Timbre and acoustics

The top that falls while the note lasts

Four earlier essays have asked where the harmonic series stops, and all four answered with a number computed for a tone that never ends. A struck C3 has 152 audible partials at the strike and eight after two-thirds of a second, so the ear's own resolution limit governs the first ten per cent of the note and the decay governs the rest. Playing ten decibels louder buys the ear a further tenth of a second, and doubling its reign would take fifty-eight.

Assumes: The series has three tops · The fourth top is the maker's

Nine rungs of this ladder have drawn the harmonic series, and every one of them has drawn it as a list of frequencies. Partial $n$ sits $1200\log_2 n$ cents above the fundamental; the spacing between neighbours narrows as $1200\log_2((n+1)/n)$ ; and the questions asked of it have all been questions about that spacing. Where do consecutive partials stop being resolved. Where does their spacing fall below a semitone. Where does it fall below the smallest pitch difference a listener can hear.

The series has three tops answers all three, and the fourth top is the maker’s adds a fourth that belongs to the instrument rather than the ear. Between them they are the closure of the question — except for one thing that none of them contains.

None of them contains any time. The three tops are computed on a tone that is sounding and goes on sounding; the fourth is the frequency at which a bell stops turning the wave round, which is a property of a tube standing still. And that is the parameter every figure on this ladder has held at the same value, by having none of it at all.

Fig. 1 The highest partial still above the threshold of hearing, moment by moment, for a note struck at 80 decibels on C3 with a 1/n spectrum and a loss rising in proportion to partial number. It starts at 152 and it is falling from the first millisecond. The horizontal lines are the three tops the ear supplies, all of which assume the note goes on for ever.

Where the three tops actually come from

The three the ear supplies are worth restating precisely, because the argument turns on what they are statements about. Each of them is a comparison between two intervals: the gap between one partial and the next, and some width belonging to the listener. When the first falls under the second, the pair stops being two things and starts being one, and the series has ended for that purpose. Which width is used decides which of the three answers comes out, and the three widths are of very different sizes — a critical band is enormous beside a difference limen — so the three tops are spread over more than an order of magnitude at every fundamental.

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 30 and 125. Every claim about how far up the series something happens is a claim about which of these three was meant. — Fig. 2 The three tops at four fundamentals two octaves apart. Every one of them is a comparison between the spacing of two neighbouring partials and a width the ear has — the critical band, a semitone, the difference limen — and none of them asks whether either partial is still there.

The resolvable top is where the spacing between neighbours falls inside one auditory filter, so the two stop being analysed separately and start beating instead. On C3 that is partial 8. The semitone top is where the spacing falls under a hundred cents, which is partial 17 at every fundamental because it is a fact about the series and not about the pitch. The limen top is where the spacing falls under the smallest frequency difference a listener can detect, which on C3 is partial 80.

All three are comparisons of one interval against another. Neither side of any of them mentions amplitude, and none of them can, because the objects being compared are frequencies.

Where the series stops being a chord and becomes a scale. The interval between each partial and the next, in semitones, against partial number. It is an octave, then a fifth, then a fourth, a major third, a minor third — and it crosses 8 semitones between partials 2 and 3. Each shaded band is one octave of pitch and holds twice as many partials as the band below it: 2 between 1 and 2, 3 between 2 and 4, 5 between 4 and 8, 9 between 8 and 16. Nothing about the series changes with height; what changes is the density, and a set of notes becomes usable as a melody when its neighbours are a step apart. — Fig. 3 The quantity all three tops are read off: the interval in cents between consecutive partials, against partial number. It is a smooth, fast-falling curve and it belongs to the series rather than to any instrument. The mark is at the resolvable top for C3.

What a struck note does instead

Put a note under it and none of that holds for long. Each partial of a struck string has its own decay rate, rising with partial number — the loss to air, to internal friction and to the bridge all rise with frequency — so partial $n$ falls at a rate proportional to $n^p$ with $p$ somewhere near one for real strings.

