x-y), 12-abs(x-y)) for x,y in zip(a,perm))
        if d < best:
            best = d
    return best

majors = [(NOTE_NAMES[r]+' maj', triad(r,'major')) for r in range(12)]
minors = [(NOTE_NAMES[r]+' min', triad(r,'minor')) for r in range(12)]
all24  = majors + minors

# ---------- Plot 1: 12x12 major-to-major heatmap ----------
mat12 = np.array([[vl_dist(majors[i][1], majors[j][1]) for j in range(12)] for i in range(12)], dtype=float)

# ---------- Plot 2: histogram of all 24x24 VL distances ----------
dists_all = []
for i in range(24):
    for j in range(i+1, 24):
        dists_all.append(vl_dist(all24[i][1], all24[j][1]))

# ---------- Plot 3: I-vi-IV-V-I in C, per-voice movement ----------
prog_labels = ['I (C)', 'vi (Am)', 'IV (F)', 'V (G)', 'I (C)']
prog_chords = [
    triad(0,'major'),   # C major
    triad(9,'minor'),   # A minor
    triad(5,'major'),   # F major
    triad(7,'major'),   # G major
    triad(0,'major'),   # C major
]

def closest_perm(prev, curr):
    best_perm, best_d = None, float('inf')
    for perm in permutations(curr):
        d = sum(min(abs(x-y), 12-abs(x-y)) for x,y in zip(prev,perm))
        if d < best_d:
            best_d, best_perm = d, list(perm)
    return best_perm

voice_paths = [list(prog_chords[0])]
for k in range(1, len(prog_chords)):
    voice_paths.append(closest_perm(voice_paths[-1], prog_chords[k]))
voice_paths = np.array(voice_paths)  # shape (5, 3)

# ---------- Plot 4: 2D projection (sum = x, diff = y) ----------
def proj(voices):
    s = sorted(voices)
    return (s[0]+s[1]+s[2])/3.0, (s[2]-s[0])

# ---------- Figure ----------
fig, axes = plt.subplots(2, 2, figsize=(13, 10))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')

# -- ax[0,0]: heatmap --
ax = axes[0,0]
im = ax.imshow(mat12, cmap='plasma', aspect='auto')
ax.set_xticks(range(12))
ax.set_yticks(range(12))
ax.set_xticklabels([NOTE_NAMES[r] for r in range(12)], fontsize=7, color='#94a3b8', rotation=45)
ax.set_yticklabels([NOTE_NAMES[r] for r in range(12)], fontsize=7, color='#94a3b8')
ax.set_title('VL Distance: All Major Triads (12x12)', color='#93c5fd', fontsize=11)
fig.colorbar(im, ax=ax).ax.yaxis.set_tick_params(color='#94a3b8')
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')

# -- ax[0,1]: histogram --
ax = axes[0,1]
ax.hist(dists_all, bins=range(0, max(dists_all)+2), color='#6366f1', edgecolor='#a5b4fc', alpha=0.85, align='left')
ax.set_xlabel('VL Distance (semitones)', color='#94a3b8')
ax.set_ylabel('Count', color='#94a3b8')
ax.set_title('Pairwise VL Distances: 24 Major+Minor Triads', color='#93c5fd', fontsize=11)
ax.tick_params(colors='#94a3b8')
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')
mean_d = np.mean(dists_all)
ax.axvline(mean_d, color='#fbbf24', linestyle='--', linewidth=1.5, label=f'mean={mean_d:.1f}')
ax.legend(facecolor='#0a0a1a', labelcolor='#e2e8f0', fontsize=8)
print(f'Mean pairwise VL distance (24 triads): {mean_d:.2f} semitones')

# -- ax[1,0]: per-voice path --
ax = axes[1,0]
voice_labels = ['Soprano', 'Alto', 'Bass']
colors_v = ['#60a5fa', '#34d399', '#f87171']
xs = range(5)
for v in range(3):
    ax.plot(xs, voice_paths[:, v], marker='o', color=colors_v[v], linewidth=2, label=voice_labels[v])
ax.set_xticks(range(5))
ax.set_xticklabels(prog_labels, fontsize=8, color='#94a3b8')
ax.set_ylabel('Pitch class', color='#94a3b8')
ax.set_title('Voice Movement: I-vi-IV-V-I (C major)', color='#93c5fd', fontsize=11)
ax.tick_params(colors='#94a3b8')
ax.legend(facecolor='#0a0a1a', labelcolor='#e2e8f0', fontsize=8)
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')
total_vl = sum(vl_dist(list(prog_chords[k-1]), list(prog_chords[k])) for k in range(1,5))
print(f'Total VL cost of I-vi-IV-V-I: {total_vl} semitones')

# -- ax[1,1]: 2D projection --
ax = axes[1,1]
for label, voices in majors:
    x, y = proj(voices)
    ax.scatter(x, y, color='#60a5fa', s=55, zorder=3, alpha=0.85)
    ax.annotate(label[:2], (x, y), color='#bfdbfe', fontsize=6, ha='center', va='bottom')
for label, voices in minors:
    x, y = proj(voices)
    ax.scatter(x, y, color='#34d399', s=55, marker='s', zorder=3, alpha=0.85)
    ax.annotate(label[:2], (x, y), color='#a7f3d0', fontsize=6, ha='center', va='bottom')
ax.scatter([], [], color='#60a5fa', s=55, label='Major')
ax.scatter([], [], color='#34d399', s=55, marker='s', label='Minor')
ax.set_xlabel('Mean pitch class', color='#94a3b8')
ax.set_ylabel('Spread (max-min)', color='#94a3b8')
ax.set_title('2D Orbifold Projection: 24 Triads', color='#93c5fd', fontsize=11)
ax.tick_params(colors='#94a3b8')
ax.legend(facecolor='#0a0a1a', labelcolor='#e2e8f0', fontsize=8)
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')

plt.tight_layout(pad=2.0)
plt.savefig('output.png', dpi=130, facecolor='#0a0a1a', bbox_inches='tight')
print('Plot saved.')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Video Lecture: The Geometry of Music

Dr. Dmitri Tymoczko presents the orbifold theory of voice leading — the mathematical framework behind this module.

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