← Back to album

The Mathematics

What each pattern actually is, how it maps to music, and what we don't claim

Every composition on this album is generated by feeding a mathematical sequence through a deterministic pipeline: pattern → pitch mapping → rhythm → voice leading constraints → orchestration → rendering. No machine learning, no neural networks, no randomness beyond the initial seed. The mathematical pattern determines the melodic contour; classical counterpoint rules (parallel fifth avoidance, leap recovery, phrase-ending resolution) shape it into something that sounds like music.

Below is an honest account of each pattern: the actual formula, how it maps to notes, and what the resulting music does and doesn't represent.

Track 01

Threshold (Fibonacci)

The Sequence

The Fibonacci sequence is defined by the recurrence relation:

F(0) = 1, F(1) = 1, F(n) = F(n-1) + F(n-2)

Producing: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

The ratio F(n+1)/F(n) converges to the golden ratio φ = (1 + √5)/2 ≈ 1.6180339...

How It Becomes Music

Fibonacci values are normalized and mapped to scale degrees in B♭ minor. Large values wrap around via modular arithmetic. The exponential growth of the sequence creates wide melodic intervals that gradually expand, the musical signature of this track.

What This Isn't

The claim that "Fibonacci is everywhere in music" is overstated in popular culture. We don't claim the sequence produces inherently beautiful music. The sequence determines pitch choices; counterpoint rules and orchestration make it listenable.

Reference: OEIS A000045. Fibonacci, L. (1202). Liber Abaci.

Track 02

The Infinite Series (Norgard)

The Sequence

Per Nørgård's infinity series (1959) is defined by:

a(0) = 0 a(2n) = -a(n) a(2n+1) = a(n) + 1

Producing: 0, 1, -1, 2, 1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, ...

The defining property: every subsequence at every scale is a transposition or inversion of the whole series. The series is genuinely self-similar at all levels.

How It Becomes Music

Series values map directly to chromatic pitch intervals from a root note in C major. This is close to how Nørgård himself used it, he composed Voyage into the Golden Screen (1968) and Symphony No. 3 (1975) using this series as melodic material.

Reference: Nørgård, P. (1959). Described in Rasmussen, K.A. (2009). Layers of continuity. OEIS A004718.

Track 03

Golden Spiral

The Pattern

Uses the golden angle: θ = 360° / φ² ≈ 137.508°

θ(n) = n × 137.5077640...° (mod 360°)

This is the same angle that governs phyllotaxis, the arrangement of leaves, seeds, and petals in plants (sunflower heads, pinecones, artichokes). It produces the most uniform angular distribution possible.

How It Becomes Music

Angular values are mapped to scale degrees in D major. The golden angle ensures that pitch choices are maximally spread across the scale, no note clusters, no gaps. The composition structure places its climax at 61.8% of the total duration (the golden section proportion).

Reference: Vogel, H. (1979). A better way to construct the sunflower head. Mathematical Biosciences, 44, 179–189.

Track 04

Harmonic Series

The Physics

When a string vibrates, it produces a fundamental frequency f plus overtones at integer multiples:

f, 2f, 3f, 4f, 5f, 6f, 7f, 8f, ...

This is the physical basis of consonance. The octave (2f), perfect fifth (3f), perfect fourth (4f), and major third (5f) all emerge from this series. Every pitched acoustic instrument produces this spectrum.

How It Becomes Music

Harmonic partials are converted to the nearest scale degrees in E minor. Higher harmonics (7f, 11f, 13f) map to chromatic intervals that sit outside the diatonic scale, creating natural tension. The result is a melody that traces the overtone series, moving from consonance to dissonance as it ascends.

Reference: Helmholtz, H. (1863). On the Sensations of Tone. Roederer, J.G. (2008). The Physics and Psychophysics of Music.

Track 05

Logistic Map (r=3.85)

The Equation

The logistic map is a one-dimensional discrete dynamical system:

x(n+1) = r · x(n) · (1 - x(n))

Originally a population growth model. As the parameter r increases:

How It Becomes Music

At r=3.85, the logistic map is deep in the chaotic regime but still contains periodic windows, moments of order within chaos. The x-values (always between 0 and 1) are mapped to scale degrees in C♯ minor. The melody alternates unpredictably between registers, but occasionally locks into brief repeating motifs, reflecting the mathematical structure of chaos.

Reference: May, R. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467. Feigenbaum, M.J. (1978). Quantitative universality for nonlinear transformations.

Track 06

Mandelbrot Boundary

The Set

The Mandelbrot set is the set of complex numbers c for which the iteration

z(n+1) = z(n)² + c,   z(0) = 0

does not diverge. The boundary of this set is infinitely complex, a fractal with Hausdorff dimension 2.

The Walk

We parametrize the main cardioid boundary using:

c(θ) = ½e^(iθ) - ¼e^(2iθ) + perturbation

and record the escape count (number of iterations before |z| > 2) at each point. Points near the boundary escape slowly, these produce the highest values.

How It Becomes Music

Escape counts map to scale degrees in E minor. Points deep inside the set (count = max) produce the highest pitches; points that escape quickly produce low pitches. The melodic contour traces the fractal structure of the boundary.

The Goosebump Engine

This track uses a research-grounded frisson trigger: an accented appoggiatura (a non-chord tone one semitone above, resolving downward) placed at exactly 61.8% of the composition's duration, the golden section. This is based on Sloboda (1991), who found that appoggiaturas accounted for 82% of reported "chill" moments in music listening.

