Shepard tone · translated to tempo

Risset Rhythm Machine

One bar, copied across five octaves of tempo. Every copy slides upward together; the fastest dissolves at the rim while a new slow one bleeds in at the centre. The beat accelerates forever and never arrives.

Vol
cycle —

Why it never arrives

Jean-Claude Risset took Roger Shepard's endlessly rising scale and moved it from pitch to pulse. The construction is the same: stack copies of one thing an octave apart, glide the whole stack up by exactly one octave, then cut back to the start — the stack you land on is identical to the stack you left.

Here, layer L runs at base × 2^(L + p), where p walks 0 → 1 across the cycle. When p hits 1, layer 2 is playing what layer 3 played at the start, layer 1 has taken over layer 2's tempo, and so on down. Only the ends disagree — and the ends are silent.

That silence is the whole trick. Gain follows a raised-cosine crossfade across the octave stack, sin²(π·u / layers) with u = L + p, so a layer is inaudible the moment it appears at the centre and inaudible again as it leaves at the rim. A gentle 2^(−0.3u) tilt keeps the fast layers from swamping the slow ones, and every voice is pitched by 2^(0.8u) — the felt hammer at the bottom of the stack becomes a glass tine at the top, exactly as Shepard's partials thin out on the way up.

The drone underneath is the original illusion, untranslated: one sine voice per layer riding the same ladder, each recycled to the bottom of the stack at the instant the crossfade has silenced it. Rhythm and pitch climb in lockstep because they read the same u.

The seam is the fragile part. Cycle length snaps so a whole number of bars fits the octave, T = N · ln2 / (base / 240), and there is exactly one integrated clock: ψ, bars elapsed by the slowest layer. The crossfade position is read back from it as p = log₂(1 + ψ/N), never counted alongside it — so nudging a slider mid-cycle moves the envelope by a hair and leaves every layer's phase untouched, instead of quietly knocking the loop point off the bar line for good. Everything else is per-hit synthesis — no samples — through a tape saturator, a dark hall, and a slow 0.37 Hz wow that keeps the mechanism from sounding quantised.

Try this

  1. Solo one layer. Hit S on any row. You hear a single loop accelerating through one octave, fading up and back out — plainly finite. The illusion lives in the sum, not the parts.
  2. Mute every layer and leave the drone running. That's a Shepard–Risset glissando on its own: rising forever, going nowhere.
  3. Kill the envelope. The stack goes flat and loud, and the octave reset turns into an audible lurch every cycle. This is the failure mode the crossfade exists to hide.
  4. Freeze the climb mid-cycle, then read the ladder. Five tempos, evenly spaced on a log axis, loudest in the middle — a Shepard spectrum drawn in BPM instead of Hz.
  5. Shorten the cycle to 8 s. The acceleration becomes obvious and the spell breaks. Around 20–40 s the ear stops tracking any single layer and the rhythm just keeps rushing.
  6. Switch to Pulse and watch the outer rings: at 1600 BPM the subdivisions have fused into a shimmer. Same bar, four octaves up.