The geometry is the music.

Notes, rhythms, and harmonics all emerge from the same thing: modular orbits on a torus. Change the geometry, the music changes with it. Every parameter you touch reshapes the mathematics - and the sound follows.

Structure Creates Sound

Place n vertices on a torus, connect every k-th point. The instrument walks this path step by step - each vertex triggers a note. The winding number k controls the interval between consecutive notes: low k produces stepwise melodies, high k creates wide leaps across the scale. The geometry is the score.

Harmony from Orbits

gcd(n, k) determines how many independent loops exist. One orbit = one voice playing all n notes in sequence. Multiple orbits = multiple voices sounding together, each on its own closed path. This is where harmony comes from: the instruments are distributed across orbits, creating chords and counterpoint from pure structure.

Geometry Becomes Melody

Where a vertex sits in its orbit determines which note plays. Position maps to scale degree, the orbit index shifts the octave register, and the 3D height on the torus shapes velocity. Movement (ω) adds a slowly shifting octave contour. Depth (ξ) pans voices across the stereo field, creating spatial texture. The melody writes itself.

Real Instruments

18 instruments from sampled grand piano, strings, and brass to synthesized pads and 8-bit bleeps. Mix any combination. Set per-voice note lengths, weights, and distribution. Solo mode lets you isolate a single instrument against the geometric pattern.

Scales from 7 to 53 Notes

Western modes, pentatonic, blues, world scales (Hirajoshi, Arabic, Persian, Byzantine), and microtonal systems up to 53-EDO. The n-EDO option uses the complexity parameter itself as the equal division - the geometry defines its own tuning system.

Share What You Find

Every parameter state encodes into a URL. Hit Share to copy a link that recreates your exact instrument configuration - voices, weights, scale, pattern, everything. Export renders 10 seconds of audio as WAV for offline use.

The Mathematics

The instrument implements the ResonanT⁴ polystar permutation: σ(p) = (p · k) mod n. Each vertex p maps to the next via multiplication by the winding number k, modulo n. The resulting orbits on a torus are the musical score.

Paper IV: Musical Structure as Torus Geometry formalizes what this instrument demonstrates - how modular orbits on a torus generate musical structure. Orbit decomposition defines rhythm, vertex position maps to pitch, and the winding number controls harmonic texture. The geometry is the music. Read Paper IV →

Instrument

Geometry

Performance

Instruments

Torus