Three strings, and the note that comes back

Two tones a few hertz apart swell and fade at their difference, and that is arithmetic anybody can hear. It is also arithmetic that assumes the two tones are two tones: added together, each with its own amplitude, neither affecting the other.

A piano has two or three strings to most of its notes, and they are not two or three sources. They are fastened to one bridge; the bridge moves; and a moving bridge is a driver on every string attached to it. What that changes is not the beating — the beating is still there when the strings are far enough apart — but everything about how the note dies.

What a bridge does to a pair #

Two strings joined at a lossy bridge have normal modes, and the modes are not the strings.

If the two move in phase, they push the bridge in the same direction and the bridge moves a lot. A moving bridge radiates, which is how the note gets out of the instrument, and radiating is losing energy: that mode dies quickly. If they move in antiphase, their forces on the bridge cancel, the bridge stays still, and the pair has almost nowhere to send its energy. That mode rings for as long as the string’s own internal losses allow.

Neither of those modes is a string. Each is a motion of the pair, and asking which string is doing which is the wrong question — in the same way that asking which of two coupled pendulums is swinging is the wrong question once they have been joined.

So a unison is one fast sound and one slow one, and they are not two strings. The fast one is what a piano technician calls the prompt sound and the slow one the aftersound, and the two-stage decay of a piano note — loud, then a long quiet tail at a noticeably shallower slope — is the join between them.

The part that is not obvious #

Here is the thing the model says that nobody would guess. A hammer striking both strings alike gives them the same initial velocity, which is exactly the in-phase motion — and the in-phase motion is the fast mode.

At a perfect unison, a symmetric strike puts one hundred per cent of the energy into the mode that drives the bridge and nothing at all into the one that rings.

That inverts the obvious expectation. A perfectly tuned unison is not the ideal case that a real one falls short of; it is the case with no aftersound.

Detuning is what rotates the modes away from the strike. As the two strings separate, the in-phase motion stops being an exact eigenvector, the slow mode starts collecting energy, and the aftersound appears. That is a purchase rather than a defect: the sustain has to be bought with a detuning.

Where the tuner actually is #

Which sets up a maximum, and the maximum can be computed.

The cliff is a real bifurcation rather than a gradual worsening, and its position is exactly the coupling strength. Below it the two modes have the same frequency and different decays — one dies, one rings, and nothing beats. Above it they have different frequencies and the same decay — the note beats at their difference and both halves die at the average rate.

A tuner is working at that boundary, and the two symptoms of overshooting it arrive together: the note starts to beat and it stops ringing. That is why a false unison sounds dead as well as sour, and why the two complaints are never separable in practice — they are the same event.

The optimum is a property of where the tail is measured #

Forty decibels is a stated choice, and the caveats below say a different threshold moves the maximum. It moves it a long way, and the way it moves is structured:

The optimum slides from the bifurcation down toward zero as the threshold deepens, by a factor of nearly three across the range, and at −20 dB it lands at 4.49 cents against a bifurcation at 4.48 — the two coincide. That is not arithmetic coincidence but the same trade seen at its limit: a shallow threshold measures the loud early part of the note, where what matters is getting as much energy as possible into the slow mode and the answer is to detune as far as the bifurcation allows; a deep threshold measures the far tail, where what matters is that the slow mode be slow, and every cent of detuning speeds it up.

That has a consequence for the conclusion drawn at the end of this essay, that the optimum detuning sits comfortably under the difference limen of about five cents. The margin is a property of the threshold. At −60 dB the optimum is a third of the limen; at −20 dB it is at it. The claim survives for any deep measure of the tail and fails for a shallow one, and which of those a listener is doing is not settled here.

And the third string #

This essay is called Three strings and the model in it has two. The caveat below says the third changes the amplitudes rather than the argument, and the model generalises in one line — every string carries its own loss and the bridge couples all of them alike — so the claim can be checked rather than assumed.

Two things change and the second is practical.

Three strings at a perfect unison die faster than two. The symmetric motion drives the bridge with three strings’ worth of force instead of two, so its decay rate is the internal loss plus three times the coupling rather than twice, and the unison case falls from 1.00 seconds to 0.68. The essay’s central point is therefore stronger on the real instrument than on the pair it models: the more strings a note has, the worse a perfect unison is, and the treble — which has three — is where a perfect unison would be deadest.

