The octave is counted best over a unison left two and a half cents wide

A firm touch buys beats until the aftersound sinks with it reduced a tuner’s touch to one number. A piano note decays in two stages, a fast prompt sound and a slow aftersound, and a mistuned octave’s countable beats run out when the aftersound runs out. If the drop between the two stages stays put as the notes are struck harder, a firm touch buys beats. If it deepens by more than 0.85 decibels for every decibel of strike, the firm touch costs them. The essay could not say which, because it had no law connecting the drop to the strike. It called the drop a measurement of pianos and stopped.

The drop is not only a measurement. It has a mechanism, and the mechanism was worked out well before the essays on counting began. Three strings, and the note that comes back found that a piano’s unison strings, joined at the bridge, do not decay independently. Two strings tuned together have two normal modes: one in which they move together and drive the bridge hard, dying in a second and a half, and one in which they move against each other, barely moving the bridge, ringing for twenty. That is a prompt sound and an aftersound, the two stages a string that decays twice is counted early put under the octave’s count. The drop between them is how much less of the second mode than of the first reaches the bridge. It is the same coupling through the bridge that makes a cellist’s wolf note, which the other wolf described as a string mode and a body mode exchanging energy; here the two partners are both strings.

Computing it from the pair turns the open question into a different one. In the coupled pair the drop does not depend on the strike at all. It depends on how far apart the tuner left the two strings.

A drop the strike cannot reach #

The pair is the one the pair tuned apart on purpose used. Each string alone would decay in twenty seconds. The bridge adds a second loss to each and couples them, and the strength of the coupling is set so that the mode driving the bridge — both strings moving together — decays in a second and a half, which is the prompt sound’s length throughout these essays. The sound is what the bridge carries: the sum of the two strings’ motions, so each mode is heard at its own excitation multiplied by how much it moves the bridge.

At a perfect unison the modes are exactly “together” and “apart”. A hammer striking both strings alike excites only the first, and the second is not heard at all: it is not struck, and if it were it would not move the bridge. That is why a perfectly tuned unison is the shortest note on the instrument, which the essay on three strings found. Detune the strings and the modes tilt. The mode that rings on now moves the bridge a little and is excited a little, so it is heard, and the more the strings differ the more it is heard. At a quarter of a cent the aftersound starts 65 decibels below the prompt sound. At a cent it is 41 below; at two cents 28; at three, 20.5; at four, 14. At 5.33 cents the pair reaches the point where its two decay rates become two frequencies, and past it the unison beats instead of decaying twice.

The strike does not appear in any of this. The pair’s equations are linear, so a harder blow puts more energy into both modes in exactly the same proportion, and the proportion is the drop. Within this model the drop deepens by nothing per decibel of strike, which is as far under the 0.85 break-even as it is possible to be. The earlier essay’s firm touch works, and works on every unison. What the model moves the question to is not the touch but the tuning: the one quantity that sets the drop is how wide the unison was left.

It also moves the aftersound’s length, and in the direction that matters. The earlier essays counted beats with a 20-decibel drop over a 12-second aftersound. The pair does not supply that combination. Its drop reaches 20 decibels at about three cents, and at three cents its aftersound lasts 9.7 seconds, not twelve, because the same tilt that lets the slow mode reach the bridge also lets it lose energy there. A loud aftersound is a shorter one. That constraint was invisible while the drop and the aftersound’s length were typed in as two independent numbers.

The count, unison by unison #

With the prompt sound, the aftersound and the drop all taken from the pair, the octave’s beats can be counted as the earlier essays counted them — the octave A3 against A4, stretched by the strings’ stiffness, every partial decaying in its two stages, each pair of partials beating at the difference of their frequencies as beats are arithmetic says, and a beat counted while its partials are audible and its filter narrow enough to separate it from its neighbours, as three beats at most set the criterion — and counted again for every width of unison.

The count follows the drop, but not in proportion. Over a unison tuned closer than a cent, the aftersound starts forty or more decibels down, below the margin a listener needs, so it contributes nothing and the count is what the prompt sound alone delivers: 1.5 beats, all in the first four-tenths of a second. Between one and two cents the aftersound climbs above that margin and the count climbs with it, to 5.8 at two cents. It peaks at 6.8 beats at two and a half cents, where the aftersound starts 24 decibels down and lasts 11.6 seconds. Past three cents the aftersound is louder still, but shorter, and the count falls: 5.6 at four cents, 3.7 at five.

