Module 7 · Weeks 14–15 · Graduate

Algorithmic Composition & Dynamical Systems

From deterministic rewriting systems to emergent complexity: algorithms that compose music and the mathematics that explains why they work.

L-System Composer

Koch curve mapped to ascending/descending pitch

Depth: 3

Tempo: 180 BPM

Scale

0 events

Rules: F → F+F-F-F+F · Axiom: F

Simulation: L-Systems & Cellular Automata Patterns

Visualise a Koch L-system melody as a piano roll, generate Wolfram CA space-time diagrams for Rules 30 and 90, and compare per-row entropy across Rules 30, 90, and 110.

Python

script.py129 lines

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

# ---------- L-System: Koch curve at depth 4 ----------
def expand_lsystem(axiom, rules, depth):
    s = axiom
    for _ in range(depth):
        s = ''.join(rules.get(c, c) for c in s)
        if len(s) > 20000:
            break
    return s

def lsystem_to_pitches(s, max_notes=200):
    pitches = []
    pitch = 4
    for c in s:
        if len(pitches) >= max_notes:
            break
        if c == 'F':
            pitches.append(pitch)
        elif c == '+':
            pitch = min(pitch + 1, 11)
        elif c == '-':
            pitch = max(pitch - 1, 0)
    return pitches

koch_str = expand_lsystem('F', {'F': 'F+F-F-F+F'}, 4)
pitches = lsystem_to_pitches(koch_str, 200)
print(f'Koch L-system string length at depth 4: {len(koch_str)}')
print(f'Melody events extracted: {len(pitches)}')

# ---------- Wolfram Elementary CA ----------
def wolfram_ca(rule_num, width, steps, init='single'):
    bits = [(rule_num >> i) & 1 for i in range(8)]
    grid = np.zeros((steps, width), dtype=np.uint8)
    if init == 'single':
        grid[0, width // 2] = 1
    else:
        np.random.seed(42)
        grid[0] = np.random.randint(0, 2, width)
    for t in range(1, steps):
        for i in range(width):
            l = grid[t-1, (i-1) % width]
            c = grid[t-1, i]
            r = grid[t-1, (i+1) % width]
            idx = (l << 2) | (c << 1) | r
            grid[t, i] = bits[idx]
    return grid

def row_entropy(grid):
    entropies = []
    for row in grid:
        p1 = row.mean()
        p0 = 1 - p1
        if p1 == 0 or p0 == 0:
            entropies.append(0.0)
        else:
            entropies.append(-p1*np.log2(p1) - p0*np.log2(p0))
    return np.array(entropies)

W, STEPS = 64, 32
ca30  = wolfram_ca(30,  W, STEPS)
ca90  = wolfram_ca(90,  W, STEPS)
ca110 = wolfram_ca(110, W, STEPS)

# entropy over time
ent30  = row_entropy(wolfram_ca(30,  128, 80))
ent90  = row_entropy(wolfram_ca(90,  128, 80))
ent110 = row_entropy(wolfram_ca(110, 128, 80))
print(f'Mean entropy - Rule 30: {ent30.mean():.3f}, Rule 90: {ent90.mean():.3f}, Rule 110: {ent110.mean():.3f}')

# ---------- 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]: piano roll --
ax = axes[0,0]
time_steps = np.arange(len(pitches))
note_names = ['C','C#','D','D#','E','F','F#','G','G#','A','A#','B']
colors_p = plt.cm.plasma(np.linspace(0.2, 0.9, 12))
for t, p in enumerate(pitches):
    ax.barh(p, 0.85, left=t, height=0.7, color=colors_p[p % 12], alpha=0.85)
ax.set_yticks(range(12))
ax.set_yticklabels(note_names, fontsize=7, color='#94a3b8')
ax.set_xlabel('Note index', color='#94a3b8')
ax.set_ylabel('Pitch class', color='#94a3b8')
ax.set_title('Koch L-System Melody (depth 4) - Piano Roll', color='#93c5fd', fontsize=11)
ax.tick_params(colors='#94a3b8')
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')

# -- ax[0,1]: Rule 30 space-time --
ax = axes[0,1]
ax.imshow(ca30, cmap='Blues', aspect='auto', interpolation='nearest')
ax.set_title('Wolfram Rule 30 - Space-Time Diagram', color='#93c5fd', fontsize=11)
ax.set_xlabel('Cell', color='#94a3b8')
ax.set_ylabel('Time step', color='#94a3b8')
ax.tick_params(colors='#94a3b8')
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')

# -- ax[1,0]: Rule 90 (Sierpinski) --
ax = axes[1,0]
ax.imshow(ca90, cmap='Greens', aspect='auto', interpolation='nearest')
ax.set_title('Wolfram Rule 90 - Sierpinski Triangle', color='#93c5fd', fontsize=11)
ax.set_xlabel('Cell', color='#94a3b8')
ax.set_ylabel('Time step', color='#94a3b8')
ax.tick_params(colors='#94a3b8')
for spine in ax.spines.values(): spine.set_edgecolor('#1e3a5f')

# -- ax[1,1]: entropy comparison --
ax = axes[1,1]
xs = np.arange(len(ent30))
ax.plot(xs, ent30,  color='#f87171', linewidth=1.5, label='Rule 30 (Chaos)')
ax.plot(xs, ent90,  color='#34d399', linewidth=1.5, label='Rule 90 (Structured)')
ax.plot(xs, ent110, color='#fbbf24', linewidth=1.5, label='Rule 110 (Complex)')
ax.set_xlabel('Time step', color='#94a3b8')
ax.set_ylabel('Shannon entropy (bits/row)', color='#94a3b8')
ax.set_title('Per-Row Entropy vs Time: Rules 30, 90, 110', 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

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