And how the three are spread matters more than how far. Spreading them evenly across a total of 2d gives an optimum at about 3.3 cents and a sustain slightly better than the pair achieves; tuning two of them together and leaving the third out gives a best of 2.6 seconds against 3.4, a fifth of the tail thrown away. Two coincident strings are one strong string as far as the bridge is concerned, so the trio behaves as a badly-detuned pair and only one of the two available slow modes is ever excited. A tuner who gets two of three exactly right and the third approximately is doing worse than one who is approximately right about all three — which is the opposite of how precision usually works, and follows from the geometry rather than from any lore about pianos.

The optimum spread is also wider on three strings than on two, at around 3.3 cents against 2.5. That narrows the margin against the difference limen from a factor of two to about one and a half, on the register that has three strings.

The same equation, with a body instead of a second string #

This site has already solved this eigenproblem once, for a different pair of objects.

A cellist’s wolf note is a string mode landing on a body resonance. The two exchange energy, the mode splits in two, and the note warbles at the difference. That is the same two-oscillator problem with the second oscillator replaced by the instrument’s body, and it lands on the other side of the same bifurcation: there the coupling is strong enough relative to the detuning that the frequencies split and the player hears a beat.

The pairing is worth stating plainly because it makes the bifurcation the object rather than either instrument. When the detuning is smaller than the coupling, coupled oscillators share a frequency and split their lifetimes. When it is larger, they share a lifetime and split their frequencies. A piano tuner lives on the first side and a cellist’s wolf note is stuck on the second, and both are consequences of one square root changing sign.

What this says about the tuning ladder #

A tuner counts beats and that is the whole method, and that rung is about the bearings — the intervals between different notes, where the two sources really are independent and the arithmetic really is subtraction. Unisons are the other half of a tuner’s work and they are not that job at all.

There is a second consequence for the tuning ladder and it is the sharper one. Every beat rate this site has computed as a target — the numbers a tuner hits when laying a temperament — is a difference between two frequencies. That arithmetic is right for a fifth between two notes. It is wrong in principle for a unison, because below the critical detuning a coupled pair does not beat at all: the two modes share a frequency exactly. The beat a tuner listens for in a unison is a symptom of having gone too far, not a quantity to be minimised.

Why three strings and not two #

The treble of a piano has three strings to a note, the middle two, and the bass one wound string. That distribution follows from the same argument and from one other number.

A single string on a lossy bridge has no antiphase mode to hide in, so it has only the fast decay. The bass gets away with it because a wound bass string is enormously more massive than the bridge admittance can drain quickly — its own losses dominate — and because a low note’s energy is much greater to begin with. In the treble the opposite holds: short, light, high strings couple efficiently to the bridge and would die almost at once, so a treble note needs the aftersound more than any other, and three strings give more ways to be out of phase than two do.

The tuning curve that stretch produces across the compass is the inter-note consequence of the same constant: every octave on the instrument is set wide because the partials that have to agree are sharp of where a 2:1 would put them. The unison problem is the intra-note consequence, and the two are usually discussed together as though they were one subject. They are two uses of one number — the inharmonicity coefficient — pointed at different pairs of frequencies, and only the second is a coupled system.

Which computation produced the numbers #

The pair is the standard two-oscillator model with dissipative coupling: each string carries an internal loss and the bridge adds a term that is imaginary rather than real, because a bridge is a place energy leaves through rather than a spring. The eigenvalues are the average frequency, minus the average loss, plus or minus the square root of (Δω/2)² − β², and the whole result is the behaviour of that square root.

The two rates are set from two decay times rather than fitted: a single string ringing for twenty seconds and an in-phase mode taken down in one and a half. Those are ordinary figures for a piano in the middle of its compass; every number in the essay scales with them, and the shape — a maximum below the bifurcation, a collapse above it — does not.

The mode amplitudes come from projecting a symmetric strike onto the eigenvectors. That is the step that produces the counterintuitive result, and it is also the step most exposed: it assumes the hammer meets both strings at the same instant with the same velocity.

