The note that gets duller as it dies

The shape of a note is most of what an instrument is, and this site draws that shape as an ADSR: an attack, a decay, a sustain level, a release. One curve, applied to the sound as a whole.

Nothing that is struck or plucked has one curve. A string’s losses — to the air it is dragging, to friction inside the wire, to the bridge it is driving — all rise with frequency, so its partials do not die together. The high ones go first and the low ones are still there, and what a listener receives is not a note fading but a spectrum collapsing.

This is not a subtlety that needs equipment to notice. Strike a low note on a piano, hold the key, and listen for four or five seconds: the sound at the end has no brightness in it at all, and the note has not changed pitch. Nothing in an ADSR curve can represent that, because an ADSR curve has no spectrum in it to change.

Sixteen decay times, not one #

Model the loss as rising with partial number: γₙ = γ₁·nᵖ, with p somewhere between about a half and just over one for real strings.

At p = 1 — decay time inversely proportional to partial number, the ordinary first approximation — a note whose fundamental takes six seconds to fall sixty decibels has a second partial that takes three, a fourth that takes one and a half, and a sixteenth that takes 0.38 seconds.

The picture a single-rate envelope would draw is a family of parallel lines and a horizontal centroid. Neither is what is there, and the difference is not a detail: it is the whole character of a plucked or struck sound.

What this adds to the identity argument #

The rung before this one found that cutting the first fifty milliseconds off a recorded note stops listeners naming the instrument while the spectrum they hear is unchanged — identity lives in the attack.

That result is about the start of the note, and it is usually taken to mean the steady part carries nothing. The decay says otherwise, and says it in a way that is easy to check by ear on any instrument to hand. How fast the colour drains is itself an identity cue, and it distinguishes exactly the instruments whose attacks are most alike.

A harpsichord and a piano have similar attacks and completely different p. A guitar’s nylon string and its steel string differ far more in this than in either their attack or their initial spectrum. A struck metal bar barely drains at all, which is why a glockenspiel note sounds the same at the end as at the beginning and a piano note does not.

What that exponent is doing is easier to see against the drawing this ladder started from, in which a note’s whole decay is a single curve and the spectrum inside it is not represented at all.

The room does it too, and the two multiply #

This site has already found the same shape in a completely different object.

So a note in a room has two spectral decays multiplied together — the string’s own and the room’s — and they point the same way. That is worth stating because it explains a familiar effect that neither explains alone: the tail of a chord in a large hall is much darker than the chord, considerably darker than either mechanism would make it, and the darkness arrives faster than the level falls.

Why the exponent is a design variable #

If one number decides how fast a note’s colour drains, an instrument builder has to have an opinion about it, and the ways of moving it are the ways instruments differ.

Material. Internal friction is a property of the wire: steel is very low-loss, gut and nylon are not, and a gut string of the same pitch and tension drains faster and further. That is a large part of why a period instrument sounds different from a modern one in a way that survives being recorded and played back at the same loudness.

Termination. The bridge takes energy out in proportion to how well the string’s impedance matches the soundboard’s, and that match varies with frequency. A stiff, light bridge takes more of the top out; a heavy one takes less and radiates less of everything. This is the same trade the bell of a wind instrument makes: whatever escapes efficiently also stops sounding sooner.

Length and tension. Air drag on a thin wire is not negligible for the high partials, and it scales with surface area against mass — so a light, thin treble string drains faster than a heavy bass one for reasons that have nothing to do with either’s material.

None of those is a free choice. Each is coupled to loudness, to sustain and to what the string’s own scaling allows, which is why the exponent varies across an instrument’s own compass as well as between instruments.

What a single-rate envelope was hiding #

Three things in this collection are affected, and it is worth naming them because each is a number that moves.

A struck chord’s roughness falls faster than its level — and running it rather than asserting it says most of that is not the spectral collapse at all. Scoring a struck major triad’s summed roughness and its level at every instant of a six-second decay:

The factor of two is already there at p = 0, where every partial decays at the same rate and nothing collapses. Roughness is quadratic in amplitude, so it falls at twice the rate of a level measured in decibels whatever the spectrum is doing — and that accounts for 77 per cent of the effect at p = 1. The spectral collapse adds the rest: 2.00 becomes 2.59, a further thirty per cent.

