Module 0 · Week 1 · Undergraduate

The Pythagorean Wager

Can all of music be reduced to ratios of whole numbers? The Pythagoreans wagered yes — and the consequences of that wager, including its spectacular failure, set the stage for two millennia of mathematics.

1. The Quadrivium

In the medieval university, the quadrivium comprised four disciplines: arithmetic (number in itself), geometry (number in space), music (number in time), and astronomy (number in space and time). This was not mere curriculum design — it reflected a deep philosophical conviction that the universe is fundamentally mathematical.

The Pythagorean school (c. 530 BCE) discovered that the intervals which sound most consonant correspond to the simplest ratios of string lengths. This was arguably the first quantitative law of nature — preceding Newton by two millennia.

“All is number” — attributed to Pythagoras. The quadrivium endured as the core of European education from late antiquity through the Renaissance.

2. Consonance and Ratio

When a string is divided into simple integer ratios, the resulting intervals are perceived as consonant. The three most consonant intervals, after unison, are:

Octave

\(\frac{2}{1}\)

Frequency ratio 2:1

1200 cents

Perfect Fifth

\(\frac{3}{2}\)

Frequency ratio 3:2

~701.96 cents

Perfect Fourth

\(\frac{4}{3}\)

Frequency ratio 4:3

~498.04 cents

The cent is a logarithmic unit: 1200 cents per octave. An interval of frequency ratio \(r\) has \(1200 \log_2(r)\) cents. This converts multiplicative ratios to additive measures.

Why do simple ratios sound consonant? When two tones with frequencies in ratio \(p:q\) are sounded together, their combined waveform repeats every \(\text{lcm}(p,q)\) cycles. Simpler ratios produce shorter repeat periods, which the auditory system processes more easily. This is a rough first approximation — the full story involves critical bands and the cochlear mechanics we will explore in Module 1.

3. The Pythagorean Comma

If we stack twelve perfect fifths, we expect to return to our starting note (seven octaves higher). But the arithmetic is merciless:

\[\left(\frac{3}{2}\right)^{12} = \frac{3^{12}}{2^{12}} = \frac{531441}{4096}\]

\[2^7 = 128 \quad\Longrightarrow\quad \frac{531441}{4096} \div 128 = \frac{531441}{524288}\]

\[\text{Pythagorean comma} = \frac{531441}{524288} \approx 1.013643 \approx 23.46 \text{ cents}\]

Twelve fifths overshoot seven octaves by about 23.46 cents — roughly a quarter of a semitone. This is the Pythagorean comma, and it means that no circle of pure fifths ever closes exactly.

Numerical verification

\((3/2)^{12} = 129.746337890625\) while \(2^7 = 128\). The ratio \(129.746.../128 = 1.01364326...\), and\(1200 \log_2(1.01364326) \approx 23.46\) cents.

Simulation: The Pythagorean Comma Spiral

The following simulation plots the spiral of fifths on a polar diagram, the cumulative comma error in cents, and a three-way comparison of Pythagorean, equal-temperament, and just-intonation tunings across all 12 pitch classes.

Python

script.py145 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec

plt.rcParams['font.family'] = 'monospace'
BG = '#0a0a1a'
ACCENT = '#60a5fa'
WARN = '#ef4444'
GOLD = '#f59e0b'
GREEN = '#34d399'
PURPLE = '#a78bfa'

fig = plt.figure(figsize=(14, 12), facecolor=BG)
fig.suptitle('Pythagorean Comma: The Inescapable Gap', color='white', fontsize=16, fontweight='bold', y=0.98)

gs = gridspec.GridSpec(2, 2, figure=fig, hspace=0.45, wspace=0.38)

# ── Plot 1: Spiral of fifths on a polar axis ──────────────────────────────
ax1 = fig.add_subplot(gs[0, 0], projection='polar', facecolor=BG)
ax1.set_facecolor(BG)

note_names = ['C', 'G', 'D', 'A', 'E', 'B', 'F#', 'C#', 'G#', 'D#', 'A#', 'E#', 'B#']
# Each fifth raises by log2(3/2) octave-classes; reduce mod 1 to stay in one octave
log2_fifth = np.log2(3/2)

angles = []
radii = []
for i in range(13):
    frac = (i * log2_fifth) % 1.0   # position in [0,1) octave
    angle = 2 * np.pi * frac
    r = 1.0 - i * 0.03              # spiral inward
    angles.append(angle)
    radii.append(r)

ax1.plot(angles, radii, color=ACCENT, lw=1.5, alpha=0.6)

for i, (a, r, name) in enumerate(zip(angles, radii, note_names)):
    color = GOLD if i == 0 else (WARN if i == 12 else ACCENT)
    ax1.scatter([a], [r], color=color, s=60, zorder=5)
    ax1.text(a, r + 0.12, name, color=color, fontsize=8, ha='center', va='center', fontweight='bold')

