Timbre and acoustics

The collapse belongs to the bass

Every envelope drawn until now has no pitch in it: the decay model counts partials rather than measuring them in hertz. Put a fundamental in and the collapse it describes shrinks monotonically up the keyboard, from 25.2 semitones of colour drain at A0 to 6.6 at C8, because a partial has to fit under twenty kilohertz to exist and the top note has four of them. Above C6 a struck note has fewer partials than the ear could have resolved, and there is nothing left for a collapse to take away.

15 min read 7 figures The instrument is not the model The limits of the listener Somebody measured it

Assumes: The note that gets duller as it dies · The middle nobody could have guessed

Every envelope this ladder has drawn is a curve against time with an amplitude up the side. Four of its figures give each partial its own decay rate; several give the whole sound one; all of them vary the loss law, the decay time and the timbre. Not one of them has a pitch in it.

That is not an omission in the drawings. It is in the model: the decay is a list of partial amplitudes and an exponent, the partials are numbered rather than measured, and every quantity it returns — the spectral centroid, its half-life, the sixty-decibel time of the $n$ -th partial — is in partial numbers. A note on C1 and a note on C8 with the same timbre and the same decay time produce identical figures.

They do not produce identical sounds, and the reason is not subtle. A partial has to sit under about twenty kilohertz to exist at all, and it has to sit above the threshold of hearing to be heard, and both of those are statements about absolute frequency. So the supply of partials a collapse can work on is a function of pitch, and it is a steep one.

Fig. 1 The same string model struck at 80 decibels on nine fundamentals an octave apart. The heavy line is how far the audible spectral centroid falls between the strike and the note’s death, in semitones. The dashed line is how many partials exist at all, and it is the reason.

The supply

The bottom note of a piano has 727 partials under twenty kilohertz. The top note has four.

That is a factor of a hundred and eighty and it is pure arithmetic — the number of whole multiples of the fundamental that fit under a ceiling — but it is arithmetic nothing on this ladder has had occasion to do, because nothing on this ladder has known what the fundamental was. It changes the character of every claim the third rung made.

Measured as the drop of the audible spectral centroid from the strike to the death of the note, the collapse runs 25.2 semitones at A0, 25.1 at C1, 23.3 at C2, 21.3 at C3, 19.1 at C4, 16.6 at C5, 13.9 at C6, 10.5 at C7 and 6.6 at C8. It falls monotonically and by a factor of 3.8 end to end.

Twenty-five semitones is two octaves of brightness lost. Six and a half is a little over a fifth. The first is a transformation and the second is a slight dulling, and they are the same model with the same loss law and the same decay time, differing only in what note was struck.

A note gets duller as it dies. Each partial of a string note against time, with the loss rising as the partial number to the power 1 — so the fundamental takes 6 seconds to fall sixty decibels and the 8th takes 0.75. The heavy line is the power-weighted centroid, falling from partial 1.77 toward the fundamental; it is halfway there after 0.15 seconds. A single-rate envelope would draw all of these as parallel lines and the centroid as a horizontal one, and a struck string does neither: what is left at the end of a long note is very nearly a sine. — Fig. 2 The drawing used ever since the collapse was found: each partial of a struck note with its own decay, the loss rising in proportion to partial number. There is no pitch anywhere in it, and there could not be — the axis is decibels below the attack and the lines are numbered rather than named.

The same collapse, drawn in hertz

The clearest way to see what is happening is to stop drawing partial numbers and draw frequencies, on one axis, for several pitches at once. Every figure this ladder has drawn puts partial number along the bottom, which is the natural axis for a claim about a loss law and is the axis on which the register dependence is invisible: partial eight is partial eight whether the note is A0 or C8. Frequency is the axis a listener has, and the audio ceiling and the threshold of hearing are both fixed lines on it. Drawing three pitches against one frequency axis puts the ceiling in the same place in all three rows, which is the whole of the argument in one picture.

Fig. 3 Every partial above the threshold of hearing at its own frequency, for the same model struck on C2, C4 and C7. The pale marks are the spectrum at the strike and the solid ones what is left just before the note dies. The axis is the same in all three rows.

The bass row is dense from its fundamental to the top of the axis and the treble row is a handful of marks in the upper half of it. The bass note’s centroid travels from 251 hertz to 65; the C7 note’s travels from 3,845 to 2,093. In hertz the second journey is the larger one — 1,752 against 186 — which is exactly why the drop has to be measured in semitones rather than in hertz to mean anything, and why a figure drawn in partial numbers can hide the whole effect.

The high note has further to fall in hertz and much less far to fall in the only unit a listener has for pitch.

Where the supply runs out

The other half of the picture is the ear, and it moves in the opposite direction. The number of partials a listener can resolve rises with pitch, because the auditory filter widens more slowly in hertz than the partial spacing does.

