A firm touch buys beats until the aftersound sinks with it

A string that decays twice is counted early put the piano’s two-stage decay under a mistuned octave. A piano string’s sound falls quickly for its first second or two — the prompt sound — and then much more slowly — the aftersound — and the essay counted how many of the octave’s beats a tuner could separate over that life. The count opens almost at once, shuts early, and holds more beats the harder the notes are struck, rising from under two beats at 60 decibels to twelve at 90.

Every one of those numbers was computed with the aftersound starting exactly twenty decibels below the strike. The essay ended by pointing out that this is the assumption with the most in it. How much of a string’s energy goes into the slow stage depends on how the string is struck — on the balance between the string’s two planes of vibration that the hammer excites, and on how the three strings of a unison share the blow — so a harder strike may well change the drop as well as the level. If it does, the rise of the count with the strike is either strengthened or cancelled, and nothing computed so far could say which.

What follows does not supply the law, which is a measurement of pianos. It does something that turns out more useful: it finds the single number the law would have to be compared against.

The count the earlier essay drew #

The octave is A3 against A4, the upper note stretched by the string’s own stiffness so that its partials and the lower note’s miss by the amounts that make the beats. Each partial of each note is its own voice and decays in two stages: a prompt stage lasting a second and a half and an aftersound lasting twelve, joined where the level has fallen by the drop. A beat between two partials — at the rate beats are arithmetic gives it — is counted while both are loud enough to be heard and their filter is narrow enough to separate them from the beats beside them, and the count at any moment is how many such beats there are.

The count’s window has two edges and they move for different reasons. It cannot open while the note is loud, because a loud note broadens every filter until neighbouring beats merge. It shuts when the partials that carry the beats fall toward the margin a listener needs above threshold, and at no moment does it hold more than the few separable beats three beats at most allowed. A harder strike pushes both edges later.

One level decides when the count ends #

The first thing to establish is which of the two quantities — the strike and the drop — the end of the count depends on.

It depends on their difference and on nothing else. Four strikes whose aftersounds start at 60 decibels — 70 decibels struck with a 10 decibel drop, 80 with 20, 90 with 30 and 100 with 40 — shut the count at 1.77 seconds, all four. Four whose aftersounds start at 50 shut it at 1.03, three at 70 shut it at 2.53, two at 80 at 3.30.

That is what the earlier essay’s check found in one case — a two-stage note closes its window where a single exponential struck the drop’s worth softer does — generalised to every strike. It has a plain reason. By the time the count shuts, the prompt stage is over and the string is ringing on its aftersound alone; the aftersound decays at its own rate from its own starting level, and how that level was reached is history the partials no longer carry. The end of the count is a fact about the aftersound’s level.

What that does to a harder strike #

The consequence for the strike is immediate, because a harder strike moves the two edges through different quantities.

With the drop fixed, striking harder raises the aftersound by exactly as much as the strike, so the shutting edge moves later — from 1.77 seconds at 80 decibels to 3.30 at 100. The opening edge moves later too, from 0.13 to 1.00 seconds, because the louder note keeps its filters broad for longer. The window shifts right and grows, and the count grows with it.

With the drop deepening one decibel for each decibel of strike, the aftersound starts at 60 decibels whatever the strike, so the shutting edge does not move: 1.77 seconds from 70 decibels to 100. The opening edge still moves later, from 0.13 to 0.30. The window shrinks from the left, and every decibel of strike costs beats. The count falls from 9.6 beats at 70 decibels to 7.5 at 100.

Why the window’s length is not the count #

The two laws also show why the break-even is below one rather than at it, which is what a first guess would put it at: if the close is pinned at one for one, surely a little less deepening than that lets the close move later and the count rise.

