Counted in the decay, or not at all

How hard the note was struck ended on a prediction it did not believe. The auditory filter’s lower skirt shallows as the level rises, so a loud note is analysed through a wider filter, every member of a mistuned octave’s beat family is diluted by the partials the wider filter lets in, and at seventy decibels the count of separable beats reaches zero. Tuners count beats on notes struck harder than that every working day, so the prediction was wrong somewhere, and three places were offered. The one argued for was the tuner’s own account: nobody listens to the strike. A struck note loses twenty decibels within a second or two, and the counting is done afterwards, in the decay, where the levels are gentler.

That is an argument about a trajectory, and it was made about a single level. A struck note is not at a level. Every partial of both strings starts somewhere and falls at its own rate, so the level the filter is given, the pedestal each member sits on, and the loudness of the beating pair itself are all functions of time — and so is the count. The reconciliation is either true of that trajectory or it is not, and finding out is a matter of running the same criteria at every instant instead of at one.

A note passes through every level once #

The decay used here is the one the note that gets duller as it dies built for a struck string, with one decision made explicit. Each partial loses a fixed number of decibels a second, and the rate rises with frequency as a power of it. The fundamental of the lower string is given a sixty-decibel decay time of twelve seconds and the exponent is 0.7, the figures carried for a piano string. A partial at 880 hertz, four times the fundamental’s frequency, therefore dies 2.6 times as fast as the fundamental does — thirteen decibels a second against five.

Both strings are struck at eighty decibels and the octave is tuned at its 2:1 stretch, which is 1.07 cents wide on a piano wire at A3 and makes the slowest member of the family stand still. What is left beating is the second member, at 1.62 beats a second where the lower string’s fourth partial meets the upper string’s second, and the third member at 6.48 a second, a fifth above that. The members above those beat faster than a beat can be counted, and none of them ever enters the count.

At every instant the arithmetic is the same arithmetic the earlier steady-tone counts used. Each member’s filter is centred on its coincidence; the share of that filter’s fluctuation belonging to the member is computed from every pair of components inside it; the member’s modulation depth after the pedestal is set against the published detection threshold at its own rate; and two members count as two only if their rates differ by the 1.618 the modulation filterbank gives. One criterion is new. A beating pair has to be heard at all, so both of its partials are required to stay twenty decibels above the threshold of hearing at their frequency. That margin is a choice, it is the choice most likely to be argued with, and it is swept below.

On the reading already published, the count never opens #

Start with the filter exactly as the level-dependent account used it: broadened by the level of the whole note. At the strike the two strings together are at 83 decibels and nothing is separable, which is the earlier result. Then the note decays, the level falls, the filter narrows, the shares climb — and the count stays at zero from the strike to the sixth second and beyond. It is the dashed line in the first drawing, and it is flat.

The reason is a race, and the drawing below shows who wins it.

On the dashed reading the second member’s share does not rise at first; it falls, from 0.08 to about 0.02 in the first second and a half, because the upper partials that contribute the fluctuation die faster than the low ones that dilute it. It then climbs, and it is still short of one half at six seconds. Its own partials were gone at 3.3. The third member’s share is under a fifth for the whole of the drawing.

The note becomes legible only after it has become inaudible, and not by a small margin. The whole-note level falls at five to seven decibels a second, because the whole note is dominated by the slowly dying fundamental; the partials that carry the beats fall at thirteen and more. A filter whose width is read off the slow quantity narrows too late to catch the fast one.

So the reconciliation argued for does not survive its own model. Listening to the decay does not rescue a count that the strike destroyed, if the level that broadens the filter is the level of the note. Something else has to give.

The level the published filter was measured against #

The coefficient that makes the lower skirt shallow is Glasberg and Moore’s, and it was fitted to notched-noise masking data in which the level varied was the level of the masker per equivalent rectangular bandwidth — the power arriving in the neighbourhood of the filter, not the power of the whole sound. For a broadband noise the two are proportional and the distinction never arises. For a note whose partials spread over four octaves at levels forty decibels apart, they are very different quantities.

Read that way, each member’s filter is broadened by what is inside it. At the strike that is 73 decibels for the second member and 70 for the third, already ten below the whole note, and it falls at the rate of the partials in that region rather than at the rate of the fundamental. The filter narrows as fast as the partials it analyses die, and the race becomes even.

The solid curves show what follows. The second member passes half its filter at 1.1 seconds and the third at about one; by two seconds the second holds more than eighty per cent of its filter and the third very nearly as much, while the second member’s partials are still more than thirty decibels above threshold. The count is two from a second after the strike until 2.2 seconds, when the third member’s partials sink under the margin, and one until 3.3, when the second member’s do.

That is the tuner’s account, computed. It holds on one reading of the filter and fails on the other, and the reading on which it holds is the one the coefficient was measured on. The earlier prediction of zero beats at seventy decibels was not wrong about the strike, which is still illegible on both readings, but it was wrong about which level to hand the filter, and the error was the one that made the decay look useless.

