The plane of the blow matters only where the unison is clean

The last three essays on counting a mistuned octave have been about one number: how far below the strike a piano note’s aftersound begins.

A string that decays twice is counted early found that a two-stage decay — a prompt sound, then a slow aftersound — opens the count sooner and closes it where the aftersound sinks below a listener’s margin. A firm touch buys beats until the aftersound sinks with it turned that into a rule for the touch: striking harder buys beats, unless the aftersound’s drop below the strike deepens by more than 0.85 decibels for every decibel of strike. The octave is counted best over a unison left two and a half cents wide then computed the drop from its mechanism, the two coupled strings of a unison, and found that it does not depend on the strike at all. It depends on how wide the tuner left the unison.

That essay named one mechanism it had left out, and it is the only remaining way the 0.85 could be exceeded. Every string vibrates in two planes: up and down, towards the soundboard, and side to side, parallel to it. The bridge moves easily up and down and much less easily sideways, so the vertical plane drains its energy into the soundboard quickly and the horizontal plane rings on. A hammer rising exactly vertically excites only the first. A hammer whose travel is tilted a few degrees, or a string not quite level where it meets the felt, puts part of the blow into the second — and if the tilt changed with the force of the blow, the drop would change with the touch.

How the count is made #

The count is the one every essay on counting a mistuned octave has used, unchanged. A mistuned octave on A3 is two notes of eight partials each, stretched by the string’s stiffness, and every pair of partials that coincide beats at the difference of their frequencies, as beats are arithmetic says. A beat is counted while both its partials are audible above a listener’s margin and while the auditory filter at the coincidence is narrow enough to separate it from its neighbours — the criterion of three beats at most — with the filter broadened by the level at which the note was struck, as how hard the note was struck found, and followed through the decay as counted in the decay, or not at all followed it. What changes from essay to essay is only the envelope each partial decays along. Here it has a prompt sound and up to three aftersounds, one for each slow mode the strings and their planes provide, each starting at its own depth below the strike and each fading at its own rate.

That is also why the result can be stated as a substitution. The count reads the envelope, and an envelope made of several slow exponentials is dominated by the loudest of them for as long as it lasts. A second aftersound ten or more decibels under the first changes the envelope by a few per cent, and the count by a fraction of a beat.

The same pair, turned on its side #

The mechanism is the coupled pair of the earlier essays, the pair tuned apart on purpose, with its members reinterpreted. There, two strings shared a bridge; here, two planes of one string share it. Each plane decays through the string’s own losses and through the bridge. The difference is the coupling: the vertical plane loses to the bridge at the rate the earlier essays used, the one that makes a unison’s bridge-driving mode last a second and a half, and the horizontal plane loses to it some number of times more slowly — call it the coupling ratio.

What the blow puts into each plane follows from its angle: the cosine of the tilt into the vertical plane and the sine into the horizontal. What the listener hears from each plane is what that plane pushes into the bridge, which is its excitation times its coupling. So the horizontal plane is heard at the tangent of the tilt divided by the coupling ratio, relative to the vertical, and the drop from the prompt sound to this second aftersound is twenty times the logarithm of the ratio times the cotangent of the tilt.

With the horizontal plane coupled ten times more weakly, a blow tilted two degrees starts the second aftersound 49 decibels down; five degrees, 41; twelve, 33; twenty, 29. The aftersound lasts 12.4 seconds at that coupling, against 2.8 for the single string’s prompt sound. A coupling three times weaker gives drops ten decibels smaller and a 6.5-second aftersound; thirty times weaker, ten decibels larger and 16.6 seconds.

The coupling ratio is not measured here, and the figure is drawn at three values for that reason. The measured behaviour of piano strings — a quick first decay and a slow second one even on a single string — says only that the horizontal plane is coupled much more weakly. Everything that follows is drawn at ten times and would shift by about ten decibels either way at three or thirty; none of the conclusions depends on which.

Two checks tie this to what was already computed. Struck straight down, the four-mode unison — two strings, two planes each — reproduces the earlier essay’s two-mode counts exactly, 1.51 beats on a half-cent unison and 6.81 on a two-and-a-half-cent one. And a single string struck straight down decays once, in the 2.8 seconds the earlier essay gave it, and delivers the same 2.81 beats.

