A damper cannot reach into the room

A damper changes the clock, not the colour is where a damper entered this account, and it ends by naming what every essay above it had held at a trivial setting:

The obvious pairing is the last two, because a piano in a hall is an ordinary object and neither effect is small. Taking the minimum of the string’s rate and the room’s, and then adding the damper’s rate to whichever won, is arithmetic that could be done this afternoon.

The arithmetic takes an afternoon and the composition in that sentence is wrong. Correcting it takes one pair of brackets, and the corrected version answers the question it asked with a flat no.

Where the damper goes in the expression #

The two interventions are different kinds of object and the difference is what decides where the brackets fall.

A room is a second system in series with the string, and how long a room rings is where its one number comes from. The string radiates, the walls return what they have, and a partial’s tail falls at whichever of the two rates is slower — the room is the slower of the two is the whole of that result, and the operation is a minimum because a hall cannot make a string decay faster and a string cannot make a hall decay faster.

A damper is not a second system. It is felt pressed against wire: an added loss on the string itself, so its rate adds to the string’s own, which is why the damper essay could show that a damped spectrum is a ringing spectrum translated bodily downward and nothing else. The tilt it leaves alone is the one the note that gets duller as it dies is about: the partials leave at different rates, and the difference between those rates is the colour. The operation is addition.

Compose them in the order the physics happens and the damper lands inside the minimum:

$$\gamma_{\text{received},n} \;=\; \min\!\left(\gamma_1 n^{\,p} + \gamma_d,\; \gamma_{\text{room}}\right)$$

The proposal put it outside — take the minimum, then add — and the difference is not cosmetic. Outside the minimum, a strip of felt on a piano string makes the walls of a concert hall absorb faster, which is not a thing that can happen. Inside it, the damper can be overruled, and the whole of this essay is the discovery that in an ordinary hall it always is.

The reason is a size comparison and it is not close. A piano damper brings a string down in something like a seventh of a second, which is 46 nepers a second. A two-second hall decays at 3.5, and the hall’s own number is a property of its walls rather than of its size alone — a room does not decay evenly is the qualification that goes with quoting one. The string’s own losses on the note drawn here run from 1.2 nepers a second at the fundamental to 9.2 at the eighth partial. So the damped string is more than a factor of five faster than the room on every partial in the note, and the minimum takes the room every time.

What that means the damper does, which is not what it sounds like #

A damper still works. What it stops is the string feeding the room, and what it cannot stop is the room returning what it already has.

In no room at all the damper removes 1.07 seconds from a note that would have lasted 1.60, which is the dry result already established and is what a piano in a studio does. In a room of three tenths of a second it removes 2.14, because the ringing note it is cutting short has itself been lengthened. From there the curve falls: 1.79 seconds in a one-second room, 1.33 at a second and a half, 0.90 in a two-second concert hall, 0.03 at three seconds — and at four and a half and beyond, nothing whatever. The damped note and the ringing note end at the same instant, to the resolution of the search that found it.

The proposal’s prediction over the same range is almost flat at about two and a half seconds, and at seven seconds it says the damper removes 6.70. So the two accounts differ by a factor of two and a half in a shoebox hall and by everything in a stone church, and the corrected one says something a player would recognise: in a very live room, staccato does not shorten anything.

The mechanism is worth stating in the terms a player has. Releasing the key stops the source. What the listener is in is a field that the source has already filled, and that field empties at the room’s rate whatever the source does. A pianist playing staccato in a cathedral is not playing short notes; they are playing short attacks into a sustain they do not control. The pedal is often described as the thing that takes the dampers out of the equation, and in a live enough room the room has already done it. Which is a sharper version of a claim made here from the other side: the room is part of the instrument, and what it has taken over here is a mechanism the instrument was built with.

