Every hyperbolic knot has an intrinsic tone. The lengths of the closed geodesics in the knot complement form a true invariant — independent of how the knot is drawn — and we map them to pitch: the shortest geodesic (the systole) becomes the fundamental, the longer ones its overtones (f = 150 × length Hz, an absolute scale, no per-knot normalization). So two different knots genuinely sound at different pitches. Below are the 110 distinct intrinsic tones of the corpus, sorted low to high — spanning 25–389 Hz (about 4.0 octaves).
A finding: the pitch tracks the hyperbolic volume, not the crossing number — the geometry, not the combinatorics, decides what a knot sings. Mirror images and machine re-parametrizations share a tone, so they appear once. (Torus knots and connected sums have no length-spectrum tone and are not listed.) See Method for how the tones are made.
These are synthesised from the knots themselves — additive synthesis from their overtones. No sampled instrument, no MIDI. The melodies are in the public domain — O’Carolan (d. 1738), Beethoven (d. 1827), traditional tunes — with the exception of the two covers at the end, which are credited to their authors.
The same 110 tones you can click on, played in order from the lowest to the highest. It was not restored from an old file — it is generated from exactly the data shown above, so what you hear is tile for tile what you see (bau_tonleiter.py, three guards: lowest partial is the fundamental · ascending · every fundamental measured back out of the finished audio).
Trainstation stands in steel; Sundown is a drawing. The pipeline finds them to be the same knot — and because the tone comes from the geometry, not from the drawing, the two recordings came out byte-for-byte identical. Nothing was matched by hand. Do not take our word for it: both lossless originals are one click away (wav), so you can run a checksum over them yourself.
One tune — Sí Bheag Sí Mhór by Turlough O'Carolan (d. 1738) — played on each sculpture. The melody is the same throughout; what changes is the timbre, and that is the knot itself.
Every voice is a different knot; no sampled instrument is involved anywhere. The House of the Rising Sun (traditional, author unknown) and Beethoven's Ode to Joy (d. 1827) are scored for nine knots each, all nine carrying a true length-spectrum tone.
An honest pair. The same sculpture, twice: once from its length spectrum — a true invariant, the same for every drawing of this knot — and once from the Fourier coefficients of this particular centre line, which is the sound of this curve, not of the knot. Both are ours; only the first one is an invariant.
Not everything here is ours. These two are cover versions — the melodies belong to their authors: Nothing Else Matters (Hetfield/Ulrich, 1991) and Road to Nowhere (David Byrne, 1985). Only the sound is ours: the same additive synthesis from knot overtones, no sampled instrument. They are early attempts from July and we have not re-checked their tuning.
Tones derived from the hyperbolic length spectrum · an absolute, parametrization-independent invariant. Knot data CC BY-NC 4.0 · W.A.Stanggassinger in collaboration with Claude Code (Anthropic).