Where it fills a gap, and where there is none to fill #

The earlier essay’s curve rose from 1.5 beats over a clean unison to 6.8 over one left two and a half cents wide, because a clean unison has no aftersound: its slow mode is neither struck nor heard. The second plane gives it one. At a tilt of ten degrees the clean unison’s count rises from 1.5 to 2.2; at twenty degrees to 4.9. One string alone, muted from its neighbours the way a tuner sets the temperament, goes from 2.8 to 3.1 at ten degrees and 4.3 at twenty.

Over the unisons that were already counted well, almost nothing happens. At two and a half cents the count is 6.81 struck straight down, 6.87 at five and ten degrees, and 7.25 at twenty. The reason is in the drop figure: a unison two and a half cents wide starts its own aftersound 24 decibels down, and at a realistic tilt the second plane’s starts 35 to 45 down. An aftersound ten or twenty decibels quieter than one already sounding adds a few per cent to it. The second plane is heard only where the first mechanism is absent.

So the two mechanisms do not add; they substitute. The unison’s detuning makes an aftersound where it is left wide, and the blow’s tilt makes one where it is left clean, and the count is set by whichever is louder.

How fast the tilt would have to straighten #

The question the firm-touch essay left was whether a harder blow could deepen the drop faster than its 0.85 decibels per decibel. With the drop equal to twenty times the logarithm of the coupling ratio times the cotangent of the tilt, the question becomes how the tilt changes with force.

For small tilts the cotangent is one over the angle, so the drop is a constant minus twenty times the logarithm of the tilt. A tilt that halves every so many decibels of strike therefore deepens the drop by twenty times the logarithm of two — 6.02 decibels — per that many decibels. The break-even rate of 0.85 is a tilt that halves for every 7.1 decibels of harder strike. That result does not depend on the coupling ratio or on the tilt at any one strike; they cancel. The coupling ratio sets how far down the second aftersound starts, but it sets it by the same number of decibels at every strike, so it moves the whole curve of drop against strike up or down without changing its slope — and the slope is all the break-even asks about. The tilt at one strike does the same: it fixes where the curve sits at 80 decibels, not how steeply it rises. Only the rate at which the tilt changes with force reaches the slope, and for small angles that rate enters as a ratio, so a tilt going from ten degrees to five is the same step as one going from two to one. The approximation fails only for tilts large enough that the cotangent stops being one over the angle, which on a regulated action would be tilts nobody would accept.

Drawn out, the four laws do what the arithmetic says. A steady tilt keeps the drop at 35 decibels whatever the strike. A tilt halving every fourteen decibels deepens it by about 0.43 decibels per decibel, under the break-even. Halving every seven decibels deepens it at 0.86 — on the dashed line. Halving every three and a half decibels deepens it at 1.72, twice the break-even.

That is a precise statement of what would have to be true of a piano action for a firm touch to cost a tuner beats by this route. Across fourteen decibels of strike — roughly the range between a moderate test blow and a hard one — the hammer’s tilt would have to fall to a quarter. Nothing in how a grand or upright action is built suggests the angle of the hammer’s travel depends on the force of the blow at all, let alone that steeply: the hammer rotates about a fixed centre and meets the string where it meets it. A string that is not level across the felt presents the same tilt at every force. What can change with force is the felt’s contact — a harder blow flattens more felt against the string — and whether that moves the effective line of the blow is a question about hammer felt that this model cannot answer.

Over a clean unison the touch depends on it #

Over a unison tuned half a cent wide the second plane is the whole aftersound, and the tilt law decides what a firm touch does. With a steady tilt the count rises from 1.6 beats at 70 decibels to 2.2 at 80, 6.1 at 90 and 10.4 at 100 — the firm-touch rule working as the earlier essays found it. With a tilt halving every fourteen decibels it still rises, more slowly, to 7.0. Halving every seven decibels holds the count between 2.2 and 2.3 from 70 decibels to 90. Halving every three and a half decibels makes it fall, from 2.2 at 80 to 1.4 at 90, before the very loudest strikes lift everything.