The prediction that was made, and the answer is no #

The damper essay did not only propose the arithmetic. It proposed what the arithmetic would show, and it is worth quoting because the answer runs the other way:

If a hall’s lengthening of the upper partials outlives the damper, then playing a passage staccato in a live room delivers a colour cue the same passage delivers nowhere else, and the instrument would sound more like itself in the hall than in the studio where its exponent was measured.

Nine tenths of a second after the strike, a damped note in no room has drained 8.82 semitones of centroid. The same note damped in a room of six tenths of a second has drained 8.00; in a two-second hall, 6.57; in a three-second one, 4.25; in a four-and-a-half-second one, 1.51; and in a seven-second one, nothing at all.

A staccato note delivers the most colour where there is no room, and progressively less as the room grows. That is the opposite of the prediction, and the reason it is the opposite is the same sentence as before, read for its other consequence. The colour cue is the difference between the rates at which the partials leave. A room slower than the string replaces every one of those rates with one number, and one number has no differences in it — which is the room is the slower of the two’s own finding, that a room cannot hold a partial up above the fundamental it is also holding, arriving one essay later and one intervention along.

So the two interventions do not fight, which is what was expected of them. They stack in the same direction: the room removes the differences between the rates, and the damper removes the seconds in which what is left of them could have been heard. In a dead room the damper does all the damage and in a live one the room has already done it, and there is no room in between where the two combine to produce a cue that neither gives alone.

The one place the interaction is real, and it is at the top #

There is an interaction, and it is a small one that only exists in a middling room.

Between about half a second and two seconds of reverberation the damped note’s end is set by the room and its colour at any given instant is still partly the string’s, because the room is only just slow enough to win. That band is also where the first reflections have not yet merged into a tail, which the first eighty milliseconds are a different room treats as a separate object and which the single rate used here does not distinguish. Over that range the damped note’s drain falls from 8.00 semitones to 6.57 while the ringing note’s holds near 9.3 — so the gap between damped and ringing, which is the thing a listener could use to tell a staccato passage from a legato one on colour alone, is largest at about two seconds and is under three semitones even there.

That is a real prediction and it is a weak one. Three semitones of centroid is the difference between two instruments at the same dynamic rather than between two articulations of one, and the identification measured at the top of this account rests on a great deal more than that. The honest summary is that a hall makes an articulation harder to hear as colour, at every reverberation time drawn, and hardest where the hall is longest.

Which computation produced the numbers #

The note is eight partials of a string-like spectrum on C3, with the string’s loss rising as $\gamma_1 n^{\,p}$ at $p = 1$ and a six-second decay at the fundamental, which is the family every essay in this account has used. The damper brings the string down in fifteen hundredths of a second, which is 46.1 nepers a second added to every partial equally. The room is one reverberation time across the whole spectrum.

A note is over when fewer than two of its partials are above the threshold of hearing. That definition is the damper essay’s and it is used here for the same reason, which is that the obvious alternative — the note is over when its fundamental goes — is wrong at the bottom of the compass: the threshold of hearing is 21 decibels at C3 and 2 at a kilohertz, so the fundamental falls under it before its own upper partials do, and a centroid taken over what is left jumps upward.

That artefact is why the colour is read at a stated instant rather than at the note’s death. Past about three seconds of reverberation the note lives long enough to meet it, and the drain read at the end comes out negative — eleven semitones of centroid gained. It is a true fact about the threshold curve and a useless statement about a note, and nine tenths of a second after the strike is early enough to be clear of it and late enough that the damper has been down for half a second.

The one number that is not carried forward is the reading instant, and it is a choice. Read at half a second the ordering above is the same and the sizes are smaller, because the damper has only just come down; read at two seconds it is the same again and the dry note has ended.

Where the model stops #

A listener may not be in the reverberant field at all. How far away the room takes over computes the distance at which it does, and every number here is for a listener past it.

A room is not one reverberation time. Every hall absorbs more at four kilohertz than at a hundred hertz, so the flat line in the hero figure ought to slope downward to the right — which would make the room lose its hold on the top partials first, exactly where the damper is being overruled hardest. The arithmetic takes a band table and the room essay drew one; using it here would move the crossing and would not move the conclusion, because the gap between 46 nepers a second and 3.5 is not a gap a band curve closes.