It does, but the count is not the window’s length. At 80 decibels the window is 1.64 seconds long and holds 8.9 beats, 5.4 beats a second. Struck at 100 with the drop fixed, the window is 2.30 seconds long and holds 11.8, 5.1 a second. Struck at 100 with the drop deepening one for one, it is 1.47 seconds and holds 7.5 — 5.1 a second again. The beats are not spread evenly through the window: its early part, while the note is still loud, has more members of the beat family above their margins and is richer in beats than its late part, where the upper partials have already dropped out.

So a harder strike trades the start of the window, which is expensive, for time at its end, which is cheap. Delaying the opening from 0.13 to 0.30 seconds loses a sixth of a second of the richest part of the count, and the close has to move later by rather more than a sixth of a second to pay for it. That is why the drop has to deepen by less than one decibel per decibel — by 0.85 — before the later close covers the later open. The asymmetry is the whole of the gap between the two numbers.

The rate at which the firm touch stops paying #

Between those two laws lies a rate of deepening at which a harder strike neither gains nor loses, and it can be found.

Striking at 100 decibels instead of 80 gains 2.92 beats when the drop is fixed. The gain falls slowly at first — it is still 2.97 at half a decibel of deepening per decibel — and then steeply: 2.05 at 0.6, 1.46 at 0.7, 0.32 at 0.8, and −0.32 at 0.9. It crosses zero at 0.85 decibels of deepening per decibel of strike. Past that the firm touch costs beats, and quickly: 1.46 at one for one, 3.73 at one and a quarter, 6.05 at one and a half.

The curve also has a maximum on the other side, and it is not at a fixed drop. The gain is largest, 3.19 beats, when the drop deepens by a quarter of a decibel per decibel. A drop that shallows with the strike makes the gain smaller again — 2.38 beats at a quarter of a decibel of shallowing, 2.00 at half — because a strong aftersound after a hard strike keeps the note loud, the filters broad and the opening late. So the aftersound can be too strong as well as too weak, and the strike that buys the most beats is one whose aftersound sinks a little as it rises.

That number, 0.85, is the answer to the question the earlier essay could not settle, put in the only form arithmetic can give it. A tuner’s firm touch buys beats on any piano whose drop deepens by less than about 0.85 decibels for each decibel of strike, and costs beats on any piano whose drop deepens faster. Whether real strings sit above or below that line is a measurement — of the prompt and aftersound levels of one note struck at several strengths — and the measurement now has a threshold to be read against.

The same number up the compass #

A threshold that held only for A3 would be a fact about one note. So the break-even was found again for octaves on C3 and A4, and for the A3 octave with each of the two stages made half and twice as long.

The octaves on C3, A3 and A4 break even at 0.85 decibels per decibel, to the second decimal place, although the three deliver very different numbers of beats — 3.8 at 80 decibels on C3, 8.9 on A3, 16.7 on A4. The number is a property of the two edges’ dependence on level and not of how many beats sit between them, and every note in the model shares that dependence because every partial loses level by the same law. The octave on A5 delivers no countable beat at any strike from 60 decibels to 100, so there is nothing for a firmer touch to buy and no break-even to find: the model finds no member of its beat family that stays both separable and above its margin at any level, which is a statement about this model at that register rather than a claim about what tuners hear there.

The stages move the number and in opposite senses. A longer aftersound raises it — to 0.93 at twenty-four seconds — because a slowly decaying tail gives each decibel of aftersound more time before the count shuts, so it takes a faster-sinking drop to cancel what the strike buys. A shorter aftersound lowers it to 0.70. The prompt sound works the other way: stretched to three seconds, it holds the note loud for longer and delays the opening edge more with each decibel of strike, and the break-even falls to 0.72; shortened to three quarters of a second, it rises to 0.95.

So the threshold is not one number for all pianos. It is one number for a given pair of decay times, and it varies between about seven and ten tenths over the range of decay times real strings show, which means a measurement of the drop’s slope has to be read against a break-even computed from that same string’s own two stages.