A partial below the threshold of hearing is not a quiet partial. It is not there: it cannot be resolved, it cannot beat against anything, it cannot contribute to a pitch estimate. So the highest partial a listener has is the highest one still above threshold, and that is a number falling from the moment of the strike.

On C3 at 80 decibels it starts at 152, which is far above every static top. It falls past the limen top after 0.035 seconds, past the semitone top after 0.324, and past the resolvable top after 0.641. From then on the series has fewer partials than the ear could have resolved, and what stops it is the clock rather than the listener.

Two-thirds of a second is roughly a crotchet at ninety beats to the minute. A piano note held under a finger sounds for several seconds. So on the instrument this anchor’s second rung is about, the ear’s own limits govern something like the first tenth of every note, and the other nine tenths are governed by a quantity none of the four tops contains.

The order in which the crossings happen is fixed and is worth stating, because it is the one part of the result that could not have come out otherwise. The three fixed tops are 80, 17 and 8 on C3, and a monotonically falling curve passes them in the order of their sizes. So there is always an interval at the beginning of a note during which the limen top is the binding one, then an interval during which the semitone top is, then one during which the resolvable top is, and then the rest of the note. What varies between instruments and between pitches is the length of those intervals, not their sequence.

The first of them is the surprising one. The limen top governs for thirty-five milliseconds — less than the fifty that the first fifty milliseconds identifies as the window carrying an instrument’s identity. The part of a struck note in which the series is long enough for the finest of the ear’s three limits to matter is entirely inside the attack, and the attack is the part of the note that is not a harmonic series at all.

The exchange rate for loudness, which is bad

The obvious objection is that the top starts high and the note is only quiet at the end, so playing louder should extend the ear’s reign. It does, by an amount that can be written down exactly rather than measured.

Loudness does not buy the ear more of the series. How long the resolvable top stays binding, against how loud the note is struck. The relation is a straight line of 0.111 seconds per ten decibels, because the level enters the audibility condition once and the loss enters it once, so nothing about it is empirical. At 50 decibels the ear governs for 0.31 seconds and at 100 for 0.86. Doubling the reign of the ear's own top would take 28 decibels more than the quietest note here, which is more dynamic range than any instrument has. — Fig. 4 How long the resolvable top stays binding, against how hard the note is struck. The relation is a straight line because the playing level enters the audibility condition once and the decay enters it once. Ten decibels buys about a ninth of a second.

The rate is 0.111 seconds per ten decibels. At 50 decibels the ear governs for 0.31 seconds and at 100 for 0.86. To double the 0.64 seconds a note struck at 80 decibels gets would take 58 decibels more, which is more dynamic range than a piano has between its quietest and loudest note, and considerably more than exists between any two dynamics in a score.

That is what makes this a structural fact about struck instruments rather than a detail of one playing level. There is no way to play a note that keeps the ear in charge of its series for as long as the note lasts.

The straightness of that line is not an empirical finding and it is worth seeing why, because the same two lines explain the whole shape of the argument. Partial $n$ leaves the strike at a level $L_n$ set by the playing level and the spectrum’s roll-off, and it falls at a rate set by the decay time and the loss law, so the moment it goes under the threshold $T$ at its own frequency is

t_n \;=\; \frac{t_{60}\,\bigl(L_n - T(n f_0)\bigr)}{60\,n^{p}}

and the note’s own top has fallen past a fixed top $N$ at the moment partial $N+1$ goes. The playing level enters that expression exactly once, additively, so the crossing time is linear in it with a slope of $t_{60}/60 n^{p}$ — which at the resolvable top of C3 is six seconds divided by five hundred and forty, or 0.0111 seconds per decibel, which is the number the sweep measures. Two routes to one figure, and the second one says the first could not have come out any other way.

The same expression says what would change the answer, and it is a short list: a longer decay time, a shallower loss law, a lower fundamental to put the partials where the threshold is kinder, or a flatter source spectrum. Loudness is on the list and it is the weakest term on it, because it sits inside a bracket that is already sixty or seventy decibels wide.