Two beats, and how many of each #

A window in which two beats are separable is not yet a window in which two beats are counted. A tuner comparing a rate against a target has to hear several whole cycles of it, and a beat at 1.62 a second takes more than half a second to complete once.

The second member — the slow one, the one a tuner actually counts — holds three and a half beats before it is gone. That is enough to say whether it is there and roughly how fast, and it is not a comfortable amount: a tuner who restrikes after four seconds is working with about three cycles each time. The third member is faster, shorter-lived and holds eight cycles, which is more material but a rate twice as hard to judge by ear, since the beats a tuner can use are the slow ones.

So the family of a mistuned octave, which every member of which is the same depth and of which at most three are separable in the middle of the keyboard, turns out to deliver its usable beat for a little over two seconds of each strike and in about three and a half cycles. The steady-tone counts said how many members a listener could separate. The trajectory says for how long and how much of each, and the answer to the second question is small.

A harder strike buys a later window, not a longer count #

The earlier argument ended with a testable difference between two habits. A tuner could strike harder, which extends the note’s life at the quiet end, or strike more gently, which gets the note into the legible range sooner. It said the arithmetic favoured the second. With a trajectory the two habits can be priced directly.

On the reading that works, a gentle strike opens the window at once and closes it early: at fifty decibels the note is legible from the strike and inaudible after a second, holding 5.6 beats in all. At seventy the window runs from a fifth of a second to 2.5 and holds 12.5. At eighty it runs from 1.1 to 3.3 and holds 11.4; at ninety from 1.6 to 4.0 and holds 11.2.

Every strike above about seventy decibels pays for its longer life with a later start, at nearly the same exchange rate, so the total barely moves. The harder strike does not buy more beats; it moves the same beats later, into a part of the decay where the tuner has waited longer for them. Below sixty the exchange fails in the other direction, and the note dies before the count has accumulated.

So the earlier claim survives in a sharper form. Gentleness is worth something up to a point and nothing after it, and the point is where the strike stops forcing the window’s opening later — about seventy decibels at the ear on this model. Above that, a tuner’s touch decides when to listen and not how much there is to hear.

The whole-note reading, for comparison, opens only for strikes of sixty decibels and below. It predicts that a tuner who strikes at the level an actual piano note is struck at never hears the beat at all, which is the first result on this page restated as a function of the strike, and it is the reason that reading is set aside.

The window is a fraction of the note’s life #

Every number so far rests on a twelve-second fundamental, and nobody has measured that value for this note on any particular piano. It would be a poor result that moved every time the decay time did. The model makes the dependence exact rather than vague.

Every loss in the model is a number of decibels proportional to time divided by the decay time. A note that dies twice as slowly passes through every one of its states twice as late, so both ends of the window scale with the decay time exactly. The window opens at 8.3 per cent of the note’s life and, with the twenty-decibel margin, shuts at 27.5 per cent. For a twelve-second note that is one second to 3.3; for a four-second harpsichord-like decay it would be a third of a second to 1.1; for a twenty-second bass string, 1.7 to 5.5.

The margin is the other choice, and it moves only the far end. A pair required to stay thirty decibels above threshold closes the window at 21 per cent of the note’s life; one required only to be audible at all closes it at 40. Neither touches the opening, because the opening is decided by the filter narrowing and not by the partials dying. That is a useful asymmetry: the one thing in the window that is a property of hearing rather than of an assumption about hearing is the moment it opens.

It also turns an unmeasured decay time from a threat into a unit. The claim that can be made without measuring any piano is that the beats of a mistuned octave become countable about a twelfth of the way through the note, and that a tuner who waits less than that is listening to nothing.

Only the middle of the keyboard has a window #

The same computation up the compass, each octave at its own stretch and each with a twelve-second fundamental, draws the geography of the result.

A1 delivers almost nothing: the members that beat slowly enough sit where the partials are crowded, and their shares never reach the criterion before the partials go. A2 has three members, each for a short stretch, holding 4.2 beats between them. A3 holds 11.8 in two members. A4 holds 23.6 in a single member beating 9.3 times a second for two and a half seconds. At A5 the octave’s members beat too fast to count at any instant.

That is the steady-tone result of three beats at most, and only in the middle of the keyboard, redrawn in time, and it keeps the shape: the middle octaves have something to count, and both ends have nothing. A real piano’s decay time falls with pitch, and because the window scales with it exactly, the upper rows would be proportionally shorter than drawn. The bass would be longer, and it would still hold very little.

The measurement the envelope never made decides the touch #

One decision was flagged at the start and has been held since: that the loss rate is a function of frequency. The alternative is that it is a function of partial number within each string — that the fourth partial of A3 and the second partial of A4 die at different rates although they sit at the same frequency, because one is a fourth partial and one a second.