The three-string figures are the same equations integrated rather than solved, because three coupled oscillators with complex damping need an eigensolver the site does not have. The integration is done in a rotating frame, with the common carrier removed so that only the detunings — a few radians a second — are left. That is not a convenience: integrating the 262-hertz carrier directly with a forward step is unstable, and a first attempt at this comparison did exactly that and returned sustains of twenty-six seconds at every detuning, a number large enough and uniform enough to look like a result. The check that caught it is the one worth keeping: the generalised model is run at n = 2 first and required to reproduce the closed form — one second at a perfect unison, a maximum near two and a half cents, a collapse past four and a half — before its n = 3 answers are read. It agrees to within a fifth of a cent on the optimum, and the three-string figures are quoted as comparisons against the two-string ones computed the same way rather than against the closed form, so that the small difference in method cancels.

What the picture cannot show #

A real hammer does not strike three strings identically. It is one piece of felt meeting three wires that are never at exactly the same height, and the asymmetry excites the antiphase modes directly. So a real piano has an aftersound at a perfect unison, from a mechanism this model deliberately excludes — and the size of that contribution is a property of the hammer’s regulation rather than of the tuning.

Only the fundamental is modelled. Each partial of a real string has its own coupling and its own decay, and the partials are stretched by stiffness, so the “unison” is only a unison at the fundamental and is progressively out at the higher partials. The two-stage decay in a real note is different for every partial, which is why the tail also changes colour.

The closed-form figures are for two strings. The three-string case has three modes, two of which are antiphase in different ways. That was previously recorded here as changing the amplitudes rather than the argument; the section above extends the model and finds it changes the numbers in the direction that matters — a worse unison, a wider optimum, and a strong penalty for spreading the three unevenly. What it does not change is the shape of the curve or the existence of the bifurcation.

The bridge is a number rather than an object. All of the coupling is folded into one loss term, which means the model cannot say anything about where the energy goes after it leaves — into the soundboard, into the case, into the room. The body is the filter on a violin and a piano’s soundboard does the same job, and that filter is exactly what this model replaces with a single constant.

Nor is the hammer’s own contribution here. A hammer is not an impulse: its contact lasts a couple of milliseconds and shapes which partials are excited at all. The strike in this model is instantaneous and identical on both strings, and both of those are simplifications on the side that makes the counterintuitive result cleaner than it should be.

And the forty-decibel criterion is a choice. A different threshold moves the maximum, by a factor of nearly three between twenty decibels and sixty, and the section above tabulates it. The optimum detuning is a property of where the tail is measured, and the tail is what a tuner is listening to — so a number quoted without its threshold is not a number.

Whose instrument, and when #

Two and three strings to a note are a property of the piano and of its immediate ancestors: a harpsichord has one or two per note with different registrations, a clavichord one pair, and a medieval psaltery whatever it had. The three-string treble is a nineteenth-century arrangement that arrived with the iron frame and the tension it made possible, which this site has priced — sixteen tonnes at A440 on one instrument.

What is not specific to the piano is the mechanism, and the reach of it is wide. Any two oscillators sharing a lossy termination behave this way: the two courses of a lute, the paired strings of a mandolin, two organ pipes on one chest, a guitar’s twelve strings in six pairs. Each of them is somewhere on the same curve, and where a tradition puts its instruments on it is a choice about what it wants — the mandolin and the celeste rank are deliberately past the bifurcation, buying a shimmer at the cost of the tail; the piano stops short of it, buying the tail.

Where this ladder goes next #

Three rungs of this ladder have treated beating as a sum. This one finds that the most common beat in a piano’s own tuning is not a sum at all, that the aftersound the whole instrument depends on has to be bought with a deliberate mistuning, and that the point where the purchase stops working is a bifurcation rather than a limit.

It also puts a number on something the instrument’s design has to respect. The aftersound is bought at about two and a half cents, and the difference limen in this register is about five — so the detuning that produces the piano’s sustain is below the threshold at which anybody could hear it as a mistuning. That is not a coincidence a designer arranged; it is a constraint the instrument had to satisfy to exist, and it is satisfied by a factor of two.

The obvious next rung is the one the coupling makes possible in the other direction: two strings tuned deliberately far apart, which is what a chorused instrument is — the mandolin, the twelve-string, the celeste rank of an organ. The same eigenproblem runs, the answer is on the other side of the bifurcation, and the thing being bought there is the beating rather than the sustain.