Why the drop might move at all #

The two stages come from two different ways the string’s energy leaves it. A string vibrating perpendicular to the soundboard drives the bridge hard and loses energy fast: that is the prompt sound. A string vibrating parallel to the soundboard, or three strings of a unison moving against one another, drives the bridge weakly and rings on: that is the aftersound. Three strings and the note that comes back worked through the unison half of that mechanism, and the pair tuned apart on purpose found that a unison’s slight mistuning is what feeds the aftersound. The drop is the ratio of the energy that goes into the slow motions to the energy that goes into the fast one.

A hammer blow is meant to be perpendicular, and the share of its energy that lands in the slow motions is set by how far it is not: the angle of the blow, an uneven hammer face that reaches one string of the three a moment before the others, the string’s own twist. A harder blow compresses the felt further and flattens its face against the strings, which is the mechanism how hard the note was struck priced for the spectrum, and it could plausibly even out the three strings’ shares — deepening the drop — or tilt the blow further — making it shallower. Both directions are physically reasonable, which is why the arithmetic has been kept out of choosing between them.

Which computation produced the numbers #

The octave, its stretch and the count are the earlier essay’s: eight partials a note falling at the model’s slope, each filter broadened by the level inside it, a pair counted while its members keep twenty decibels above threshold and stay separable from their neighbours, beats summed over the counted stretch of each member. The prompt stage has a sixty-decibel time of one and a half seconds and the aftersound twelve. The drop is 20 decibels at a strike of 80 and moves by the stated rate per decibel of strike, floored at one decibel. Strikes run from 60 to 100 decibels on each note.

The shutting edge is compared across strikes with the same aftersound level by grouping the twenty-five strikes on the difference and checking that each group’s times agree to within the trajectory’s own resolution of three hundredths of a second.

Where the model stops #

The drop is one number for every partial. A real string’s upper partials lose their energy to the bridge at different rates, and the share that goes into the slow motions is not the same for the fundamental and the eighth partial. A drop that deepens with the strike for high partials only would move the break-even, because the beats a tuner counts in an octave come mostly from its lowest coincidences, and a drop that differs by partial would shift which of them outlast the others.

The prompt stage’s length does not change with the strike. It is held at a second and a half, and the opening edge is the one that depends on it.

The strike is the same on both notes. A tuner playing an octave strikes both keys, usually with similar force, but not necessarily; an octave struck unevenly has two drops.

What no rate of deepening can tell a tuner #

Whether real pianos are above or below 0.85. That is a property of hammers, voicing and unisons, and it probably differs across a keyboard and between instruments. What the arithmetic supplies is a threshold that turns a vague question — does a firm touch help? — into a comparison of one measured slope with one number.

Whether a tuner counts the beats a strike delivers. Counted in the decay or not at all found the count opening only after the strike has faded, and a tuner who listens late in the note, as many are taught, is using the aftersound whatever the prompt sound did. On this arithmetic that tuner is listening to exactly the quantity — the aftersound’s level — that decides when the count ends.

Whose touch #

Piano tuners, whose whole method a tuner counts beats set out, are commonly taught to strike firmly and hold the key when setting unisons and octaves, and the advice is given as a way of making beats clearer. On the arithmetic here that advice is right on any instrument whose aftersound keeps pace with the strike, and it has a specific failure: on an instrument voiced so that a hard blow sends proportionally less energy into the slow motions, the firm touch shortens the part of the note where the beats can be counted. Voicing a piano — needling and hardening the hammer felt — changes exactly that proportion, so whether the advice holds is a property of how that piano was voiced.

Still open: the drop measured on one string at several strikes #

This is a debt no arithmetic can pay. The quantity it needs is the aftersound’s starting level relative to the strike, for one note struck at a range of forces, and it is read straight off the decay curve of a recording: the level at which the fast initial slope gives way to the slow one. Plotted against the strike’s peak level, its slope is the number the figures above were drawn to be compared with.

What makes it worth doing is that the answer is binary in effect. A slope under 0.85 says a tuner’s firm touch works as taught. A slope over it says the touch that makes the attack of a beat clearest is the one that buries its end, and it would say that to every tuner who has been told to play harder.