bracelet — spatial necklaces #

A rhythm you hear as a place. ~1:49, 138 BPM, A minor, binaural. Headphones required — on speakers it collapses to mono and the track is gone.

node pop/bracelet/c/render.mjs           # → out/bracelet.mp3
node pop/bracelet/c/render.mjs --play    # and open it

The thesis #

In Toussaint's taxonomy two rhythms are the same necklace if one is a rotation of the other, and the same bracelet if one is a rotation or reflection. Both equivalences are asserted on paper and never heard: in mono, a rhythm and its retrograde are plainly different objects.

Put the cycle in physical space around the listener — step i of n at azimuth 2πi/n, step 0 dead ahead, in the horizontal plane — and both become physical operations:

• rotation → the sound field spins. Precession in time and rotation in space become the same operation.
• reflection → a left/right mirror across the median plane.

E(k,n) stops being a number and becomes an actual regular k-gon of sound sources around your head. A lopsided rhythm is heard as a lopsided room.

The horizontal plane is not a stylistic choice. Left/right is the best-resolved axis humans have; front/back and elevation sit inside the cone of confusion. Reflecting about the frontal plane instead would put the whole experiment in the ambiguous axis and prove nothing.

What the platter turned up, and what it cost #

Building this against papers/rhythm-platter/ produced one finding that rewrote the form:

Most canonical timelines are achiral. son, bossa, shiko, tresillo, cinquillo and bembé are each their own reflection up to rotation — mirroring them only turns the ring. Only rumba, soukous and gahu are chiral.

And chirality alone is not enough. The ring must also carry its balance centroid off the mirror axis, or the flip changes the pattern without moving the field. soukous sits at 180°, dead on the axis. So the mirror section uses gahu: chiral, centroid at +17.8° — nearly frontal, where the minimum audible angle is ~1° — and the strongest pull of any candidate (0.710). The flip swings it to −17.8°, a 35° sweep across the best-resolved part of the field.

Picking soukous, the obvious choice, would have been a null test.

Form #

The morph starts from a deliberate non-timeline: five onsets crammed into the first half, gaps (2,2,2,2,8) — a badly lopsided pentagon heard as a clump in the front-right with the whole left side silent. Twelve moves later it is the regular pentagon. You hear the polygon become regular. The canonical claves are too close together to morph between (son → bossa is a single swap), which is itself Toussaint's point about how tightly that family clusters.

Build #

Rhythm from pop/lib/c/ac_necklace.h, binaural from pop/nullabye/c/ac_hrtf.h. The engine prints its own claims — spacing, precedence margin, per-ring chirality, evenness, off-beatness, and the centroid bar by bar — to stderr at render time.

Elevation carries off-beatness: generator pulses (gcd(p,16)=1) lift +22.5° above the plane. That is the one measure rotation changes, so it is the axis that visibly moves as the ring precesses.

The centre bus is never spatialised. Kick and sub live below ~150 Hz where localisation is weak, so placing them in the ring would be a claim the ear cannot check. They are the fixed floor everything else turns against.

Ring voices open with a broadband transient. HRTF cues live in spectral notches and interaural timing; a pure tone gives the ear almost nothing to localise. The chiff is what makes a bead a place.

Verification #

Per-strike measurement of the still section, ILD only (ITD is the stronger cue and does not appear in RMS):

Symmetric and correctly signed. Note that the whole-bar average is near mono by construction — a complete ring sweeps every azimuth, so it cancels. Measuring the bar average is the wrong test and will make a working spatialiser look broken.

Mastering #

Deliberately not the minitek club chain. The left/right differences are the content, so: no stereo widening, no mid-side EQ, channel-linked compression only, and −14 LUFS rather than −9 to keep transient cues intact. See c/render.mjs.

Limitations #

ac_hrtf.h is a procedural binaural model — ITD, head shadow, elevation-dependent pinna combs — not a personalised measured HRIR set. Localisation is real but generic, front/back confusions are expected, and results vary by listener and by headphone. The lane's claim is that the equivalence classes become audible operations, not that the imaging is accurate.

Status: first cut rendered 2026-08-07.