# Mark the comma gap
ax1.annotate('', xy=(angles[12], radii[12]), xytext=(angles[0], radii[0]),
             arrowprops=dict(arrowstyle='<->', color=WARN, lw=1.5))

ax1.set_yticklabels([])
ax1.set_xticklabels([])
ax1.spines['polar'].set_color('#1e3a5f')
ax1.tick_params(colors=BG)
ax1.set_title('Spiral of Fifths
(B# does not land on C)', color='white', fontsize=10, pad=12)

print('Plot 1: spiral of fifths rendered')
print(f'  Start angle (C):   {np.degrees(angles[0]):.2f} deg')
print(f'  End angle (B#):    {np.degrees(angles[12]):.2f} deg')
print(f'  Comma gap:         {(angles[12] - angles[0]) / (2*np.pi) * 1200:.2f} cents')

# ── Plot 2: Cumulative comma error in cents ───────────────────────────────
ax2 = fig.add_subplot(gs[0, 1], facecolor=BG)

fifths = np.arange(0, 13)
# After k fifths, cents above the closest lower octave
true_cents = [(k * 1200 * log2_fifth) % 1200 for k in fifths]
equal_cents = [(k * 700) % 1200 for k in fifths]    # 12-TET uses 700 cents per fifth
error = [1200 * log2_fifth * k - round(1200 * log2_fifth * k / 100) * 100 for k in fifths]

# Simpler: cumulative error = k * (fifth_in_cents - 700)
fifth_cents_pyth = 1200 * log2_fifth   # ~701.955
cum_error = [(k * (fifth_cents_pyth - 700)) for k in fifths]

ax2.axhline(0, color='#1e3a5f', lw=1, ls='--')
ax2.axhline(23.46, color=WARN, lw=1, ls=':', alpha=0.7, label='Pythagorean comma (23.46c)')
ax2.plot(fifths, cum_error, color=ACCENT, lw=2, marker='o', ms=5, label='Cumulative error vs 12-TET')
ax2.fill_between(fifths, 0, cum_error, alpha=0.15, color=ACCENT)
ax2.scatter([12], [cum_error[12]], color=WARN, s=80, zorder=5)
ax2.annotate(f'{cum_error[12]:.2f}c', (12, cum_error[12]), textcoords='offset points',
             xytext=(-30, 8), color=WARN, fontsize=9)
ax2.set_xlabel('Number of fifths stacked', color='#94a3b8', fontsize=9)
ax2.set_ylabel('Cents error vs equal temperament', color='#94a3b8', fontsize=9)
ax2.set_title('Comma Accumulation', color='white', fontsize=10)
ax2.legend(fontsize=8, facecolor='#0f172a', labelcolor='white', edgecolor='#1e3a5f')
ax2.set_facecolor(BG)
ax2.tick_params(colors='#94a3b8', labelsize=8)
for sp in ax2.spines.values():
    sp.set_color('#1e3a5f')

print('Plot 2: comma accumulation rendered')
print(f'  Pythagorean fifth: {fifth_cents_pyth:.4f} cents')
print(f'  Error per fifth:   {fifth_cents_pyth - 700:.4f} cents')
print(f'  After 12 fifths:   {cum_error[12]:.4f} cents (= Pythagorean comma)')

# ── Plot 3: Three-way tuning comparison ──────────────────────────────────
ax3 = fig.add_subplot(gs[1, :], facecolor=BG)

note_labels = ['C', 'C#', 'D', 'Eb', 'E', 'F', 'F#', 'G', 'Ab', 'A', 'Bb', 'B']
n_notes = 12

# 12-TET: exactly 100 cents per semitone
tet_cents = [i * 100 for i in range(n_notes)]

# Pythagorean: build from circle of fifths, sort by semitone position
pyth_ratios = {
    0: 1, 1: 256/243, 2: 9/8, 3: 32/27, 4: 81/64,
    5: 4/3, 6: 729/512, 7: 3/2, 8: 128/81, 9: 27/16,
    10: 16/9, 11: 243/128
}
pyth_cents = [1200 * np.log2(pyth_ratios[i]) for i in range(n_notes)]