The series has three tops. Where the harmonic series stops, asked three ways, at four fundamentals. Consecutive partials stop being separately resolvable around partial 8 — that one depends on the fundamental, since a critical band is a fixed width in hertz and the spacing is not. They stop being a semitone apart at partial 17 at every fundamental, because the ratio (n+1)/n does not know what n is measured in. And they stop being distinguishable in pitch at all between partials 11 and 190. Every claim about how far up the series something happens is a claim about which of these three was meant. — Fig. 4 The three tops the ear supplies, at four fundamentals across the compass. The resolvable top rises from partial 3 at C1 to partial 10 at C7, while the number of partials that exist falls from 611 to 9 across the same span.

So the supply falls by a factor of a hundred and eighty and the demand rises by a factor of ten, and somewhere they cross. On these numbers the crossing is between C6 and C7: at C6 a struck note has 19 partials and the ear could have resolved 10, and at C7 it has 9 and could have resolved 10.

Above about that point a struck note has fewer partials than the ear could have told apart even at the strike. There is nothing for a collapse to take away that a listener was hearing as a separate thing, and the note’s change of colour is a change in a spectrum that was never resolved into components in the first place.

That is the same crossing the fifth top finds in time, arriving in pitch. In the top octave of a keyboard the two coincide: the note has almost no partials, the ear could have used almost all of them, and the ear’s own limits are irrelevant for the whole of the note rather than for nine tenths of it.

The bottom of the keyboard is the mirror image and it is stranger. At A0 the critical band is wider than the fundamental itself, so the resolvable top is partial 1 — a listener never hears any partial of the lowest note as a separate thing. That note nevertheless carries seven hundred of them, and its colour drains by two octaves. So the extreme bass is a case where an enormous spectral transformation happens entirely inside a region the ear analyses as one object, and the extreme treble is a case where there is nothing to transform. The register in between, which is where music is mostly written, is the only part of the compass where the collapse is both large and resolved.

What the loss law does to the register dependence

The register dependence above is drawn at one loss law, and it is worth asking whether it is an artefact of that choice. It is not, but the two interact in a way that is worth stating, because it decides which instruments the finding is about.

Fig. 5 The same three pitches at a much steeper loss law, with a partial’s decay rate rising as the square of its number rather than in proportion. The bass row loses almost everything it had; the treble row, which had almost nothing, ends very close to where it started.

A steeper exponent takes the upper partials away faster, so it increases the collapse at every pitch that has partials to lose and does nothing at all where there are none. The register dependence therefore widens with the exponent rather than washing out: the bass note gets a bigger transformation and the treble note stays where it was.

The parameter that flattens it is the source spectrum. A note that starts with a steep roll-off has little energy above its first few partials wherever it is played, so there is not much difference between having four partials and having seven hundred. That is the case the last figure on this page draws, and it names the class of instruments the finding is about: those with a lot of spectrum at the strike. A struck string is one. A bowed string held under the bow is not a struck instrument at all and has no collapse to have a register dependence.

Where the model gets it backwards

Putting a pitch into a model that has never had one is not a free operation, and the interesting failure is not in the collapse at all. It is in how long the note lasts.

Fig. 6 How long a struck note stays above the threshold of hearing across the compass, under a decay time held fixed for the whole instrument and under one falling with pitch. The two curves disagree in sign.

With the fundamental’s decay time held at six seconds — which is what every envelope figure on this ladder does, because the model has no fundamental — the note’s audible life rises with pitch, from 1.3 seconds at A0 to 8.3 at C8. The mechanism is unarguable within the model: the only way pitch enters is through the threshold of hearing, and the threshold at 27.5 hertz is some sixty decibels above the threshold at four kilohertz, so a bass note struck at the same level starts much closer to inaudibility.

Every strung instrument does the opposite. A piano’s bottom notes ring for half a minute and its top notes are gone in under a second, and the ratio is on the order of a hundred rather than the six the model predicts in the wrong direction.

Letting the decay time fall as the fundamental to the power 0.9 — which is what the ladder’s own account of the losses implies, since a loss that rises with frequency should do so whether the frequency belongs to a partial or to a fundamental — turns the answer round: 9.8 seconds at A0 and 0.7 at C8, with the peak in the second octave where the threshold is still expensive. That is the right shape.

So the ladder’s decay time is not a constant of the instrument and every figure on it has treated it as one. The exponent, which the third rung swept carefully across its plausible range, describes how loss rises with frequency within one note. The same physical claim, applied across the compass, is a second exponent that nothing has ever fitted — and the two ought to be the same number, because the mechanism is the same.

Whether they are the same number

That is a real prediction and it is worth writing down as one, because it is falsifiable and this collection cannot currently settle it.

If the loss on a wire is a function of frequency and nothing else, then the sixty-decibel time of a note’s fundamental should fall with the fundamental exactly as fast as the sixty-decibel time of a partial falls with the partial number — the same exponent, in the same units, for the same reason. The third rung puts the within-note exponent between about a half and just over one and argues for something near one. A compass exponent near one predicts a hundredfold ratio of decay times across a seven-octave keyboard, which is the right order of magnitude for the instrument this ladder keeps citing.

It is only the right order of magnitude, and there are two obvious reasons it should not be exact. A piano’s strings are not one wire: they change gauge, they change from plain to overwound, and the bass strings are a different object from the treble ones. And the bridge’s own admittance varies along its length, so the coupling loss — which is the dominant term for a strung instrument — is a property of where on the bridge the string is anchored rather than only of its frequency.