And the exchange rate for the loss law, which is worse

The other free parameter is the exponent — how fast a partial’s decay rate rises with its number. The note that gets duller as it dies puts it between about a half and just over one for real strings, and argues that it is a design variable a maker moves.

Fig. 5 The three crossing times against the loss law, from a very shallow exponent to a steep one. Every curve falls, and nothing in the range keeps the resolvable top binding for as long as a crotchet at ninety.

At an exponent of 0.4 — a chime rather than a string — the resolvable top governs for 2.40 seconds. At 1.0 it is 0.64, at 1.4 it is 0.27, and at 2.2 it is 0.046. The relation is steeply monotone, which is what one would expect: a faster loss can only take partials away sooner.

What that does say, and it is the useful half, is that an instrument whose colour drains slowly is an instrument the ear stays in charge of for longer. A glockenspiel or a struck bar keeps nearly every partial it started with. A piano does not. The identity cue the third rung of the envelope ladder found — how fast an instrument’s colour drains — is the same quantity as the one that decides how long the ear’s own tops apply, and they are the same computation read two ways.

The same question two octaves down

None of the numbers above is a constant of the instrument, and the clearest way to see how far they move is to ask them of a note at the other end of the keyboard. A bass string is longer, heavier and less stiff for its length; its fundamental decays for far longer, and its loss law is shallower because the frequencies involved are lower.

Fig. 6 The same computation for a bass note: C2, struck at 90 decibels, with a twelve-second fundamental and a loss rising as the partial number to the power 0.7. It starts at partial 305 and the ear’s resolvable top — which on C2 is partial 6 — stays binding for three and a half seconds.

Every reign lengthens: the limen top holds for 0.24 seconds against 0.035, the semitone top for 1.63 against 0.32, and the resolvable top for 3.48 seconds against 0.64. That is a fivefold improvement, and it comes from three changes acting in the same direction rather than from any one of them.

It is also the wrong way round from what the sound suggests. A bass piano note is the one a listener would describe as losing its brightness fastest — it starts with a spectacular clangour and settles into something almost sinusoidal — and the ear is nevertheless in charge of its series for five times as long as it is on C3. Both things are true because the bass note started with three hundred partials: it can lose two hundred and ninety of them and still have more than the six the ear could have told apart.

A treble note is the other extreme and it barely needs a figure. On C5 with a two-second fundamental the resolvable top is partial 9, the note starts with 38 partials above threshold, and the crossing happens at 0.2 seconds. Above that the series is shorter than the ear’s own limit for essentially the whole of the note, which is one reason why the top octave of a piano is the part of it that sounds least like a string and most like a struck bar.

So the fifth top is not a correction of a fixed size. It is a quantity that varies across the compass by an order of magnitude, in a direction opposite to the one a listener’s impression of brightness suggests, and no rung of this ladder could have found it because no rung of this ladder has had a note in it.

What this changes about the four tops

Nothing is retracted. The three the ear supplies remain exactly right about a sustained tone, and there are plenty of those: an organ pipe, a bowed string held under the bow, a wind note all reach a steady state and stay there. On any of those a listener has all the time there is, and the top of the series is whichever of the three fixed numbers binds first.

The fourth, the maker’s, is the frequency at which a bell stops turning the wave round, and it belongs to a tube rather than to a note. It is unaffected too.

What is added is that all four are the tops of a sustained note, and a large part of what music is made of is not sustained. Everything struck, plucked or hammered has a fifth top instead, and it is not a number: it is a curve, it starts higher than any of the four and it ends below all of them, and it crosses each one at a computable moment.

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 1.2 — so the fundamental takes 3 seconds to fall sixty decibels and the 8th takes 0.25. The heavy line is the power-weighted centroid, falling from partial 1.77 toward the fundamental; it is halfway there after 0.06 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. 7 The mechanism underneath, in the envelope drawing used throughout: each partial of a struck note with its own decay, the loss rising as the partial number to the power 1.2. The upper partials do not fade with the note, they leave it.

