About

Fractals encode infinite complexity in finite rules. This tool computes escape-time fractals (Mandelbrot, Julia sets) and geometric fractals (Sierpinski, Koch) then maps their mathematical structure to audible frequencies via additive synthesis. The iteration count n at each sampled point determines pitch on a logarithmic scale between 20 Hz and 8000 Hz. Escape velocity modulates amplitude envelopes. Fractal density across a scan line controls harmonic content through up to 8 partials. The result is not random noise. It is a deterministic acoustic signature of the fractal geometry.

Misunderstanding the mapping between iteration depth and frequency produces harsh, unmusical output. This tool applies logarithmic frequency scaling consistent with human pitch perception (Weber-Fechner law). It also normalizes amplitude across regions of varying density to prevent clipping. Note: results approximate the fractal’s structure at the chosen resolution. Increasing maxIter reveals finer detail but increases computation time proportionally. Audio output depends on your system’s sample rate, typically 44100 Hz or 48000 Hz.

Formulas

The core escape-time algorithm iterates the recurrence relation:

z_n+1 = z_n² + c

where z₀ = 0 for Mandelbrot and z₀ = pixel coordinate for Julia sets. Iteration stops when |z| > 2 (escape radius) or n reaches maxIter. The escape count n is then mapped to audio frequency:

f = f_min ⋅ (f_maxf_min)^nmaxIter

This logarithmic mapping ensures perceptually uniform pitch distribution. The amplitude for each sampled point uses normalized iteration ratio:

A = A_max ⋅ (1 − nmaxIter)

Points inside the set (n = maxIter) produce silence. Harmonic content uses additive synthesis with k partials at integer multiples of the fundamental:

signal = K∑k=1 Ak ⋅ sin(2π ⋅ k ⋅ f ⋅ t)

where f = fundamental frequency in Hz, t = time in s, A = amplitude, K = number of harmonics (derived from local fractal density), k = harmonic index, f_min = 20 Hz, f_max = 8000 Hz, maxIter = maximum iteration depth.

Reference Data

Fractal Type	Formula	Dimension	Audio Character	Iteration Range	Typical Frequency Spread
Mandelbrot Set	z_n+1 = z_n² + c	2.0 (boundary)	Rich, evolving tones with harmonic clusters at set boundary	50 - 500	80 - 6000 Hz
Julia Set (c = −0.7 + 0.27i)	z_n+1 = z_n² + c	1.2 - 2.0	Symmetrical, bell-like timbres with mirrored spectral content	50 - 300	100 - 5000 Hz
Burning Ship	z_n+1 = (\|Re(z)\| + i\|Im(z)\|)² + c	2.0	Harsher, asymmetric textures with abrupt transitions	50 - 400	60 - 7000 Hz
Sierpinski Triangle	Chaos game: 3 vertices	1.585	Tritone intervals, sparse rhythmic pulses	1000 - 50000 points	200 - 3000 Hz
Koch Snowflake	L-system: F → F+F−−F+F	1.2619	Self-similar melodic patterns at multiple octaves	1 - 7 depth	100 - 4000 Hz
Cantor Set	Remove middle third recursively	0.6309	Sparse, gapped frequency bands with silence intervals	1 - 12 depth	50 - 2000 Hz
Mandelbrot (Power 3)	z_n+1 = z_n³ + c	2.0	More symmetric harmonics, organ-like sustained tones	50 - 300	100 - 5500 Hz
Multibrot (Power 4)	z_n+1 = z_n⁴ + c	2.0	Four-fold symmetric, crystalline overtone series	30 - 200	120 - 4500 Hz
Tricorn (Mandelbar)	z_n+1 = z_n² + c	2.0	Conjugate symmetry produces detuned, beating frequencies	50 - 300	80 - 5000 Hz
Newton Fractal (z³−1)	Newton’s method root finding	Basin boundaries	Rapid convergence → short attack; slow convergence → ambient pads	20 - 100	150 - 3500 Hz
Logistic Map	x_n+1 = r ⋅ x_n(1 − x_n)	N/A (1D)	Bifurcation cascades create rhythm from order to chaos	100 - 1000	40 - 2000 Hz

Frequently Asked Questions

Higher maxIter values resolve finer detail at fractal boundaries, producing more distinct pitch gradations. At 50 iterations, you get approximately 50 discrete frequency steps between 20 Hz and 8000 Hz. At 500 iterations, the resolution increases tenfold but computation time grows proportionally. The boundary region (where most variation occurs) generates the richest harmonic content because adjacent pixels escape at different rates.

Points inside the Mandelbrot or Julia set never escape - they reach maxIter without |z| exceeding the escape radius of 2. The amplitude formula maps these to zero: A = A_max × (1 − 1) = 0. This is intentional. The interior of the set is mathematically "trapped," and silence represents that convergence. Zooming into boundary regions maximizes audible output.

The harmonic count K is derived from local fractal density - specifically, the variance of iteration counts in a small neighborhood around each sampled point. High variance (chaotic boundary regions) produces more harmonics (up to 8), creating complex timbres. Low variance (smooth exterior or deep interior) reduces to 1 - 2 harmonics, producing pure or near-pure tones. This maps the fractal's geometric complexity directly to acoustic complexity.

The tool scans the fractal image column by column (or row by row, depending on scan direction). Each column contains multiple sampled points. Their iteration values are aggregated into a chord: multiple simultaneous frequencies played through additive synthesis. The scan speed parameter controls how fast the "cursor" moves across the fractal, effectively setting the tempo. A 512-pixel-wide fractal at 4 columns per second plays for 128 seconds.

Yes. The Julia set constant c = a + bi determines the fractal's topology. Connected Julia sets (where c lies inside the Mandelbrot set) produce dense, harmonically rich audio. Disconnected sets (Cantor dust, where c is outside the Mandelbrot set) produce sparse, staccato bursts. Moving c along the Mandelbrot boundary creates the most musically interesting transitions because the Julia set undergoes rapid topological changes.

Zooming into a fractal region changes the iteration distribution of the visible area. Deep zooms into boundary filaments reveal self-similar structures that produce recurring melodic motifs at different pitch registers - a direct acoustic manifestation of fractal self-similarity. Zooming into the interior produces more silence. Zooming into the exterior produces uniform low-iteration tones. The most musically varied output comes from boundary regions at moderate zoom levels (10× - 1000×).