The collapse belongs to the bass recorded exactly this as a measurement nobody has made: whether the exponent that governs decay across a keyboard is the exponent that governs it within a note. If it is, the loss is a function of frequency alone. If it is not, the two partials of a beating pair die at different rates, the pair stops being equal, and its depth falls through the note.

For a gentle strike the two laws give identical counts, because the window is closed by the lower string’s partial dying and the two laws assign that partial the same loss. For a hard strike they diverge completely. Under the frequency law ninety decibels delivers 11.2 beats; under the partial-number law it delivers 0.7. The upper string’s partial outlives the lower string’s at the same frequency, so the pair’s own fluctuation — which is the product of the two — shrinks with the weaker one, and its share of the filter does not reach one half until 3.6 seconds. By then the window has less than half a second left before the pair falls under the margin.

So whether a hard strike costs a tuner the beat depends on a fact about piano wire that nobody has measured, and it is the same fact the envelope’s decay arithmetic has been waiting for. That is the connection this page did not expect to make. Two questions that looked unrelated — how the colour of a note drains across a keyboard, and how hard a tuner should strike an octave — are one measurement: the sixty-decibel time of a few dozen partials on one instrument, taken at enough pitches to see whether a fourth partial and a second partial at the same frequency die together.

What the arithmetic rests on #

The filter is Patterson’s rounded exponential with its width tied to the equivalent rectangular bandwidth, and its lower skirt shallows by 38 per cent of its 51-decibel value every ten decibels, as Glasberg and Moore parameterised it. The spectrum of each string falls as one over the partial number, on a stiff wire whose stiffness comes from a piano scaling. The decay is exponential per partial with the rate rising as the 0.7 power of frequency, the depth criterion is the temporal modulation transfer function, with a threshold index of 0.03 at slow rates rising above fifty hertz, and the separation between two fluctuations is the 1.618 a bank of quality-one channels gives. The threshold of hearing is ISO 226’s, and the beating pair must stay twenty decibels above it.

Of those, the filter coefficient and the threshold are published measurements, the depth threshold is published for noise carriers rather than tones, the separation is derived from a published channel shape, and three things are choices: the share criterion of one half, which was invented for the steady-tone counts and never measured; the margin; and the decay law. The first two are swept on this page and the third is drawn both ways. The decay time is not a choice anybody could defend, and the page does not need one, because the window is a fixed fraction of it.

What one decay leaves out #

A real string decays twice. Even with its unison partners muted, a piano string vibrates in two polarisations that couple to the bridge differently, and its sound falls quickly for a second or two and then slowly for much longer — the prompt sound and the aftersound that three strings and the note that comes back derived for the unison. A single exponential per partial is the first approximation to that and not the thing itself. The fast early drop is exactly what a tuner’s “twenty decibels in a second or two” describes, and a model with it would put the level into the legible range sooner.

Nothing here listens. The window is where three criteria and a margin are satisfied at once, and whether a listener counting a slow swell in a sound that is also getting quieter uses the stretch the criteria mark, or a shorter one, or integrates across the edges, is a question for an experiment. The drawing says where the beats are available and not that they are used.

The attack is not modelled at all. The first tens of milliseconds of a struck note are dominated by the hammer’s own noise and by partials that the decay model starts at full strength, and forward masking from that transient reaches into the first fifth of a second. Since the window does not open until a second in, this does not move any number above, but it would matter for the gentle strikes whose windows open at zero.

And the level at the ear is not the level at the string. Eighty decibels is a level at the listener; a tuner’s head is a foot from the soundboard and every figure here would shift with the distance, which the scaling by decay time does not absorb.

Whose practice this describes #

The practice is aural piano tuning of octaves in the middle of the keyboard, with the unison strings muted so that one string of each note sounds. It does not describe a harpsichord, whose decay is short enough that the window’s opening at a twelfth of the note’s life arrives within a few hundred milliseconds, or an organ, whose steady tone never passes through the race at all and whose beats should be as countable loud as quiet. The earlier prediction for the organ stands and the harpsichord now has one too: its tuner, if the scaling holds, listens earlier in absolute time and to the same fraction of the note.

Still open: a note that decays twice #

The window this page draws opens at 8.3 per cent of a single exponential decay and shuts at 27.5, and a real piano string does not decay once. Its level drops quickly while one polarisation drains into the bridge and slowly while the other rings on, and the fast stage is the part tuners describe. What that does to the window can be stated before it is computed: the fast stage pulls the level inside each filter down sooner, so the filter narrows earlier, while the slow stage keeps the partials above the margin longer — which would open the window sooner and shut it later, and put more of the second member’s cycles inside it. The coupled-string arithmetic that produced the prompt sound and the aftersound already exists, and running the trajectory through its two-stage envelope needs nothing further. It would also say whether the ninety-decibel strike still delivers eleven beats once the fast stage has taken its twenty decibels, or whether a hard strike on a real string finally buys something.