Acoustics & Music Theory
why it sounds like that
Concept

Coupled oscillator — where it appears

Two resonators joined firmly enough that neither keeps a frequency of its own, so the pair responds at two frequencies instead of one. Past a threshold set by the strength of the joining, the energy that used to decay away travels back and forth instead — which is what a wolf note is, and what a marimba bar and its tube turn out to be.

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 Where a bar and its pipe stop being two things. The two normal modes of a bar and a resonator tuned to it, against how strongly they are coupled, at 262 hertz. Below a threshold the pair has one frequency and two different decay rates — the pale curves, which are the damping splitting rather than the pitch — and above it the frequencies separate. The threshold is exact and it is not a matter of degree: it is where the coupling rate equals half the difference between the two damping rates, which for a bar of Q 197 against a tube of Q 80 is a coupling of 0.37 per cent. A marimba's own coupling is 0.62 per cent — 1.67 times the threshold, and not free: it is fixed by how much louder the tube makes the note, since the coupling that splits the pair is the coupling that carries the energy out. So the resonator model's assumption that the tube is a filter downstream of the bar is wrong at middle C, and it is wrong by less than a factor of two.

A bar and its pipe are one object

Three earlier essays treat a marimba's resonator as a filter the bar's output passes through, and both of them said in their own caveats that the coupling was not modelled. It is here, and the debt was right: the coupling is 1.67 times the threshold at which the pair acquires two frequencies instead of two decay rates, so the tube is not downstream of anything. Every consequence of that is smaller than the peaks it would have to be seen between.

instruments · Struck bar A string that decays twice opens its count at once and shuts it early. The count of separable beats a mistuned octave on A3 delivers after both notes are struck at 80 decibels, with each filter read at the level inside it, under one exponential decay of 12 seconds and under two stages — a prompt sound of 1.5 seconds carrying all but the last 20 decibels, and an aftersound of 12 seconds. One exponential: open from 1.00 s to 3.30 s, 11.8 beats. Two stages: open from 0.15 s to 1.77 s, 8.8 beats — and the single exponential struck 20 decibels softer closes at 1.77 s.

A string that decays twice is counted early

A mistuned octave's beats were found countable only between a twelfth and a quarter of a note's life, on a note decaying once. A piano string decays twice, a fast prompt sound over a slow aftersound, and the prediction was that this would open the count sooner and close it later. It opens sooner — at a seventh of a second rather than a second — and closes exactly where a single decay struck twenty decibels softer closes, so at 80 dB it holds 8.8 beats instead of 11.8. The count now rises with the strike to 90 dB, and a tuner who strikes hard is right.

tuning · Beating

Named alongside it

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

BeatingPartialSustainTuning by earAuditory filterChorus effectDampingDecayDetuningGamelanNormal modeQuality factor

All concepts