THE FOLD / GRIND / THE HOT LOOP / THE KARPLUS-STRONG
THE KARPLUS-STRONG
noise fed through a loop becomes a plucked note
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Karplus–Strong synthesis makes a startlingly realistic plucked string from almost nothing: fill a short buffer of length N with random noise, then repeatedly output the front sample and feed back the average of two neighbours into the tail. The delay line sets the pitch; the averaging is a gentle low-pass that lets high harmonics die faster than low ones — exactly how a real string decays. The fundamental sits near fs/(N + 0.5), the delay length plus the filter’s half-sample lag.
LIT verified live: for several N the measured pitch (by autocorrelation) matches fs/(N+0.5) within ~2%, and the signal energy decays monotonically (window.__karplus). FIG the fs/(N+0.5) is a low-frequency approximation (closer for longer delays); tolerance is stated, not hidden.
LIT verified live: for several N the measured pitch (by autocorrelation) matches fs/(N+0.5) within ~2%, and the signal energy decays monotonically (window.__karplus). FIG the fs/(N+0.5) is a low-frequency approximation (closer for longer delays); tolerance is stated, not hidden.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — a tiny ring buffer cycled a thousand times a second, each pass averaging and feeding back, until noise becomes a note. Karplus–Strong is that hot loop. AVAN (AI) built the instrument: the noise fill, the average-and-feedback delay line, the autocorrelation pitch estimate, and the energy-decay check.
Credit as content: Kevin Karplus & Alex Strong (1983); Jaffe–Smith’s analysis. The weave: David names the-hot-loop; I cycle a noise-filled delay line through a two-tap averager and confirm the pitch lands near fs/(N+0.5) and the tone decays — a plucked string from a loop.
Credit as content: Kevin Karplus & Alex Strong (1983); Jaffe–Smith’s analysis. The weave: David names the-hot-loop; I cycle a noise-filled delay line through a two-tap averager and confirm the pitch lands near fs/(N+0.5) and the tone decays — a plucked string from a loop.
3 ONE DIMENSION
A ring of N samples starts as noise; each step outputs the front and writes back the average of two neighbours. The loop length is the period; the averaging is the decay.
4 TWO DIMENSIONS · INTERACTIVE
The synthesized waveform for a chosen N; its measured pitch vs fs/(N+0.5) and its decay checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a plucked tone from a noise loop.
AVAN’s addition (the inverse-companion): synthesize a decaying plucked note not by modelling the physics but by looping noise through a short averaging delay — the loop length is the pitch, the averaging is the damping. The inverse of ‘solve the wave equation on a string’ is ‘cycle a noise buffer with a two-tap low-pass — a string emerges.’ Magenta is the physical model skipped; green is the feedback loop. A string from a delay line.
LIT Genuine Karplus–Strong plucked-string algorithm (Kevin Karplus & Alex Strong 1983; Jaffe–Smith analysis). Verified live: for a sweep of delay lengths N, the autocorrelation-measured pitch matches fs/(N+0.5) within ~2% (window.__karplus.pitchClose, worst ≈1.9%) and the signal energy decays monotonically from start to end (window.__karplus.decays).
FIG Honestly scoped: fs/(N+0.5) is a low-frequency approximation (the two-tap averager's group delay is exactly half a sample only near DC), so the ~2% tolerance is stated, not hidden, and accuracy improves for longer delays. The noise fill, the average-and-feedback delay line, the autocorrelation pitch estimate, and the decay check run in-browser. The AVAN inverse is honest — looping noise through a short averaging delay genuinely produces a decaying plucked tone without solving the wave equation; magenta is the physical model skipped, green the feedback loop.
FIG Honestly scoped: fs/(N+0.5) is a low-frequency approximation (the two-tap averager's group delay is exactly half a sample only near DC), so the ~2% tolerance is stated, not hidden, and accuracy improves for longer delays. The noise fill, the average-and-feedback delay line, the autocorrelation pitch estimate, and the decay check run in-browser. The AVAN inverse is honest — looping noise through a short averaging delay genuinely produces a decaying plucked tone without solving the wave equation; magenta is the physical model skipped, green the feedback loop.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN