A string that decays twice is counted early

Counted in the decay, or not at all ran the beats of a mistuned octave through a decaying note instant by instant and found a window. The beats are countable only once each auditory filter has narrowed enough to separate them, which happens about a twelfth of the way through the note’s life, and only until the beating partials sink under a margin above the threshold of hearing, about a quarter of the way through. On a twelve-second note that was one second to 3.3, and the count it held was small.

That window was computed on a note that decays once: every partial losing a fixed number of decibels a second. A piano string does not. Three strings and the note that comes back derived why — two strings, or two polarisations of one, coupled through a moving bridge, share out their decay into a mode that drives the bridge and dies quickly and a mode that barely moves it and rings on. The first is the prompt sound, the second the aftersound, and the fast drop between them is what a tuner means by a note losing twenty decibels in a second or two.

The earlier essay stated what the second stage should do before computing it. The fast stage pulls the level inside each filter down sooner, so the filters narrow earlier; the slow stage keeps the partials above the margin longer. So the window should open sooner and shut later, and hold more beats. The first half is right and the second is not.

Two stages, stated #

The two-stage note is built the way the single one was, with one change. Every partial of both notes starts at its level from the strike and loses energy along two exponentials at once. Most of its energy is in the prompt stage, which for the fundamental has a sixty-decibel time of 1.5 seconds; a share twenty decibels down — one per cent of the energy — is in the aftersound, whose sixty-decibel time is the twelve seconds the single exponential used. Each stage’s rate rises with frequency by exactly the law the single stage used, the 0.7 power of frequency, so the only thing that changes between the two computations is the shape of the envelope.

At the strike the two notes are indistinguishable. Over the first second the prompt stage takes twenty decibels off every partial, much faster at high frequencies than low, and then the aftersound is all that is left, falling as the single exponential did but starting twenty decibels lower.

Everything downstream is the earlier computation unchanged: each member’s filter centred on its coincidence and broadened by the level inside it, the member’s share of that filter’s fluctuation, its depth against the detection threshold at its own rate, the separation between rates that one fluctuation or two derived from the modulation filterbank, and the requirement that the beating pair stay twenty decibels above the threshold of hearing. The count is still at most the handful of members three beats at most found separable in the middle of the keyboard; what the envelope changes is when and for how long.

It opens at once #

The prompt stage does what the prediction said it would at the start. The level inside the second member’s filter at the strike is the same for both notes, but on the two-stage note it falls twenty decibels in the first second or so, and a filter whose width is set by the level inside it narrows as fast as that level falls. The second member reaches half its filter’s fluctuation almost immediately.

So the window opens at 0.15 seconds instead of 1.00. The beats of a real string are available within a seventh of a second of the strike, which on the single exponential was the privilege of a strike ten decibels gentler.

It shuts where a softer strike shuts #

The other end of the window is not decided by the filter. It is decided by the beating partials sinking under the margin, and after the prompt stage has gone the partials are the aftersound’s — starting twenty decibels below where the single exponential’s started, and falling at the single exponential’s rate.

That makes the closing time a single exponential’s closing time for a softer strike, exactly. A two-stage note struck at 80 decibels shuts its window at 1.77 seconds, where one exponential struck at 60 shuts it. The same holds at every strike: a two-stage note struck at 70 closes where a single one struck at 50 does, at 1.03 seconds, and one struck at 90 where a single one at 70 does, at 2.53.

The count held between the two ends is the consequence. Opening seven eighths of a second sooner does not make up for closing a second and a half sooner, and at 80 decibels the two-stage note holds 8.8 beats against the single exponential’s 11.8. The second member — the slow beat at 1.62 a second that a tuner actually counts — holds 2.7 of them, not the 3.6 the single exponential gave it.

A harder strike keeps buying beats #

The earlier essay found that above about seventy decibels a harder strike bought nothing: it moved the same beats later without adding to them, and below sixty the note died before a count accumulated. Gentleness was worth something up to a point, and a tuner’s touch decided when to listen rather than how much there was to hear.

On a string that decays twice that conclusion moves twenty decibels. The two-stage note delivers under one beat at 50 decibels and under two at 60, because its aftersound starts under the margin almost at once. Its count rises with every step of the strike to ninety decibels, where it holds 12.3 beats — nearly the 12.5 the single exponential holds at its best — and only falls a little at a hundred.

So on a real string a hard strike is what buys the beats, and the reason is that the part of the note a tuner counts in is the aftersound, which is the strike less its drop. A tuner who strikes an octave hard is putting its aftersound where a single-stage note struck at a moderate level would already be.

That reverses the practical advice the single exponential gave, and it agrees with what tuners do. Aural tuners are commonly taught to strike test notes firmly, and the single-exponential arithmetic had to call that a habit that costs nothing and buys nothing above seventy decibels. With the prompt sound put back, it buys the count.

Where the two notes change places #

The two curves in the strike figure cross, and where they cross is the useful number. At 80 decibels the single exponential holds 11.8 beats and the two-stage note 8.9; at 90 the single exponential holds 10.9 and the two-stage note 12.3. The difference swings from 2.9 beats one way to 1.4 the other across those ten decibels, so the two cross at about 87 decibels a note. Below that strike the model with one decay is the more optimistic about the count, and above it the model with two is.

The softer-strike law accounts for most of the gap on either side. A two-stage strike at 70 decibels holds 4.7 beats and a single exponential at 50 holds 5.2; at 80 the two-stage note holds 8.9 against the single one’s 10.1 at 60; at 90, 12.3 against 12.5 at 70. The two-stage count is always a little short of its softer twin, and the shortfall is the opening: the softer single note is legible from the strike, while the two-stage note spends its first tenth or two of a second waiting for its filters to narrow.

The member that matters most tells the same story more sharply. The second member beats 1.62 times a second, slowly enough to be the beat a tuner can actually use, and the single exponential gave it 3.6 beats at 80 decibels. The two-stage note struck at 90 gives it 3.6 beats as well, separable from a third of a second to two and a half. A firm strike on a string that decays twice buys back exactly the slow-beat count one exponential promised at a moderate one — and it delivers it a second earlier and a second shorter.

That is a practical statement about touch and it can be put plainly. On the single exponential the best strike for counting the slow beat was around seventy or eighty decibels at the ear. On two stages with a twenty-decibel drop it is around ninety, and the strikes a tuner would call gentle deliver almost nothing to count.

The drop decides the end, the prompt the beginning #

The two constants of the two-stage envelope are both stated rather than measured, and they turn out to govern different ends of the window.

The closing time depends on the drop and on nothing else. At a ten-decibel drop the window shuts at 2.54 seconds for every prompt length; at twenty, 1.75; at thirty, a second. That is the softer-strike law again, read the other way: the aftersound of a note struck at 80 with a drop of d is a single exponential struck at 80 − d, and its closing time is that note’s.

The prompt sound’s length moves only the opening. A one-second prompt sound opens the window at 0.13 seconds and a three-second one at 0.29, because a slower prompt stage narrows the filters more slowly. It barely changes the count, since the window’s end is fixed.

So of the two things about a piano string’s envelope that nobody has pinned down for this note, one — how far the aftersound sits below the strike — decides almost everything about the count, and the other barely matters. At a ten-decibel drop the two-stage note holds 12.4 beats at 80 decibels, slightly more than one exponential; at thirty it holds 4.1.

The reading that never opened #

The earlier essay set one reading of the auditory filter aside: the reading in which each filter is broadened by the level of the whole note rather than the level inside it. On a single exponential that reading never opened at eighty decibels, because the whole note is dominated by its slowly dying fundamental and its level falls too slowly for the filters to narrow before the beating partials were gone.

On two stages the whole note’s level falls twenty decibels in the prompt stage along with every partial, and that is enough. The whole-note reading now opens at 0.60 seconds and holds 1.9 beats before the window shuts at the aftersound’s 1.77. It is a much poorer count than the in-filter reading gives, but it is not nothing, and the decision between the two readings is now a matter of degree rather than of existence. A two-stage string lets the published filter parameterisation be read either way and still predict a count.

