The pair tuned apart on purpose

The previous rung solved a two-by-two eigenproblem and found something a piano depends on: two strings joined at a bridge are not two strings. Below a certain detuning they share a frequency and split their decays, so one normal mode drives the bridge and dies while the other cancels at it and rings on — and that second mode is the aftersound the whole instrument is built around. Past that detuning the modes swap what they share, the frequencies split instead, and the aftersound collapses.

The rung ended by naming the obvious next question: two strings tuned deliberately far apart, which is what a chorused instrument is. Same eigenproblem, the other side of the bifurcation, and the thing being bought is the beating rather than the sustain.

The answer turns out to be a design constraint, and it explains the construction of every chorused instrument there is.

The cost is a step, not a slope #

The shape of that curve is the whole result and it is not the shape anybody would guess.

A tuner working below the bifurcation is on a steep slope: at zero detuning there is no aftersound at all — a symmetric hammer blow puts all its energy into the mode that drives the bridge — and the aftersound is bought by detuning, peaking at about two and a half cents. Past four and a half it is gone.

Past four and a half cents, the sustain is 2.8 seconds and stays 2.8 seconds. Twenty cents, thirty cents, a hundred: the two modes have already swapped what they share and both of them now carry the bridge’s loss. There is nothing further to lose.

So an instrument maker choosing to chorus a pair on one bridge faces a decision with a very odd shape. The first four and a half cents cost a factor of seven in sustain and buy a beat of less than a hertz — barely a beat at all. Everything beyond that is free.

Which means there is no such thing as a slightly chorused string pair. Either the detuning is under the bifurcation, in which case the beat is too slow to hear and the sustain is intact, or it is over, in which case the sustain has already been paid for and the sensible thing is to take a proper beat rate. The intermediate case does not exist, because the intermediate case is where the sustain has been spent and the beating has not yet arrived.

Which is why none of them shares a bridge #

Now look at what the chorused instruments actually are.

An organ’s voix céleste is a second rank of pipes, tuned sharp, standing beside the rank it beats against. An accordion’s musette is two or three separate reeds in separate chambers. A Javanese gamelan’s paired instruments are two whole instruments, tuned apart, played by two people. A twelve-string guitar’s courses are tuned in unison or in octaves rather than detuned, and its chorusing comes from the octave pairs and the slight inevitable imprecision rather than from a deliberate offset.

Not one of them puts the deliberate detuning on a shared bridge.

That is the eigenproblem’s doing. Two sources that do not share a bridge have no coupling term, so there is no bifurcation to be past and no aftersound to lose: each rings for its own twenty seconds and their sum beats at their difference. The dashed line in the figure above is what that costs, and it costs nothing.

An instrument maker who wanted a chorused piano would have to give each string its own bridge, which is very nearly a description of two pianos. The honky-tonk sound is made by detuning the unisons of an ordinary piano, and the ordinary piano is exactly the shared-bridge case whose hammer strikes all its strings together: a honky-tonk piano beats and does not sustain, and the second half of that sentence is usually attributed to age.

The crossover, in an instrument nobody builds #

There is a third possibility the eigenproblem allows and no instrument uses, and it is worth naming because its absence is evidence.

A pair could be tuned exactly at the bifurcation. There the discriminant is zero, the two eigenvalues collide, and the pair has a single repeated mode — a degenerate case that decays neither promptly nor slowly but with the arithmetic mean of the two, and beats at nothing. It is the worst of both: no aftersound, no beat.

At 4.5 cents on middle C that is a beat of 0.7 hertz, which is below the rate a listener follows as a beat at all and above the detuning that keeps the sustain. A tuner who lands there has produced a note that is short and plain, and the only way to find out is to strike it.

That is a real hazard rather than a curiosity, because 4.5 cents is inside the range a hurried unison tuning lands in — the difference limen in this register is about five cents, so a tuner cannot hear the difference between 2.5 and 4.5 as a pitch and can only hear it in the decay.

Three strings, and the bifurcation stops being one #

Everything above is a two-by-two eigenproblem, and a piano’s middle register has three strings to a note. Making the matrix three-by-three is the same construction — internal loss on the diagonal, the dissipative bridge coupling on every entry — and it changes two things, one of them expected and one not.