A damper is not a step. The felt takes tens of milliseconds to seat, during which the added loss is rising rather than switched on, and a piano’s dampers are graded — heavier in the bass, lighter in the treble, and absent above the break. The step used here overstates how abruptly the string stops feeding the room, which makes the numbers above a best case for the damper: a real one removes rather less than 0.90 seconds in a two-second hall, not more.

And the listener is at one distance. The whole account takes the reverberant field as what arrives, which is what the room essay’s own closing paragraph recorded as its largest simplification. A listener close to the instrument receives the direct sound at the string’s own rates, damper and all, and would hear the articulation the model says is buried. Where that listener is sitting is the quantity this pair of essays still has no number for.

What the picture cannot show #

Whether any of it is audible as articulation. The centroid is a summary of a spectrum and not a percept, and nothing here says how many semitones of centroid drop a listener needs before two notes sound like different articulations rather than different lengths. The three-semitone gap identified above is either a usable cue or nothing, and the arithmetic cannot tell which.

Nor what a player does about it. Pianists play differently in dry and live rooms and describe the adjustment in terms of touch and pedalling rather than of decay rates. The model says the room takes the articulation out of the colour; whether players compensate by attacking harder, by pedalling less or by choosing tempi that leave more silence is a question about practice that no amount of this arithmetic reaches.

Whose instruments, and where #

A damper is a keyboard object. A harpsichord has none, a clavichord’s tangent is both hammer and damper, and a piano’s are the mechanism that makes the instrument’s sustain a decision rather than a fact — which is why the damper essay could say that a note’s length is written on the page as a note value and a tempo.

The rooms swept here run from a studio to a cathedral, and the repertoire is not evenly spread across them. The piano’s own repertoire was written for rooms between about one and two seconds, which is precisely the band where the damper still removes something and the colour cue is already half gone. The organ, which has no dampers at all and whose notes end when the key returns, belongs to the four-second end where this arithmetic says a damper would have made no difference anyway — and it is worth noticing that the instrument built for those rooms is the one that never acquired the mechanism.

The middle of the range has a second claim on it. The room chooses the harmonic rhythm found that a hall sets an upper bound on how fast chords can change before they smear, and the bound falls in the same one-to-two-second band. So a room of that length is simultaneously deciding how often the harmony may move and how much of an articulation survives as colour, by the same one number, and a composer writing for a particular building is choosing both at once without either being written anywhere.

There is a symmetry with the bowed case worth drawing out, because it runs the other way. A bow holds the number a blow hides found that a continuously driven string states its loss law in its steady spectrum for as long as the bow moves — a standing offer rather than a window. A room does nothing to that offer, because the room is not slower than a sustain that never ends; it is slower only than a decay. So the intervention that erases a struck note’s identity cue leaves a bowed note’s intact, and the instruments that lose most to a live hall are exactly the ones whose identity is in how they stop.

Still open: whether the direct sound puts the articulation back #

The simplification with the most in it is the same one the room essay recorded against itself, and adding the damper has sharpened it rather than settled it.

Everything above assumes the listener receives the reverberant field. A listener also receives the direct sound, arriving first and carrying the string’s own rates — and under a damper the direct sound is the one thing that does stop. So the damped note’s early arrival is short and its late arrival is the room’s, and the question is at what distance the short part stops being loud enough to matter.

The quantity to compute is the same one already named: the direct sound at the level the inverse-square law gives it, plus the room’s copy at the level the critical-distance ratio gives it, with the rates this essay has worked out. What would come out is a distance rather than a curve — the distance at which a staccato passage stops sounding staccato — and the prediction from the sizes above is that it is uncomfortably short, because the direct sound under a damper lasts a seventh of a second and the reverberant field lasts two seconds. A listener in the fourth row may be hearing an articulation that nobody in the twentieth row is.