So the firm-touch essay’s warning survives in one corner. On a piano whose unisons are tuned truly clean, a tuner who strikes the octaves hard is relying on the blow’s tilt for every beat above the prompt sound’s one and a half, and if the tilt somehow straightened fast with force, a firm touch would cost beats. The corner is small: it needs clean unisons, a tilt law nobody has observed, and a single test note.

Over a wide unison nothing can #

On a unison left two and a half cents wide the four curves are indistinguishable. With the tilt steady the count runs 1.5, 2.9, 6.9, 10.7 and 11.6 beats from 60 decibels to 100; with it halving every three and a half decibels, 1.6, 4.1, 6.9, 10.4 and 11.8. The largest difference anywhere is about a beat, at 70 decibels, where the fast-straightening law has tilted a soft blow to thirty degrees; from 80 decibels up the four agree to within a third of a beat.

This closes the question the firm-touch essay opened. The unison essay showed that the drop from the pair of strings cannot move with the touch. This one shows that the drop from the pair of planes could move with it in principle, but that on a unison left open by the amount that gives the longest note and the most countable octave, it cannot reach the count, because the unison’s own aftersound is ten to twenty decibels louder. Over a well-left unison, the firm touch buys beats, and nothing about the blow can take them away.

The connection that makes the substitution make sense #

The two mechanisms are the same mechanism, and the coupled pair is the reason. Three strings, and the note that comes back introduced it: two oscillators sharing a lossy support split into a mode that drives the support and dies fast and a mode that barely moves it and rings on. The unison’s two strings are one such pair; each string’s two planes are another. The listener hears, in either case, the slow mode’s share of what reaches the bridge.

What differs is how the slow mode gets its share. In the unison, detuning tilts the modes so that the slow one is struck a little and heard a little. In the string’s two planes, the modes are already the planes, and what gives the slow one its share is the tilt of the blow. Both are ways of breaking a symmetry — the earlier essay said so about an uneven hammer, and the tilt is a third way — and both deliver an aftersound for the same reason. That is why they substitute rather than add: each is a way of feeding the same kind of slow mode, and once one of them has fed it enough, feeding it a little more from the other changes little.

What the model assumes #

The planes are independent. The bridge is taken to move vertically for the vertical plane and horizontally for the horizontal plane, with no cross-talk. A real bridge moving obliquely couples the planes, so energy put into one leaks into the other; that would make the second aftersound present even for a perfectly vertical blow, and shift every curve here upward at small tilts.

The two horizontal planes of a unison are not solved as a pair. Their coupling is weak enough that their own split point falls near half a cent, and near a split a sum of separate exponentials misrepresents the envelope. They are taken to decay independently through their own weak bridge loss and to be heard together. A full four-mode solution would change the numbers over the cleanest unisons and not the conclusion about wide ones.

The coupling ratio is chosen, not measured. Ten is an illustration; three and thirty move every drop by about ten decibels. The seven-decibel break-even does not depend on it.

The tilt is the only thing the force changes. The felt’s stiffness, its contact time and the balance of partials it sends into the string all change with force, and all are outside a linear model. The earlier essays priced the touch with the same assumption.

What the picture cannot show #

Whether hammers strike at an angle, and by how much. A tilt of a few degrees from string irregularities and action geometry is plausible; the figures here span zero to twenty degrees because the actual distribution of tilts on real actions is not known here. On a clean unison that distribution decides the count.

And what a tuner hears as “clean”. A unison half a cent wide is below anything a tuner hears as beating, and the earlier essay found tuners disagreeing about how open to leave unisons. On the clean ones, this essay says, the octave’s countability is in the hands of the hammer rather than the tuner.

Still open: whether the tuner’s own test blow is tilted #

The strip mute that tuners thread through the middle section, so that the temperament is set on single strings, makes every note a single string for the whole of that work — and a single string’s only aftersound is the second plane. So the one situation in which the blow’s tilt carries the whole count is the one in which the octaves and the temperament are actually set. The computation that would say how much this matters is the single string’s count at the tilts a real action produces, set against the count over the tuned unisons that replace the mute afterwards, and it needs one number this model does not have: the tilt of a hammer’s blow on a regulated piano, which is a measurement rather than an arithmetic.