Equations, evolved until they breathe.
Sixteen strange attractors and three dynamical systems. Rendered and sonified in real time.
Best with headphones, or speakers capable of low frequencies.
Strange Attractors · Sonified
Joshua Borsman · 2026
Equations, evolved until they breathe.
Sixteen strange attractors and three dynamical systems, integrated in real time and sounded by a generative engine. Five hundred particles trace each attractor’s trajectory through a fourth-order Runge-Kutta integrator. Their positions become pitch, velocity, and stereo field. Nothing is sequenced. The score is the differential equation.
Each attractor is a system of ordinary differential equations with chaotic solutions — deterministic but non-repeating. The Lorenz system spirals between two lobes. The Aizawa wraps a torus. Chua’s circuit folds a double scroll. The topology is different in each case, but the property they share is sensitivity to initial conditions: nearby trajectories diverge exponentially, which is what makes the sound interesting. The same equation, run long enough, never produces the same phrase twice.
Three additional systems round out the piece. Gray-Scott reaction-diffusion grows coral, labyrinths, and worm-like structures on a GPU-computed field. A 350-pendulum ensemble starts from near-identical initial conditions and diverges into chaos. Kuramoto coupled oscillators demonstrate spontaneous synchronisation — order emerging from noise as the coupling strength increases.
The piece is in A major pentatonic. Each note is synthesised additively, as a stack of sines at inharmonic ratios — 1.0, 2.76, 5.40, 8.93 — the classical partial series of a struck metal vessel. The 2.76 ratio is the one that gives a bell its character. Attack is near-instantaneous; the fundamental droops slightly on the strike and settles over a hundred and fifty milliseconds. Upper partials decay faster than the bottom, darkening the tone as it rings. Lower pitches sustain longer. Through the convolution reverb the strikes overlap into a wash that almost never empties.
Underneath, a quiet drone holds the room — A1, E2, A2 — tonic, fifth, octave. Filtered low enough that you feel it more than hear it. Slow vibrato, just enough to keep it alive.
The reaction-diffusion systems run in B♭ major pentatonic — a half-step away from A. Both major, so the shift feels like entering a different room of the same space rather than leaving it.
The piece is meant to play for hours. There is no track length and no loop; the score is whatever the particles happen to be doing.
Five leader particles, spread across the swarm, are tracked for sonification. When a leader crosses zero on any axis, or reverses direction, it triggers a pluck. The Y-coordinate sets the pitch, quantised to the nearest scale degree. Speed determines velocity and filter brightness — faster particles ring brighter. The X-coordinate sets stereo position. The result is a rhythm that follows the attractor’s topology: the Lorenz system triggers in pairs as particles switch lobes; the Thomas attractor, with its slow sinusoidal orbits, plucks gently and evenly; Halvorsen fires in rapid bursts.
Phase Space can be heard in two ways. The default is the browser engine — synthesis is internal, mixed to your speakers or headphones. An alternate analog engine emits eight DC control signals through a multichannel audio device: pitch at one volt per octave, gate, velocity, and five modulation channels derived from the simulation. Patched into an outboard rack, the geometry plays the room.
Lorenz · Rössler · Thomas · Aizawa · Halvorsen · Chen · Dadras · Sprott B · Burke-Shaw · Rabinovich-Fabrikant · Nose-Hoover · Newton-Leipnik · Chua’s Circuit · Arneodo · Finance · Four-Wing · Gray-Scott reaction-diffusion · double pendulum ensemble · Kuramoto coupled oscillators
Joshua Borsman is an artist and engineer. His work treats physical phenomena — tides, gravity, telemetry, light — as collaborators rather than subject matter, and the pieces are usually instruments built to make those phenomena audible or visible at human scale.
Phase Space is one of those instruments. Its score is the geometry of chaos; the composer is the differential equation. The work joins a small group of Borsman’s pieces in which a mathematical or physical system is made audible: a telescope tuned to a stellar spectrum, a string plucked when the International Space Station passes overhead, ocean weather translated into drones and plucks along the Pacific Northwest coast. In Phase Space, the system being made audible is deterministic chaos itself.