A damper changes the clock, not the colour

The eight essays here have now put two things between the string and the listener that were not in the model. Energy can be supplied while the note sounds, which moves the loss law from the middle of the note to the slope of its sustain. A room can decay more slowly than the string, which replaces the string’s rates with the room’s. This one puts in the third, and it is the one nearly every note on a piano actually meets: the player lets go of the key.

A damper is a strip of felt lowered onto the wire. What it does to the arithmetic is not obvious in advance, and the two plausible guesses are both wrong. It matters because the identity the account here has been chasing since its first essay is carried by the part of the note a damper ends. It is not a second decay in series, as a room is, so the rates do not resolve to a minimum. And it is not a broadband fade that erases the spectrum’s shape, which is what “damping” suggests. It is an added loss, and added losses add — which makes it a different object from both the drive and the room, and the only one of the three that leaves the spectrum’s arithmetic alone.

Adding a constant preserves every difference #

Write the string’s own rate for partial $n$ as $\gamma_n = \gamma_1 n^{\,p}$, and the damper’s as $\gamma_d$, the same for every partial because felt against wire is a resistive termination with no strong frequency dependence over the range in question. Under the damper each partial falls at $\gamma_n + \gamma_d$, and the quantity that makes a spectrum tilt is the difference between two partials’ rates:

$$(\gamma_m + \gamma_d) - (\gamma_n + \gamma_d) = \gamma_m - \gamma_n$$

The damper cancels out of it entirely. On the string drawn above, the gap between the eighth partial’s rate and the fundamental’s is 8.06 nepers a second while ringing and 8.06 nepers a second under the damper. The ratio of the two rates collapses from 8.0 to 1.17, which is why the damped curve looks nearly flat — but a ratio is not what tilts a spectrum, and the thing that does is untouched.

The consequence can be stated exactly rather than approximately. The damped spectrum at time $t$ is the ringing spectrum at time $t$ minus one level, the same level for every partial, growing at 400 decibels a second. Sampled half a damper-time after the key comes up it is exactly 20 decibels down at every partial; a tenth of a second later, exactly 40; a tenth after that, exactly 60. There is no residue and no approximation in that statement — it is a translation, and it holds to machine precision at every level and damper time drawn here.

So a damped note is getting duller as it dies at precisely the rate it always was. What the damper takes away is the time in which to do it.

What the note runs out of #

If the drain is unaffected and only the clock changes, then the share of the colour a listener receives is decided by one quantity: how long the note stays above the threshold of hearing.

A note on C3 held for four tenths of a second and then released onto a damper of 0.15 seconds has delivered 7.7 of the 9.9 semitones the free note would eventually have delivered — 78 per cent — and is inaudible 0.12 seconds later, having added one more semitone on the way down. Held for a tenth of a second it delivers 36 per cent; held for 1.6 seconds, 99.

The tail after the key comes up is short and it is the same length whatever the player does. The damper takes the whole spectrum down at 400 decibels a second, so a note that was ten decibels louder survives the damper for twenty-five extra milliseconds. A pianist cannot buy colour with dynamics; playing forte raises the level of everything, including the part of the spectrum that is about to be removed, and the extra audible tail it buys is two hundredths of a second against a drain that takes seconds.

That leaves the note’s written length as the only thing that decides how much of the cue goes out, and the written length is a tempo and a note value.

A semiquaver at two hundred crotchets a minute lasts seventy milliseconds and sends 29 per cent of the colour drain. A semibreve at sixty lasts four seconds and sends all of it. Between those corners the grid is monotone in both directions, which is what makes a single cell readable: a crotchet at a hundred sends 87 per cent, a quaver at the same tempo 70, a semiquaver 48.

So the identity cue the third essay built is a property of slow music. The same instrument playing the same notes at a fast tempo is sending a third of what identifies it, not because anything about the instrument changed but because the notes are shorter than the cue is. That is a strange kind of statement to be able to make about timbre, and it follows from one constant added to a set of rates.

The shorter the note rings, the more of it gets out #

The grid above is drawn at one pitch and it is very nearly the same grid at every pitch: a semiquaver at 120 sends 42 per cent of the drain at C2, at middle C and at C6 alike. That is not a discovery about registers; it is the model saying that the only timescale in it is the string’s own sixty-decibel time, which the account here has held at six seconds everywhere because it has never had a fundamental to make it depend on.