What settles it is a measurement this collection does not have: decay times of a real instrument’s fundamentals across its compass, which is one afternoon with a microphone and is the smallest measurement this ladder has ever needed.

Fig. 7 The same sweep with a steeper source spectrum, a shorter decay and a faster loss law. The collapse shrinks to about seven semitones everywhere and its dependence on register nearly vanishes, which says the finding above is a statement about instruments with a lot of spectrum to lose.

Which computation produced the numbers

The source spectrum is $1/n$ over every partial under twenty kilohertz, normalised so its total power sets the stated playing level at the fundamental. The decay is this collection’s own model, with partial $n$ ’s rate rising as $n^p$ . A partial is present when its level exceeds the ISO 226 threshold at its own frequency, evaluated pointwise rather than at a tabulated frequency.

The collapse is measured as the drop of the power-weighted centroid of the audible partials, in semitones, between the strike and the moment just before the fundamental itself goes under. Restricting the centroid to audible partials is what makes the measurement a statement about a listener; computing it over all the partials in the arithmetic gives a much larger number everywhere and hides the register dependence, because inaudible partials at the top of a bass note’s series dominate the average.

The nine fundamentals are A0 and the eight Cs, which is the compass of a piano, and every one of them is computed with identical parameters.

The threshold curve is the one the quietest thing audible established for this collection, and it is doing more work here than in most places it is used. At the bottom of the keyboard it is the binding constraint on how many partials are audible and on how long the note lasts; in the middle of the keyboard it barely enters; at the top it decides nothing at all, because every partial that exists is comfortably above it. A model that had used a flat threshold instead would have found no register dependence in the note’s lifetime and the same one in its collapse, which is a useful way of separating the two findings: the first belongs to the ear and the second to the audio ceiling.

Where the model stops

Twenty kilohertz is a convention. It is a round number for a young adult and it falls with age faster than any other audiometric quantity. Taking it to fourteen kilohertz — a common middle-aged ceiling — removes a third of the partials from every note and compresses the whole register dependence, which means the finding here is one a listener’s own hearing modifies.

The source spectrum is held. A real instrument’s spectrum at the strike is not the same shape across its compass: where the hammer lands is a fixed fraction of the string but the string’s stiffness and the hammer’s compliance both change with register, so the treble notes start with steeper spectra than the bass ones. That works in the same direction as the finding and would strengthen it.

And the register dependence is a supply argument, not a loss argument. Nothing here says a treble string loses its upper partials more slowly. It says there were fewer of them to lose. Those are different claims and only the second is computed.

What the picture cannot show

It cannot show inharmonicity. A stiff string’s partials are stretched, and the stretching is far worse in the treble, so the top of the compass has fewer partials and they are not where a harmonic series would put them. Whether that makes the surviving spectrum more or less like a struck bar is a question with an answer and it is not this one.

Nor the difference between a piano and a harpsichord, which is the comparison the third rung of this ladder used to argue that the drain rate is an identity cue. That comparison is now known to be confounded: two instruments compared at different pitches differ in drain by an amount register alone would produce, and the confound is larger than the effect over most of the keyboard. The cue survives, conditioned on register — a listener knows the pitch — but it is a cue about a note rather than about an instrument.

And it cannot show a chord. Every figure here is one note. A chord spread across three octaves contains notes whose collapses differ by a factor of two in extent and, on any realistic account of the decay times, by an order of magnitude in duration. What a listener receives from a struck chord is therefore not one collapsing spectrum but three or four of them running at different speeds, and the low one is still transforming when the high one has finished.

It cannot show the onset either. A low note takes longer to start and a higher note speaks sooner and takes longer to reach its level, so the beginning of a struck note has its own register dependence, in the opposite direction to the one computed here and on a timescale a thousand times shorter. Neither is in the other’s figures.

And it cannot show the hammer’s own mass. A hammer heavier than the string it strikes is a different exciter from one lighter than it, and the ratio changes systematically across a keyboard because the strings get lighter far faster than the hammers do. That is a third register dependence acting on the spectrum at the strike, which is the left-hand end of everything on this page.

Where this ladder goes next

Five rungs. The shape of a note is most of what an instrument is; the identity is in the first fifty milliseconds; the colour drains at a rate that is a second cue; the middle carries the whole of that rate and both endpoints carry none of it; and now the pitch, which every figure above held at one value by having none of it, and which moves the collapse by a factor of four across a keyboard and reverses the sign of the note’s own lifetime.

What is owed after this is the decay time as a function of pitch, and it is the first debt this ladder has recorded that arithmetic cannot pay. Everything above can be computed from a loss law and a spectrum; the compass exponent is a property of a particular instrument’s wire and bridge, and the honest statement is that it should equal the within-note exponent and that nothing here has checked. The measurement is small — the sixty-decibel time of thirty or forty fundamentals on one instrument — and it would settle whether this ladder has one exponent or two. Until it is made, every figure on this ladder that quotes a decay time is quoting a number chosen to make the drawing legible rather than one measured from anything, and that should be said in every caption that uses one.