Reference: Mandelbrot, B. (1980). Fractal aspects of the iteration of z → λz(1-z). Annals of the New York Academy of Sciences. Sloboda, J.A. (1991). Music structure and emotional response. Psychology of Music, 19, 110–120.

Track 07

Rössler's Strange Attractor

The System

The Rössler system (1976) is a system of three coupled ordinary differential equations:

dx/dt = -y - z dy/dt = x + ay dz/dt = b + z(x - c)

With parameters a = 0.2, b = 0.2, c = 5.7, the system produces a chaotic attractor: a single-lobe spiral in the x-y plane with occasional excursions in the z-direction. It is the simplest known continuous-time chaotic system.

How It Becomes Music

The x-coordinate of the trajectory is sampled at regular intervals and mapped to scale degrees in B♭ minor. The spiral motion creates a slowly rising and falling melodic contour, interrupted by sudden z-axis excursions that produce melodic leaps. The result has a meditative, circling quality punctuated by surprises.

Reference: Rössler, O.E. (1976). An equation for continuous chaos. Physics Letters A, 57(5), 397–398.

Track 08

Cantor's Dust

The Construction

The Cantor set is constructed by recursive middle-third removal:

Start: [1] Depth 1: [1, 0, 1] Depth 2: [1, 0, 1, 0, 0, 0, 1, 0, 1] Depth 3: [1,0,1, 0,0,0, 1,0,1, 0,0,0, 0,0,0, 0,0,0, 1,0,1, 0,0,0, 1,0,1]

Each 1 is replaced by [1, 0, 1]; each 0 by [0, 0, 0]. The resulting set has measure zero but is uncountably infinite, it contains "as many points as the real line" yet takes up no space. Its Hausdorff dimension is log(2)/log(3) ≈ 0.6309.

How It Becomes Music

The binary pattern (1 = note, 0 = rest) is used as a rhythm mask. The fractal structure creates characteristically sparse rhythms, clusters of notes separated by expanding silences. As the Cantor construction recurses deeper, the gaps dominate. Melodic pitch is generated separately; the Cantor set only shapes when notes occur, not which notes.

Reference: Cantor, G. (1883). Über unendliche, lineare Punktmannigfaltigkeiten V. Mathematische Annalen, 21, 545–591.

Track 09

Zipf's Law

The Distribution

Zipf's law describes a power-law relationship between rank and frequency:

P(k) ∝ 1/k^s   (s ≈ 1)

The most common item appears roughly twice as often as the second most common, three times as often as the third, and so on. This distribution appears in natural language (word frequencies), city populations, website traffic, earthquake magnitudes, and, as Zipf himself noted, in music (some intervals appear far more often than others).

How It Becomes Music

We define 12 pitch categories (one per scale degree in C minor) and draw from a Zipf distribution with exponent s = 1.0. The tonic appears most often, followed by the fifth, then the third, creating a natural tonal hierarchy that mirrors how traditional composers intuitively use pitch. The melody sounds "normal" precisely because natural music already follows this distribution.

Reference: Zipf, G.K. (1949). Human Behavior and the Principle of Least Effort. Voss, R.F. & Clarke, J. (1975). '1/f noise' in music and speech. Nature, 258, 317–318.

Track 10

Thue-Morse Resolution

The Sequence

The Thue-Morse sequence can be defined two equivalent ways:

Construction: Start with 0, repeatedly append the bitwise complement. 0 → 01 → 0110 → 01101001 → ... Equivalently: T(n) = (number of 1-bits in binary(n)) mod 2

This sequence has zero autocorrelation at every lag, no subsequence ever repeats exactly. It is the canonical "most aperiodic" binary sequence, used by Johnson (2003) specifically for anti-repetitive musical rhythms.

How It Becomes Music

After nine tracks built from self-similar patterns (Fibonacci, infinity series, golden ratio, fractals), the final track uses Thue-Morse to maximally avoid self-similarity. Binary values (0, 1) alternate between two musical behaviors: note vs. rest, high vs. low register, or tonic vs. dominant. The melody refuses to settle into any repeating pattern, it is mathematically guaranteed to never repeat.

What "Zero Autocorrelation" Means

For any shift k, if you overlay the sequence with a shifted copy of itself, the number of matches equals the number of mismatches. No rhythmic or melodic pattern ever occurs more often than chance would predict. This is the opposite of how most music works, which is exactly the point of closing the album with it.

Reference: Thue, A. (1906). Über unendliche Zeichenreihen. Morse, M. (1921). Recurrent geodesics on a surface of negative curvature. Johnson, T. (2003). Self-Similar Melodies.

The Pipeline

How Patterns Become Compositions

Every track passes through the same deterministic pipeline:

1. Pattern generator → sequence of numbers 2. Pitch mapping → map to scale degrees (normalize or modular) 3. Rhythm → Euclidean rhythms + pattern-specific timing 4. Voice leading constraints: - No parallel fifths/octaves - Leap recovery (large jump followed by step in opposite direction) - Phrase endings resolve to tonic 5. Bass voice → harmonic series pattern, chord grammar FSM 6. Orchestration → violin (melody) + cello (bass) 7. Rendering → FluidSynth + MuseScore General SoundFont 8. Mastering → normalized to -14 LUFS (Spotify target)

The pattern determines melodic contour. Classical counterpoint rules make it musical. The composer engine adds nothing creative, it applies rules mechanically. All musicality emerges from the interaction between mathematical structure and 500 years of counterpoint conventions.

Every note is reproducible. Given the same seed, key, and pattern, the output is identical every time. There are no hidden random draws, no neural network weights, no human intervention post-generation.