Put a real decay time in and the grid moves, in a direction that is the opposite of the obvious one. A short-ringing string delivers more of its colour cue, not less, because the whole drain is over before the key comes up.

On a string with a six-second decay, a note held four tenths of a second delivers 78 per cent of the drain. On a two-second string the same note delivers 97 per cent, and on a one-second string 100. The instrument that rings least gives away most of what it has, and the instrument that rings longest is the one whose cue an articulated passage keeps cutting off.

That inverts an intuition worth naming, because it is the intuition these essays started from. A long-ringing instrument sounds like the one that gives a listener time to hear its colour change. It does — if the notes are held. What a long decay actually buys is a slower drain, and a slower drain is one that a note of any given length gets less far into.

The comparison the third essay made is confounded twice #

The essay that found the colour draining argued that the drain rate is an identity cue by setting a piano against a harpsichord: two struck strings with different loss laws and audibly different ways of dying. the seventh essay found that such comparisons made from recordings are comparisons of two rooms as well as two instruments. This one adds a second confound and it is inside the instrument rather than around it.

A harpsichord’s dampers sit on the jack and fall the instant the key returns, which is a shorter and more uniform release than a piano’s; its strings are thinner, under less tension and stopped by a plectrum rather than struck by a hammer, and they ring for one or two seconds rather than six. Every one of those differences changes how much of the drain reaches the listener, and they do not all pull the same way. The shorter decay delivers more; the earlier damper delivers less.

So the two instruments differ in the quantity the essay was measuring, the quantity this essay has just shown decides how much of the measured quantity gets out, and — from the fifth essay — the register the comparison was made at. A single perceptual difference is being attributed to one of at least three things, and nothing in these essays currently separates them.

What would separate them is not a new instrument but a stated protocol: two notes at the same pitch, at the same level, held for the same written length, in the same room, differing only in the decay time and the exponent. That is a synthesis task rather than a recording task, and it is what the fourth essay’s listening experiment was already asking for. This essay adds one parameter to the specification — the note has to be gated at a stated length, and the length has to be reported, because a result obtained on held notes says nothing about a passage.

Where the dampers stop #

Every piano has a register with no dampers at all. Makers fit them from the bottom of the compass up to somewhere between C6 and F6, and above that the wires are left free: a key release does nothing, and the note rings until it stops by itself.

The usual explanation is a practical one — the treble notes die so fast that a damper would have nothing to do, and eighteen fewer moving parts is eighteen fewer things to regulate. That explanation is correct and it has a consequence nobody states, which the essay that put a pitch into the decay model supplies.

Those are exactly the notes with almost no colour drain to deliver. A C6 drains 13.9 semitones of centroid, a C7 10.5 and a C8 6.6, against 25.2 at the bottom of the keyboard — because a partial has to sit under twenty kilohertz to exist, and C8 has four of them. The register whose drain a player cannot end is the register that has barely any drain to end, and the registers where the cue is largest are precisely the ones a damper cuts short.

That is one mechanism seen twice rather than a coincidence. A treble string’s upper partials leave quickly and there are few of them, which is simultaneously why the note needs no damper and why it has little colour to lose. The maker’s decision and the acoustic fact are the same fact, arriving at the workshop as a question about felt and here as a question about centroids.

It also makes a prediction about the two ends of the keyboard that is worth stating plainly. In the bass, what a listener receives of an instrument’s colour drain is under the player’s control, note by note, through the written length. In the top two octaves it is not under anyone’s control and there is very little of it. So on an instrument played with any articulation at all, the colour cue is delivered by the middle of the compass, at slow tempi, on notes the player chooses to hold — which is a much narrower claim than “a piano gets duller as it dies”.

Three interventions and three different variables #

The last three essays each put one thing between the string and the listener, and the useful thing about them together is that they touch three different quantities.

A drive removes the decay altogether while it lasts and puts the loss law into the steady spectrum’s slope, so the cue stops being a rate and becomes a ratio. A room replaces the rates with its own wherever its own are slower, so the cue keeps its destination and loses its speed. A damper changes neither the rates nor the destination, and truncates the journey.

Rate, speed, duration. Nothing in these essays before them changed any of the three, because nothing in these essays before them had anything in it but a string.