Whose note, and where the numbers came from

The spectrum is $1/n$ , which is a sawtooth’s and is close to what a plucked or struck string radiates before its bridge filters it. Steeper roll-offs shorten every reign here: at $1/n^2$ the resolvable top on C3 governs for 0.43 seconds rather than 0.64.

The playing level is the level of the fundamental at the strike, in decibels of sound pressure, and the threshold of hearing is this collection’s own ISO 226 curve evaluated at each partial’s frequency rather than at a tabulated one. The decay is the site’s own model with the fundamental’s sixty-decibel time at six seconds.

The three static tops are computed by the same function the three-tops essay used, unchanged, so the horizontal lines in these figures are that essay’s own answers and not a re-derivation of them.

Where the model stops

Threshold is not the only way a partial leaves. A partial can also be masked by a louder neighbour without falling under the absolute threshold, and masking spreads upward more than downward — which on a falling spectrum is exactly the wrong direction to be helpful. Including it would move every crossing time earlier, so the numbers here are ceilings.

The threshold curve is for a tone, not a partial. ISO 226 is measured on isolated pure tones, and a partial inside a complex tone is not one. The convention is this collection’s standing one and it is stated wherever it is used.

And a bass note is a special case. At A0 the critical band is wider than the fundamental, so the resolvable top is partial 1 and the crossing happens at time zero. The three tops are not equally meaningful across the compass, which is a fact about the ear rather than about this argument, and it is why the figures here are drawn at C3.

What the picture cannot show

It cannot show what a listener does with a partial they still have. Whether a top is used is a separate question from whether it exists, and the partials in the dominance region around the third to the fifth are the ones a pitch is mostly built from. Those are low, so they survive longest, which is the reassuring half of the result: a note keeps the evidence for its own pitch far longer than it keeps its brightness.

Nor a repeated note. Every figure here is one note into silence. A pianist plays six or eight notes a second and each of them rings for seconds, so the spectrum reaching a listener is a sum of notes at every stage of their own collapse, and the top of that is not the top of any of them.

And it cannot show the room. A hall does not decay evenly, and its own treble decay is faster than its bass, so the room subtracts from the top of the series a second time and over a longer span. The two compound sequentially rather than simultaneously, because the string has finished most of its collapse before the first reflection arrives.

It also cannot show a played instrument’s noise floor. The threshold used here is the threshold in silence, and a concert hall with an audience in it is nowhere near silent. Raising the floor by twenty decibels — which is a quiet hall rather than a noisy one — takes 0.22 seconds off every crossing on C3 at a stroke, by exactly the exchange rate the level sweep gives. So the reigns computed here are the best case, and the practical answer in a room is shorter than any number on this page. That is the same direction masking pushes, and the two are not independent: a hall’s own low-frequency rumble masks upward into precisely the region where a struck note’s surviving partials live.

Where this ladder goes next

Nine rungs. A string does everything at once; a stiff string’s partials are not quite the series; the series is not a chord; it has three tops that are the ear’s; a fourth that is the maker’s; the shape that makes the series a series; that shape re-chosen with a mouthpiece in front of it; the same mouthpiece against seven lengths; and now the fifth top, which is the clock’s, and which governs nine tenths of every struck note.

What is owed after this is the sum. Every rung above is one note in silence, and the object a listener has in front of them is a texture — several notes at once, each with its own series, each series with its own top falling at its own rate. Whether two notes an interval apart still have partials to coincide on by the time a listener has heard them both is a question with a definite answer and this collection has both halves of it: the coincidence arithmetic that decides which partials of two notes meet, and the decay law that says which of them are still present. What that would settle is whether the consonance of an interval on a struck instrument is a different quantity a second after the strike from what it is at the strike — which is a claim about counterpoint on a piano that nobody has priced, and it needs only arithmetic.