Two reference points sit beside the curve. The first is the count the earlier essays assumed, 8.9 beats, which no unison reaches: the pair cannot give a 20-decibel drop and a 12-second aftersound at once. The earlier counts were generous by about a quarter, and the reason is the constraint in the paragraph above. The second is one string alone. A lone string on the bridge loses energy to its own damping and to the bridge once, not twice, so it decays in a single stage of 2.8 seconds — faster than the aftersound of any unison, slower than any prompt sound — and it delivers 2.8 beats. That is more than a clean unison gives and less than half what a well-left one does.

The count opens with the strike and shuts with the aftersound #

The two edges of the count move for different reasons, and the unison reaches only one of them.

The count opens as soon as the loud strike has decayed enough for the filters to narrow and separate the beats: a tenth of a second or so after the blow over every unison, a little later over the widest, where the aftersound adds level early and keeps the filters broad. The filter is the level-broadened one of how hard the note was struck, followed through the decay as counted in the decay or not at all followed it. That edge belongs to the strike, and every unison is struck the same.

It shuts when the partials carrying the beats fall toward the listener’s margin, and that edge belongs to the aftersound. The earlier essay found that the moment the count shuts is set by the level the aftersound starts at and nothing else. Here the level is set by the unison: 27 decibels at half a cent, 46 at a cent and a half, 56 at two and a half, 70 at five. The count shuts at 0.40 seconds over the cleanest unisons, when the prompt sound runs out; at 1.4 seconds over unisons two and a half to three cents wide; and earlier again over the widest, where the aftersound starts loud but dies within four seconds. The window is widest where the aftersound is both loud enough to clear the margin and long enough to stay above it.

The note and the count want the same unison #

That window has a familiar shape, because the essay on three strings drew one very like it for a different quantity: how long the note lasts.

A clean unison makes a note that falls sixty decibels in a second and a half: all prompt sound, no aftersound. At a cent it lasts 5.7 seconds, at two cents 7.2, and from there it shortens again, to 3.3 seconds at five cents, as the aftersound gets louder and briefer. The octave’s beats follow almost the same curve, displaced by half a cent: they need the aftersound a little louder than the note’s length does, because a beat must clear the listener’s margin — the depth the beat a tuner can actually use found a listener needs — while a note need only stay above silence.

The detuning that gives a piano note its sustain is the detuning that gives a tuner an octave to count. That is not a coincidence of two curves. Both are the aftersound, one measured by how long it lasts and the other by how long it stays loud enough to hear a beat in. A unison left two to two and a half cents wide serves both, and a unison tuned dead clean serves neither.

This is where the arithmetic meets a piece of tuning practice. A tuner setting the temperament and the octaves, counting beats interval by interval, does not usually hear unisons at all. A felt strip mute is threaded between the strings of the middle section so that each note sounds on one string, the octaves and temperament are set on those single strings, and only afterwards is the mute pulled and each unison brought in to its tuned string. So the octaves are counted over single strings, 2.8 beats’ worth in this model, and then the unisons are tuned, usually by ear toward the absence of beating between them. If they are tuned truly clean, the notes the octaves were set on lose their aftersound and a later check of an octave, with all strings sounding, has 1.5 beats to count. If they are left two or three cents wide, the check has 6.8.

Tuning the unisons “clean” is therefore not the same instruction as tuning them “right”. Weinreich, who worked out the coupled pair, drew the first half of that conclusion: the aftersound, and with it the sustain, depends on a mistuning too small to be heard as beats. The second half is new here. The same slight detuning is what lets the octave’s beats survive long enough to count.

What the touch buys, unison by unison #

The earlier essay’s question can now be answered in the model’s own terms. The drop does not move with the strike; the unison fixes it. So a firm touch always gets the fixed-drop behaviour, and that behaviour depends on which drop the unison fixed.

The three curves are three different pianos to a tuner’s hand. Over a unison a cent wide the aftersound starts 41 decibels down, and until the strike puts it above the listener’s margin a harder blow buys nothing: the count sits at 1.6 from 60 decibels to 80. From 85 it climbs quickly, to 5.2 at 90 and 12 at 100. On such a unison a tuner who plays softly hears almost no octave beats and one who plays very hard hears plenty.