The expected change is the direction: three strings give the energy another mode to hide in, so there is more aftersound. The size is worth having.

At the detuning where a pair has finished collapsing, a triple still has eleven and a half seconds of aftersound. And the collapse is no longer a bifurcation: the pair reaches within one per cent of its floor at 4.48 cents and is exactly flat thereafter, while the triple takes 15.8 cents to come within ten per cent and 42.7 to come within one. The knee that made the previous section’s argument — spend it all or spend none of it — is a two-string phenomenon. A triple has a slope.

The unexpected change is what happens when two of the three agree. A pair of strings at exactly equal frequency has an antisymmetric combination that the bridge cannot see at all: it is in the null space of the coupling, so it decays at the string’s own loss and at nothing else, however far away the third string is tuned.

The aftersound of a three-string note is carried by whichever two strings are closest together. Two in exact unison and a third anywhere at all gives the full twenty seconds — the single-string decay, undiminished, at a spread where the evenly-tuned triple has lost four fifths of it. What a tuner is doing on a three-string unison is therefore not what it looks like: the job is not to bring three strings into agreement but to bring two of them into exact agreement, and the third is worth much less than the first two.

The prompt sound moves the other way. Three strings drive the bridge harder than two, so the prompt decay goes from 1.50 seconds to 1.03. A three-string note has a shorter attack and a longer tail than a two-string note — a bigger contrast between the two halves of its decay, which is the quality the middle of a piano is prized for and the one the bass, on two strings, does not have.

The tuning is a beat rate, and cents do not record it #

There is a second decision behind every chorused rank, and this collection has already made the argument in a different room.

A gamelan’s tuning is not a table of cents because the paired instruments are tuned to a beat rate, and a beat rate and a cent value are different policies: a constant detuning in cents doubles the beat every octave, while a constant beat rate means the detuning in cents halves every octave.

The numbers put the size of the disagreement beyond argument. Two beats a second needs 52 cents at C2 and 3.3 cents at C6 — a factor of sixteen across four octaves. Ten cents throughout gives 0.4 beats a second at C2 and 6 at C6.

An organ tuner setting a celeste works by ear, at a rate, up the compass — and therefore lays a curve in cents that no cent table would have suggested. That is the same finding the gamelan rung reached from the opposite direction, and the reason it appears twice is that it is a fact about beating rather than about either instrument.

Three reeds, and why not four #

The other question a chorused voice has to settle is how many of them, and the answer has a computable edge.

The choir rung found that a unison’s fluctuation does not go away as voices are added, but its rate does: two voices put nearly all the modulation into one line, sixteen spread it across a band with no line in it, and that is what a choir sounds like. A chorused instrument is a unison of two or three, which is the same axis at its other end.

Adding a fourth voice at the same total spread does not add a fourth rate; it adds three more pairs, and the pairs start to fill in between the lines the first three left. That is where the count stops being a tuning and starts being a texture.

Eighty-seven per cent, sixty-seven, thirty-six. Two and three keep a rate; four does not. Every chorused stop and every accordion register in ordinary use has two or three voices, and the arithmetic says where the boundary is.

The symmetry is worth twenty points, and not more #

The obvious follow-up is whether the symmetry of the detunings is doing any work, since a musette’s reeds are placed deliberately and a choir’s singers are not.

It is, and less than expected. Three voices at −15, 0 and +15 cents keep 67 per cent of the fluctuation in one line. Three voices drawn at random from a distribution of the same width average 45 per cent, and the individual draws run from 30 to 79 — so some random triples do better than the symmetric one.

The symmetry buys about twenty points on average and the distributions overlap. At four voices it buys nothing measurable: the symmetric four sits at 36 per cent and random fours average 38.

That is a weaker result than the design of the instruments would suggest, and it is worth reporting as weak. What makes a musette a musette is not that its reeds are symmetric; it is that there are three of them and they are close enough together for every pairwise rate to be countable.