They also fail in different places, which matters more than the tidiness. A drive is available only to instruments that have one. A room’s effect is largest in the bass and in large halls, and absent in a studio. A damper’s effect is largest at fast tempi and absent on a held note. An account of how a listener identifies a struck instrument has to name which of these is in force, and none of the three is a correction to be applied afterwards: each of them changes which part of the note the information is in.

Which computation produced the numbers #

The string is this collection’s eight-partial model at a stated pitch and level, with partial $n$ decaying at $\gamma_1 n^{\,p}$ and $\gamma_1$ set by a six-second sixty-decibel time on the fundamental. The damper is one further rate, $\ln(10^3)/T_{\text{damper}}$ with $T_{\text{damper}}$ of 0.15 seconds, added to every partial’s rate from the moment of release and to none before it.

The colour gone by time $t$ is the fall of the power-weighted centroid of the audible partials, in semitones, from its value at the strike — the same measure the third, fifth and seventh essay use.

The note’s end is taken as the moment fewer than two partials remain above the threshold of hearing, rather than the moment the fundamental goes under. That is a correction to an earlier convention and the reason is worth recording: the threshold of hearing at 131 hertz is some 21 decibels and at a kilohertz about 2, so under a uniform decay a bass note’s fundamental crosses its own threshold before its upper partials cross theirs. A centroid taken over what is left then jumps upward, and a measure of colour drain that ends by reporting the note getting brighter is measuring the threshold curve rather than the note.

The damper time is the one number here that is chosen rather than derived. Published figures for a piano damper put the release under two tenths of a second across most of the compass; 0.15 is inside that range, and every conclusion above is insensitive to it because the damper’s rate cancels out of every difference. What it does change is the length of the tail, and the sensitivity there is the twenty-five milliseconds per ten decibels quoted above.

Where the model stops #

Felt is not perfectly broadband. A damper is a soft porous absorber pressed against a wire, and soft porous absorbers are more effective at high frequencies than at low ones — which is the same fact that makes a room’s decay time fall with frequency. So $\gamma_d$ probably rises somewhat with partial number, the cancellation above is approximate rather than exact, and the direction of the error is that a real damper takes a little more colour off than this one does.

The damper is not instantaneous either. It is lowered by a key mechanism with its own travel, so there is a few milliseconds during which it is touching the wire without full pressure, and the rate rises through that interval rather than stepping. At 400 decibels a second the step is a good approximation over any interval a listener could resolve.

And a real piano has three strings per note. Three strings tuned very slightly apart exchange energy through the bridge and give the note two decay rates rather than one, and a damper lands on all three at once but not with identical pressure. Whether that matters here depends on whether the double decay’s second stage is still running when the key comes up, which on most notes at most tempi it is. The hammer’s own contribution to the starting spectrum, which depends on where it lands, is upstream of everything here and unaffected by any of it.

What the picture cannot show #

It cannot show the sustaining pedal, which is the obvious next dial and is a different object rather than a setting of this one. Lifting every damper at once does not merely leave each note ringing; it lets every other string in the instrument resonate sympathetically with what is being played, so the spectrum a listener receives contains partials from wires nobody struck — a sympathetic field of the kind three strings on one note describes at the scale of a single note.

Nor what a listener does with a truncated cue. Everything above is a share of a computed quantity. Whether receiving 29 per cent of a colour drain is 29 per cent as useful, or nearly as useful, or useless, is a listening question, and the answer is unlikely to be linear.

It cannot show the release noise. Felt landing on a moving wire makes a sound of its own, brief and broadband, and on a well-regulated instrument it is quiet. It is nevertheless a real event at the moment the figure draws a wall, and it is information about the instrument that this model has no term for at all — the mirror of the attack noise an attack time is not a single number is about.

And it cannot show what a player is doing instead. A pianist shortening notes is not usually trying to shorten notes; they are articulating a line, and the colour consequence is a side effect of a decision made for a different reason. Nothing here says the shorter note is worse, only that it is carrying less of one particular quantity.

Still open: whether the three interventions interact #

The three essays above each hold the other two at their trivial settings. A driven note is drawn in no room and with no damper; a room is drawn around a struck note held for ever; a damper is drawn in no room at all.

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 — and the two interventions ought to fight: the room holds up the upper partials and the damper removes the seconds in which they would have been heard being held. Whether what survives is the room’s rate on a short note or nothing at all is the sort of question the arithmetic here can answer and has not.

What it would settle is a practical claim rather than a theoretical one. 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.