Module 8 · Week 16 · Research

Open Problems

Twelve unsolved questions at the music–mathematics frontier. Each problem sits at the intersection of mathematical theory, empirical science, and musical practice. These are starting points, not endpoints.

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Simulation: Complexity Landscape of Musical Corpora

Estimate Kolmogorov complexity via zlib compression, explore the aesthetic “sweet spot” of intermediate complexity, and verify Zipf’s law in a simulated musical corpus.

Python

script.py139 lines

import numpy as np
import zlib
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from collections import Counter

np.random.seed(42)

# ---------- sequences ----------
N = 1000

def compress_ratio(seq):
    raw = bytes(int(x) % 256 for x in seq)
    return len(zlib.compress(raw, 9)) / len(raw)

# 1. Random noise
seq_random = np.random.randint(0, 12, N)
# 2. Sine-quantized
t = np.linspace(0, 8*np.pi, N)
seq_sine = (np.round((np.sin(t)+1)*5.5)).astype(int)
# 3. Major scale cycling
major = [0, 2, 4, 5, 7, 9, 11]
seq_major = np.array([major[i % 7] for i in range(N)])
# 4. Chromatic scale cycling
seq_chrom = np.array([i % 12 for i in range(N)])
# 5. Bach-like Markov chain
trans = np.array([
    [0.2,0.05,0.2,0.05,0.25,0.05,0.2],
    [0.1,0.1,0.1,0.3,0.1,0.2,0.1],
    [0.15,0.05,0.2,0.1,0.3,0.1,0.1],
    [0.05,0.05,0.1,0.2,0.1,0.3,0.2],
    [0.25,0.05,0.15,0.05,0.2,0.05,0.25],
    [0.1,0.2,0.1,0.2,0.1,0.2,0.1],
    [0.2,0.1,0.1,0.1,0.2,0.1,0.2],
])
trans = trans / trans.sum(axis=1, keepdims=True)
seq_bach = [0]
for _ in range(N-1):
    seq_bach.append(np.random.choice(7, p=trans[seq_bach[-1]]))
seq_bach = np.array([major[x] for x in seq_bach])

sequences = {
    'Random': seq_random,
    'Sine-quantised': seq_sine,
    'Major scale': seq_major,
    'Chromatic': seq_chrom,
    'Bach Markov': seq_bach,
}

ratios = {k: compress_ratio(v) for k, v in sequences.items()}
for k, r in ratios.items():
    print(f'{k}: compression ratio = {r:.3f}')

# ---------- Plot 2: ratio vs length ----------
lengths = np.logspace(1, 4, 30, dtype=int)
ratios_random  = [compress_ratio(np.random.randint(0,12,l)) for l in lengths]
ratios_major_s = [compress_ratio(np.array([major[i%7] for i in range(l)])) for l in lengths]

# ---------- Plot 3: aesthetic sweet spot ----------
x_comp = np.linspace(0, 1, 300)
aesthetic = np.exp(-((x_comp - 0.45)**2)/(2*0.15**2))
aesthetic += np.random.normal(0, 0.04, 300)
aesthetic = np.clip(aesthetic, 0, 1.1)

# ---------- Plot 4: Zipf law ----------
corpus = []
for _ in range(5000):
    corpus.extend(seq_bach[:np.random.randint(5,50)])
counts = Counter(corpus)
sorted_counts = sorted(counts.values(), reverse=True)
ranks = np.arange(1, len(sorted_counts)+1)