Over a unison two and a half cents wide the count rises steadily with the strike, from 1.5 at 60 decibels to 10.4 at 90 and 11.8 at 100, which is the behaviour the earlier essay found with its fixed drop. Over a unison four cents wide the aftersound is already loud and short, a firm touch buys a little up to 85 decibels, and past that it costs slightly, because the loud start broadens every filter for longer while the short aftersound gives nothing back.

So the firm touch works as taught on every unison between 70 and 90 decibels — the rise is there in all three curves — and how much it buys is the unison’s decision. On the cleanest unisons a hard strike is not a refinement of the technique but the only way to hear the octave’s beats at all.

A hammer that strikes one string softer #

Everything so far assumes the hammer strikes both strings alike, and hammers are voiced and aligned so that they do. A hammer that has worn unevenly, or sits slightly across the unison, strikes one string harder than the other, and that excites the slow mode directly.

At an exactly clean unison it makes no difference: the slow mode, however strongly struck, still does not move the bridge and is not heard. Once the strings are a cent apart it does move the bridge a little, and an uneven blow puts much more energy into it. A blow half as hard on one string raises the octave’s count at a cent from 1.6 beats to 6.9, and at two cents from 5.8 to 9.3 — more than the best even blow achieves at any tuning. Over a wide unison the slow mode is already well excited by the detuning alone, and the unevenness changes the count by under a beat.

This has an uncomfortable reading. A hammer out of alignment is a regulation fault, and its effect here is to make a nearly clean unison sustain and make the octave above it easier to count. A tuner whose hammers are perfectly even and whose unisons are perfectly clean has, in this model, the shortest notes and the fewest octave beats in the building. None of that says the fault should be introduced. It says that sustain and countable beats both come from breaking the symmetry of the pair, and that the tuner’s detuning and the hammer’s unevenness are two ways of breaking it.

What the model leaves out #

Two strings, not three. The treble of a piano has three strings a note and the pair here is the two-string case. Three strings have three modes, one driving the bridge and two nearly silent, and the arithmetic of the drop is the same in kind but not in number. Part of the bass has two strings a note, and there this is the model.

One polarisation. A piano string also vibrates in two planes, vertical and horizontal, and the plane parallel to the soundboard couples to the bridge more weakly. That is a second source of two-stage decay, independent of the unison and present on a single string. It is not in the pair, so the single string here decays once, and a real single string’s aftersound would add to its 2.8 beats. How much depends on the angle of the hammer’s blow, which is the other half of the mechanism the earlier essay named and not one this model computes.

A resistive bridge. The bridge is taken to absorb energy without storing any. A real bridge also has a reactive part, which shifts the two modes’ frequencies as well as their decay rates and can put the point at which the unison splits somewhere other than 5.33 cents. Weinreich’s analysis includes it; this one does not.

Stages on the fundamental. The prompt sound, the aftersound and the drop are those of the fundamental, and each partial’s stages are scaled from them as in the earlier essays. A real pair’s upper partials have their own coupling and their own detuning in hertz, which grows with the partial number — every partial beats at its own rate — so the drop at the partials that carry the octave’s beats is not quite the drop at the fundamental.

What the picture cannot show #

Whether a tuner leaves the unison wide. The figures say what a unison two and a half cents wide would do. They do not say that tuners do it, and the evidence that they do is anecdotal: some tuners describe a unison “left open” slightly for bloom, others insist on clean ones. The arithmetic gives a reason for the first practice and a cost for the second; it does not show which is common.

The strike as a pianist makes it. The strike here is a level at the string. A real blow of different force also changes the hammer’s contact time and felt stiffness, and so the balance of the partials each string receives. Those are nonlinear effects the pair cannot see, and they are the only route left in this model by which a firmer touch could change the drop.

Still open: the plane of the blow #

The unison explains a drop that does not depend on the strike. The string’s two planes of vibration explain a second one that might: a hammer rising almost vertically excites mostly the vertical plane, but a slight tilt in its travel, or a string not quite level where it crosses the hammer, puts some of the blow into the horizontal plane, whose weak coupling to the bridge makes it the second aftersound. Whether that share changes with the force of the blow is the one remaining way the earlier essay’s 0.85 could be exceeded. The computation is the same pair with its two members reinterpreted as the two planes of one string — different couplings to the bridge, a mixing set by the blow’s angle — and it would say how steep a change of angle with force would have to be to make a firm touch cost beats on a piano whose unisons are left wide.