# 5-limit Just Intonation
just_ratios = {
    0: 1, 1: 16/15, 2: 9/8, 3: 6/5, 4: 5/4,
    5: 4/3, 6: 45/32, 7: 3/2, 8: 8/5, 9: 5/3,
    10: 9/5, 11: 15/8
}
just_cents = [1200 * np.log2(just_ratios[i]) for i in range(n_notes)]

x = np.arange(n_notes)
width = 0.28

bars1 = ax3.bar(x - width, tet_cents, width, label='12-TET', color=ACCENT, alpha=0.85)
bars2 = ax3.bar(x,          pyth_cents, width, label='Pythagorean', color=GOLD, alpha=0.85)
bars3 = ax3.bar(x + width, just_cents, width, label='Just Intonation (5-limit)', color=GREEN, alpha=0.85)

ax3.set_xticks(x)
ax3.set_xticklabels(note_labels, color='#94a3b8', fontsize=9)
ax3.set_ylabel('Cents above C', color='#94a3b8', fontsize=9)
ax3.set_title('Cents Comparison: 12-TET vs Pythagorean vs Just Intonation', color='white', fontsize=11)
ax3.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#1e3a5f')
ax3.set_facecolor(BG)
ax3.tick_params(colors='#94a3b8', labelsize=8)
ax3.set_ylim(0, 1280)
for sp in ax3.spines.values():
    sp.set_color('#1e3a5f')

print('Plot 3: tuning comparison rendered')
print('  Interval | 12-TET |  Pyth  |  Just')
print('  ---------|--------|--------|-------')
for i, note in enumerate(note_labels):
    print(f'  {note:8s} | {tet_cents[i]:6.1f} | {pyth_cents[i]:6.1f} | {just_cents[i]:6.1f}')

plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor=BG)
print('Saved output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

4. The Core Impossibility

The comma is not an accident of the number 12. It reflects a deep number-theoretic impossibility:

Theorem

There exist no positive integers \(a, b > 0\) such that \(2^a = 3^b\).

Proof

Suppose \(2^a = 3^b\) for positive integers \(a, b\). The left side is even; the right side is odd. Contradiction. \(\square\)

Equivalently, \(\log_2 3\) is irrational. If \(\log_2 3 = p/q\) for integers \(p, q\), then \(2^p = 3^q\), contradicting the above. Since \(\log_2(3/2) = \log_2 3 - 1\), the number of fifths in an octave is also irrational. No equal division of the octave into fifths is possible.

This elementary fact — that powers of 2 and powers of 3 never coincide — is the original sin of Western tuning theory. Every tuning system is a different strategy for managing this impossibility.

5. The Comma Spiral

Stacking perfect fifths around the circle of pitch classes, we spiral outward instead of closing. After 12 fifths, we arrive at B♯ instead of returning exactly to C — overshooting by the Pythagorean comma.

The spiral of fifths. Starting from C and stacking 12 pure fifths (ratio 3:2), we arrive at B♯ — not back at C, but ~23.46 cents sharp. The gap (red) is the Pythagorean comma.

6. The Harmonic Series

A vibrating string produces not just its fundamental frequency \(f_1\), but also integer multiples: \(f_1, 2f_1, 3f_1, 4f_1, \ldots\) These are the harmonics(or overtones/partials). The relative amplitudes of harmonics determine the timbre of a sound.

Harmonics and intervals

Fundamental

1:1

Octave

2:1

Oct + Fifth

3:1

2 Octaves

4:1

2 Oct + M3

5:1

2 Oct + 5th

6:1

b7 (flat)

7:1

3 Octaves

8:1

The harmonic series connects music to number theory: the consonant intervals of Western music are precisely the ratios of small integers that appear early in the series. The mathematical study of these ratios, their approximations, and their incompatibilities is the subject of this entire course.

Formally, the \(n\)-th harmonic has frequency \(f_n = n \cdot f_1\). The interval between harmonic \(n\) and harmonic \(m\) is the ratio \(n/m\), measured in cents as \(1200 \log_2(n/m)\).

7. Course Roadmap

This course follows a trajectory from the concrete to the abstract, building mathematical machinery as the musical questions demand it:

Acoustics & Fourier

Wave equation, eigenvalue problem, Fourier series/transform, psychoacoustics

Tuning & Diophantine

Continued fractions, why 12 tones, just intonation, equal temperament

Group Theory

Dihedral groups, Tonnetz, neo-Riemannian theory, generalized interval systems

Rhythm & Combinatorics

Euclidean rhythms, necklaces, rhythmic canons, DFT on finite groups

Information & Expectation

Shannon entropy, Markov models, Bayesian surprise, neural correlates

Topology of Chord Space

Orbifolds, voice-leading geometry, persistent homology

Algorithmic Composition

L-systems, cellular automata, stochastic music, deep generative models

Open Problems

12 unsolved questions at the music-mathematics frontier

Each module includes interactive components, detailed mathematical derivations, and connections to the broader mathematical landscape. The course spirals through three depth levels: undergraduate, graduate, and research frontier.

The Math Connecting Music, Astronomy & Quantum Physics

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