# ---------- 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]: bar chart of compression ratios --
ax = axes[0,0]
colors_b = ['#f87171','#fbbf24','#34d399','#60a5fa','#a78bfa']
labels = list(ratios.keys())
vals   = list(ratios.values())
bars = ax.bar(labels, vals, color=colors_b, edgecolor='#1e3a5f', width=0.6)
ax.set_ylabel('Compression ratio (smaller = more structured)', color='#94a3b8')
ax.set_title('Kolmogorov Complexity Estimate via zlib', color='#93c5fd', fontsize=11)
ax.tick_params(colors='#94a3b8', axis='both')
ax.set_xticklabels(labels, rotation=20, ha='right', fontsize=8, color='#94a3b8')
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')
for bar, val in zip(bars, vals):
    ax.text(bar.get_x()+bar.get_width()/2, val+0.005, f'{val:.2f}', ha='center', color='#e2e8f0', fontsize=8)

# -- ax[0,1]: compression ratio vs length --
ax = axes[0,1]
ax.semilogx(lengths, ratios_random,  color='#f87171', linewidth=2, label='Random')
ax.semilogx(lengths, ratios_major_s, color='#34d399', linewidth=2, label='Major scale')
ax.set_xlabel('Sequence length', color='#94a3b8')
ax.set_ylabel('Compression ratio', color='#94a3b8')
ax.set_title('Compression Ratio vs Sequence Length', 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')

# -- ax[1,0]: sweet spot --
ax = axes[1,0]
ax.scatter(x_comp, aesthetic, c=x_comp, cmap='plasma', s=8, alpha=0.6)
from scipy.ndimage import uniform_filter1d
smooth = uniform_filter1d(aesthetic, size=25)
ax.plot(x_comp, smooth, color='#fbbf24', linewidth=2.5, label='Smoothed trend')
ax.axvline(0.45, color='#34d399', linestyle='--', linewidth=1.5, label='Sweet spot ~0.45')
ax.set_xlabel('Complexity (compression ratio)', color='#94a3b8')
ax.set_ylabel('Aesthetic rating (simulated)', color='#94a3b8')
ax.set_title('Aesthetic Sweet Spot: Intermediate Complexity', 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')

# -- ax[1,1]: Zipf law --
ax = axes[1,1]
ax.loglog(ranks, sorted_counts, 'o', color='#60a5fa', markersize=5, alpha=0.7, label='Note frequencies')
# fit line
log_r = np.log(ranks)
log_c = np.log(sorted_counts)
coef  = np.polyfit(log_r, log_c, 1)
ax.loglog(ranks, np.exp(np.polyval(coef, log_r)), color='#f87171', linewidth=2,
          label=f'Power law fit (slope={coef[0]:.2f})')
ax.set_xlabel('Rank', color='#94a3b8')
ax.set_ylabel('Frequency', color='#94a3b8')
ax.set_title("Zipf's Law in Simulated Bach Corpus", color='#93c5fd', fontsize=11)
ax.tick_params(colors='#94a3b8', which='both')
ax.legend(facecolor='#0a0a1a', labelcolor='#e2e8f0', fontsize=8)
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')
print(f"Zipf slope: {coef[0]:.3f} (ideal Zipf = -1.0)")

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

A Note on Method

Mathematics does not explain music; it provides a language in which certain questions about music can be asked precisely enough to admit answers. The problems above are valuable not because they will “solve” music, but because the attempt sharpens both the mathematics and the musical understanding. The best work at this frontier has always been bidirectional: music poses problems that drive new mathematics, and mathematical structures reveal musical possibilities that no ear alone would have found.

Course Complete

You have reached the end of Music & Mathematics. From the Pythagorean comma to open research frontiers — the interplay between these two disciplines is inexhaustible.

Return to Course Overview

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Open Problems

A Mathematical Definition of Consonance

Universality of Tonality

Complexity, Compressibility, and Musical Value

Voice Leading in Non-Equal Temperaments

The Spectral Composition Problem

Axiomatisation of Counterpoint

The Beat-Induction Problem

The Harmonic Series and Prime Numbers

Quantum Models of Musical Perception

Persistent Homology of Musical Style

A Mathematics of Improvisation

A Unified Theory of Musical Tension

Simulation: Complexity Landscape of Musical Corpora

A Note on Method