What changes about the fraction of a note’s life #

The earlier essay’s most portable claim was that the window is a fixed fraction of a note’s life, because every loss scaled with one decay time. A two-stage note has two decay times and a drop, and the claim does not survive in that form.

What survives is a pair of claims, one for each end. The window’s opening is set by the prompt sound, and scales with the prompt sound’s length. Its closing is set by the aftersound, and is the single exponential’s closing for a strike lowered by the drop — so it scales with the aftersound’s decay time, at the fraction a softer strike would give. The opening is a fraction of the fast stage and the closing a fraction of the slow one, and the count between them depends on how far apart those two decay times and the drop put them.

The model the two stages are drawn from #

Each partial’s energy at a time after the strike is the sum of two exponentials: all but the drop’s share decaying at the prompt stage’s rate, and the drop’s share at the aftersound’s. The fundamental’s prompt stage has a sixty-decibel time of 1.5 seconds, its aftersound twelve, and the drop is twenty decibels, and every other partial’s two rates are the fundamental’s multiplied by its frequency ratio to the 0.7 power, as the single exponential’s rates were. Everything else — the stiff-string spectrum falling as one over the partial number, the rounded-exponential filter with its lower skirt shallowing with the level inside it, the depth threshold, the separation of rates and the twenty-decibel margin — is the earlier computation’s.

The coupled-string model three strings and the note that comes back derived puts the aftersound’s share at something set by how the strings are detuned and how strongly they couple — an aftersound is bought with a small detuning, which is the same fact the pair tuned apart on purpose found making a piano’s unisons sustain. The drop here is a stated constant in its place, and the drop sweep is the range it could plausibly take.

What two stages still leave out #

The drop and the prompt time for this note. Neither has been measured on a particular A3 string, and the drop decides the count. A string whose aftersound is ten decibels down behaves nearly like the single exponential; one thirty decibels down delivers a third as many beats.

Two stages that differ by partial. The coupling that produces the prompt sound depends on how strongly each partial drives the bridge, and that varies up the spectrum. Here every partial has the same drop and the same ratio between its two stages, which is the simplest assumption and not a measured one.

The unison partners. A tuner setting an octave mutes the other strings of each note, and the two-stage envelope here is a single string’s two polarisations. A note with all its unison strings sounding has the richer decay the coupled-string essay drew, with beating between the strings themselves on top.

And the same measurement the envelope essays keep asking for. The collapse belongs to the bass recorded that nobody has measured whether a partial’s decay depends on its frequency or on its number. On two stages that question now has two halves, one for each stage.

What a count through a decay cannot establish #

That a tuner counts in the window drawn. The window is where the criteria are met, and a tuner listening to a sound that is losing twenty decibels in its first second may attend only to the aftersound, which on this model is exactly the part that shuts earliest.

That a harder strike is heard as better. The count rises with the strike to ninety decibels; whether a tuner finds a ninety-decibel octave easier to judge, or merely louder, is not in the arithmetic.

Whose tuning #

The practice is the aural tuning of octaves on a piano with the unison strings muted, the case the earlier essay computed and the case a two-stage decay was always going to matter to. On the usual account it applies with more force to the treble, whose prompt stage is shorter and whose aftersound is proportionally weaker, and with less to the bass, whose strings ring long in both stages. It does not apply to a harpsichord, whose plucked string decays without a strong prompt stage, and it does not apply to an organ, which does not decay at all.

Still open: the drop that a tuner’s strike sets #

The drop has been a constant on this page and it is not one: how much of a string’s energy goes into the aftersound depends on how the string is struck, since a hammer that excites the two polarisations unequally changes the share that rings on. If a harder strike changes the drop as well as the level, the rise of the count with the strike found here is either strengthened or cancelled. How hard the note was struck priced the level of a strike; the same computation with the drop made a function of the strike — through the hammer’s contact and the angle of its blow — would say whether a tuner’s firm touch buys the aftersound it appears to on a string whose two stages are fixed.