Which computation produced the numbers #

The sustain curve is the previous rung’s eigenproblem with no change. Two strings with internal loss γ and a dissipative bridge coupling β have eigenvalues (ω₁+ω₂)/2 − i(γ+β) ± √((Δω/2)² − β²), the square root changes character at Δω = 2β, and the sixty-decibel time drawn is 6.9078 divided by the smaller of the two modes’ decay rates. γ and β are set from two published decay times — twenty seconds for a single string and one and a half for the prompt sound — and the bifurcation at 4.5 cents falls out of them rather than being placed.

The modulation spectra are the choir rung’s measurement applied to chosen detunings instead of drawn ones: sum the components, take the envelope, remove its mean, and take the discrete Fourier transform up to 30 hertz, past which a fluctuation is heard as roughness rather than as a beat. Concentration is the share of the modulation power in the largest single component.

The comparison against random spreads uses the same standard deviation as the chosen offsets, averaged over five seeds, so the two are matched on width rather than on range.

One number is a choice and it changes the fourth figure: the four-voice case is drawn at −15, −5, +5, +15, which is even spacing over the same total width. Even spacing is not the only symmetric arrangement of four, and a maker who put them at −15, −15, +15, +15 would get two voices’ behaviour with four reeds’ loudness. Nothing here searched the arrangements.

Whose music, and when #

The celeste rank is a nineteenth-century French organ-building device, and its name says what it was for. The musette tuning of a French accordion is early twentieth century and is a regional signature: the same instrument built for a Scottish or an Italian market is tuned with a different spread, and players describe the difference in beats rather than in cents.

Javanese paired tuning is older than either and is the case where the effect is not an ornament but the point — ombak, the wave, is a named quality of a gamelan and a thing a tuner is judged on. It is also, in the terms the scale rung set out, the case that makes the shared-bridge argument concrete: the pair is two separate instruments because it was never going to be anything else.

The honky-tonk piano is the exception that proves the constraint. It is the only common chorused instrument built on a shared bridge, it is made by detuning an ordinary piano’s unisons, and it is short.

What the picture cannot show #

The coupling model has one bridge parameter and a real bridge has a frequency response. β is taken as a constant, so the bifurcation is at the same detuning in cents at every pitch. A real bridge is stiffer at some frequencies than others and the bifurcation moves with it, by an amount this collection has no measurement for.

The three-string result depends on the coupling being purely dissipative. The exact twenty seconds for two strings in unison is a null-space statement: the antisymmetric combination is invisible to a coupling proportional to the all-ones matrix, and it is invisible exactly. A real bridge has a reactive part as well, and two real strings differ slightly in impedance, so the null space is approximate and the twenty seconds is an upper bound rather than a measurement. The direction of the result — that the closest pair carries the tail — survives any of that; the exactness does not.

The modulation measure is about the envelope and not about the ear. A listener hearing 67 per cent of the fluctuation in one line does not necessarily hear a beat at that rate; whether the transition from beat to texture happens at the same place in a listener as in a Fourier transform is a listening question.

And nothing here says a chorused sound is good. The whole essay is about what it costs and how it is built. Why two slightly different copies of a note should be worth building an instrument around is a question about preference, and this collection’s honest position is that it has no model of it.

Where this ladder goes next #

Five rungs. Beats are arithmetic anybody can hear; a tuner counts them and that is the whole method; a cellist’s wolf is the same arithmetic on a coupled pair; a piano’s unison buys its sustain at a detuning below a bifurcation; and a chorused voice is the same pair past it, which is why none of them shares a bridge.

The rung after it is the one every figure in this ladder has quietly assumed. Every beat computed here is between two fundamentals, and two real notes beat between all their partials at once: a pair fifteen cents apart beats at 1.9 hertz on the first partial, 3.8 on the second, 5.7 on the third, and by the eighth the rate is past the fifteen or so per second at which a beat stops being a beat and becomes roughness. So a chorused note is a beat at the bottom of its spectrum and a roughness at the top of it, simultaneously, with a crossover partial that can be computed from the detuning — the same boundary roughness needs a few hundred milliseconds to establish sits on — and every figure in this ladder has drawn only the bottom.