So the claim is true and the mechanism named for it is the smaller half. What the collapse contributes is real and it is an increment on something that would happen to a chord of pure tones fading at one rate.

The perceptually useful form is the ratio at each instant. Taking roughness per unit level, a struck triad is at 0.78 of its initial harshness after a tenth of a second, 0.44 after half a second and 0.24 after one — so a struck chord is about four times smoother for its loudness one second in than at the attack, which is the quantity a listener is actually comparing when a chord “settles”.

One consequence of the half-second timescale is worth taking to the room figures below. The string’s collapse is over in half a second and a hall’s reverberation time is measured in seconds, so the two do not overlap in the way “multiplied together” suggests. What reaches the room is a sound that has already lost its top; the room then takes the top off what is left, over a much longer span. They compound, but sequentially, and the darkness of a long tail is mostly the room’s because the string had finished its part of the job before the first reflection arrived.

Any figure that draws a spectrum draws an instant. Every bar chart of partials on this site is the spectrum at some unstated moment, and for a struck instrument that moment matters more than the instrument does. The convention here has been to draw the spectrum at the attack, which is the brightest instant there is.

And a decaying note’s pitch salience changes. The pitch of a complex tone is carried mostly by the partials in the dominance region around the third to fifth. Those decay faster than the fundamental, so the evidence for the pitch is strongest early and thinnest late, and a note that has decayed for several seconds is being heard on its fundamental alone.

A spectrum is an instant that is never labelled. On a bowed or blown instrument the label does not matter because the note holds still; on a struck one the eighth partial is gone before the first second is out, so any single list of amplitudes describes a moment that has already passed.

Which computation produced the numbers #

The decay law is γₙ = γ₁·nᵖ, which is the standard one-parameter family: p = 1 is loss proportional to frequency, p = 2 is the thermoelastic and viscoelastic behaviour of a stiff wire, and p = 0 is the single-rate envelope every other figure here uses. Measured piano strings sit between about 0.5 and 1.2 depending on the register and the string’s construction, and p is stated rather than fitted to anything.

The centroid is the power-weighted mean partial number over the timbre’s own partial list, computed at each instant from the amplitudes drawn.

The starting spectra are this site’s own TIMBRES table, unchanged, so the first instant of every figure here is the spectrum every other figure in this family draws.

The sixty-decibel decay time of six seconds is a plausible figure for a piano note in the middle of the compass. Everything scales with it: halving it halves every time in the essay and moves no ratio.

The roughness figures use the same summed Plomp–Levelt kernel every consonance figure on the site uses, applied to three notes each carrying its own decaying partial list rather than a single one — which is the whole of what makes the comparison possible, since a chord scored from one shared spectrum has no way to be at different points of its own decay. The level is the root-mean-square over the same partials, so the two quantities are read off the same amplitudes at the same instant and the ratio between their half-lives is not a comparison of two models.

The one thing the collapse makes easier #

A decaying spectrum is usually a nuisance for analysis and it turns out to be a help for one thing: it makes the fundamental easier to find late in the note than early.

At the attack the partials in the dominance region are at full strength and the fundamental is one line among many. What is worth having is the timescale, because it is much shorter than the tail this section is named for. At the attack the fundamental carries 63 per cent of the note’s energy and the third to fifth partials carry 13; at a quarter of a second those are 84 and 4; at half a second, 92 and 1.2; at one second, 97.5 and 0.1.

The collapse is essentially finished half a second in. The fundamental passes ninety per cent of the energy at 0.44 seconds at p = 1, at 1.12 seconds at p = 0.5 and at 0.23 at p = 1.5 — so the whole transition from complex tone to nearly a sine happens inside the first note of a phrase rather than in the four-second tail. The note’s own pitch evidence narrows that fast: less of it, but far less ambiguous, and the narrowing is over before a player has reached the next note.

