Acoustics & Music Theory
why it sounds like that
Concept

Detuning — where it appears

A deliberate difference in pitch between two sources meant to sound as one, measured in cents or in beats per second. Below a few cents it buys sustain and above a dozen it buys width, and the two policies cannot both be held across a range because cents and hertz scale differently.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

What a chorus costs on one bridge. The sixty-decibel time of the longest-lived mode of two coupled strings at 262 hertz, against how far apart they are tuned. Below the bifurcation at 4.5 cents the pair splits its decays and one mode rings on; above it the pair splits its frequencies instead, both modes carry the bridge's loss, and the sustain sits at 2.8 seconds however much further the tuning is opened. The flat line is what the same two strings would do if they did not share a bridge. A chorused sound and a long one cannot be had from one bridge, and the first four and a half cents cost all of it.

The pair tuned apart on purpose

A piano's unison buys its sustain with a detuning of two and a half cents and loses it past four and a half. Read from the other side that is a design constraint on every instrument that wants the beating instead: past the bifurcation the sustain is gone and no further detuning costs anything more, which is why every chorused voice in the world — the celeste rank, the musette reeds, the paired gamelan — is built out of sources that do not share a bridge.

tuning · Beating One pair, 12 beat rates. Two notes at 220 hertz, 15 cents apart, drawn as the beat rate between each pair of partials. The fundamentals beat at 1.91 hertz and the k-th partials at k times that, so the series of rates is a straight line and it crosses 15 hertz at the 8th partial. Below the line the fluctuation is counted; above it the same physical fluctuation is heard as roughness. 1 per cent of this spectrum's energy is on the roughness side. The bars are drawn at each partial's own amplitude, so a spectrum with little energy high up crosses the line where nothing is listening.

Every partial beats at its own rate

Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.

tuning · Beating A wrong bar costs the same whichever instrument it is on. What moving one degree of a measured slendro by 10 cents, flat or sharp, adds to the roughness per second of a two-instrument texture — a ringing part at 0.15 s a note over a damped one four times slower — on the ringing instrument and on the damped one. The ensemble in tune scores 7.47. Degree 1 (0¢): ringing 0.142 flat and 0.202 sharp, damped 0.163 and 0.185, of which beating 0.178; Degree 2 (231¢): ringing 0.119 flat and 0.068 sharp, damped 0.096 and 0.090, of which beating 0.093; Degree 3 (474¢): ringing 0.106 flat and 0.021 sharp, damped 0.069 and 0.057, of which beating 0.063; Degree 5 (717¢): ringing 0.079 flat and 0.072 sharp, damped 0.075 and 0.078, of which beating 0.077; Degree 6 (955¢): ringing 0.023 flat and 0.018 sharp, damped 0.019 and 0.022, of which beating 0.020. Over all ten errors the ringing instrument's cost 0.85 and the damped one's 0.85, and the wrong bar beating against the other instrument's right one comes to 0.86 on either — as much as the whole, because the intervals the error changes add as often as they save. One error costs about 1.1% of the texture's roughness.

A wrong bar beats the same on either instrument

The ringing instrument carries a gamelan scale's standing, so a tuning error on it was predicted to cost more than the same error on the damped instrument. Put in the beating the model lacked, and the prediction fails: moved ten cents, a degree costs the same 0.85 summed over the scale on either instrument, because almost all of the cost is the wrong bar beating against the right one on the other instrument, and a beat belongs to both of the bars that make it. Moving a whole instrument costs exactly what its five degrees cost separately, and it takes nothing from the scale's standing.

scales · Beyond twelve

Named alongside it

The objects these essays reach for when they reach for this one.

BeatingChorus effectRoughnessCoupled oscillatorCritical bandwidthGamelanInharmonicityIntonationNormal modePartialScale degreeSlendro

All concepts