That is the opposite of the usual account, in which a note’s pitch is established in its first tens of milliseconds and then merely persists. Both are true and they use different evidence — the residue mechanism reading the partials at the start, and something much simpler reading a nearly pure tone at the end.

It also explains a small practical fact about tuning by ear. A piano tuner listens to a note’s tail, not its attack, and the reason usually given is that the attack is too noisy. The stronger reason is here: the tail is where the partials that would confuse a beat count have already gone, so the beat between two notes’ fundamentals is available on its own. And “tail” turns out to mean something less patient than it sounds — half a second after the strike the note is already ninety per cent fundamental, which is about how long a tuner waits.

So the decay carries less identity than the attack does, and the dulling this essay is about is a shared trajectory rather than an instrument’s signature. What distinguishes a struck note from a plucked one is over in a twentieth of a second; what they do afterwards is nearly the same thing at different rates.

Whose instruments, and when #

The effect belongs to anything struck or plucked and is absent from anything continuously driven. A bowed string, a blown pipe and a sung note are in a steady state, and their spectra hold as long as the player holds them — which is why the whole argument here has nothing to say about the voice or about wind instruments.

The one generalisation worth carrying is a warning about the site’s own habit. A spectrum is a function of time, and this collection has drawn it as a fact. Every consonance verdict, every roughness curve and every timbre comparison here is evaluated on one instant of one sound, chosen implicitly and never named. On sustained instruments that is harmless. On struck ones it is a choice with a decibel value attached.

What the picture cannot show #

The three loss mechanisms are lumped into one exponent. Air drag, internal friction and bridge coupling have different frequency dependences and different sizes, and separating them needs measurements on a real string. What the model gets right is that all three rise with frequency; what it cannot do is say which dominates where.

The two-stage decay is absent. A piano’s unison rings in two stages because the coupled pair has a fast mode and a slow one, and each partial has its own pair. The real envelope of a real partial is therefore two exponentials, and this figure draws one.

Nothing here is measured. Every curve is a model evaluated on this site’s own timbre table, and the numbers that would settle the exponent are recordings of individual notes, which this collection does not have.

And the room figures are Sabine’s, with the same limits the room ladder recorded: an even distribution of absorption, a diffuse field, and no early reflections. The multiplication of the two decays is an ordering claim rather than a computed product.

What it would take to settle the exponent #

The honest position of this rung is that it has a family of curves and no measurement to pick one out of it, so it is worth saying exactly what would.

A single recorded note, analysed with a short-time transform, gives every partial’s level against time directly. Fitting a straight line to each on a decibel axis gives sixteen decay rates; plotting those against partial number on log axes gives the exponent as a slope, with the goodness of the straight line as a check that a power law is the right shape at all.

That is a small experiment and this collection cannot do it, for the reason it cannot do several others: there are no recordings here. Everything on this site is computed from a stated rule, which is what makes its figures checkable and is also what puts a wall between it and any question whose answer is in a sound somebody made.

So the exponent stays a stated parameter, the essay says so, and what is claimed is the shape rather than the number: the partials separate, the centroid falls, the note ends as a sine, and how fast is an instrument-by-instrument fact that this site has bounded rather than measured.

One more consequence is worth stating because it is a warning about this site’s own sound. Every button in this collection that plays a partial list plays it steadily: the amplitudes are fixed for the duration of the note. For a bowed or blown timbre that is right. For the string and bell timbres it is wrong in exactly the way this rung describes, and a reader pressing a button labelled “a struck string” hears a sustained spectrum that no struck string produces.

Where this ladder goes next #

Two rungs of this ladder have been about the beginning of a note, and both found the same thing from different directions: identity is in the transient. This one is about the end of it, and finds a second identity cue that no attack-based account contains — the rate at which the colour drains, which is one number and which sorts instruments the attack cannot.

The rung after it is the one both halves point at. If the attack carries the identity and the decay carries a second cue, the question is what the middle carries, and for a struck instrument the honest answer may be nothing at all: there is no steady state, only a continuous slide from one spectrum to another. Whether a listener uses any of that slide, or only its endpoints, is a listening experiment and not a calculation.