Acoustics & Music Theory
why it sounds like that
Concept

Spectral centroid — where it appears

The amplitude-weighted mean frequency of a spectrum, which is the commonest single-number summary of brightness. It moves with the dynamic on any instrument whose excitation is nonlinear, which is why a loud note is a different timbre rather than a louder one.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

Two clocks on one contact, and which of them runs out first. Two times that could end a hammer's contact with a string, across the compass. The flat line is the felt's own recoil, which this site models as a fixed 1.6 milliseconds at a stated force and which does not know what note is being played. The falling line is the string's: a mass m driving a string of impedance Z loses its momentum in m/2Z, which is 11.9 milliseconds at the bottom of the keyboard and 0.76 at the top. They cross at G3. Below the crossing the felt lets go first and above it the string gets there first, and over six octaves the two are within a factor of two of each other — so the contact time is a coupled quantity and not a property of the felt.

The hammer that is heavier than its string

Three earlier essays varied where the hammer lands, how long it stays and how wide it is, and the third named the fourth variable: the exciter's own mass. It inverts across one instrument — the hammer is lighter than the wire it hits at the bottom of a piano and thirteen times heavier at the top — and it turns the contact time into a quantity the string decides rather than the felt.

instruments · Excitation point Where each exciter's contact corner sits against its comb. The merger's four corners, drawn for three exciters. The comb is the strike point and is a horizontal line, because it is a fraction of the string and has no pitch in it. The contact corner is the exciter's, and it falls up the compass because the contact time is a fixed number of milliseconds against a shrinking period. Where they cross, the binding term changes. piano hammer crosses at E3; harpsichord plectrum crosses nowhere in the compass; dulcimer beater crosses at A5. So the crossing found earlier is not a fact about pianos. It is a fact about compliance: a felt hammer stays on the string for about 1.6 milliseconds and a plectrum for 0.05, and a difference of that size is a corner in a different place.

The crossing belongs to the felt

On a piano the strike point decides the top of the spectrum below E3 and the hammer's contact decides it above. The merger's own bookkeeping asked whether that crossing is a fact about pianos or about hammers. A plectrum never crosses at all and a hard beater crosses two octaves higher, and the boundary is a contact time of about an eighth of a millisecond — ten times shorter than felt.

timbre · Excitation point Intonation is a unison problem and nothing else. The roughness between two instruments on one note, against how far apart they are in cents, drawn for a unison and for the intervals beside it. A perfect unison is 0.0007 — the partials coincide and there is nothing to beat. Five cents apart it is 0.0465, 65 times as rough, and ten cents apart it is rougher than a major third played exactly. The mechanism is that partial n of a note mistuned by c cents is mistuned by c cents as well, which is n times as many hertz — so the top of the spectrum enters the critical band long before the fundamental does. The other curves are flat, because a third's roughness is set by which partials nearly coincide and a few cents does not change which.

Two players on one note

Six essays have put one instrument on each note of a chord, and the commonest thing an orchestrator actually does is put two on the same note. Two independent sources add in power, so the composite is neither of them — except that it nearly always is one of them, because the level at which ownership changes hands is rarely at zero. And a unison ten cents out is rougher than a major third dead in tune.

timbre · Spectrum How much of each instrument is narrow enough to steepen a wave. For each bore, the quantity that decides how nonlinear it is: the narrowest radius divided by the radius at each station, integrated along the tube. The shading is that integrand, so a bar that stays dark is a tube still doing damage to the wave and a bar that fades is a flare that has thinned it out. Divided by the instrument's own length the integral is a pure number: a plain cylinder 1.00, a tenor trombone 0.88, a trumpet 0.86, an F horn 0.59, a cone of a trumpet's length 0.24. A plain cylinder is 1 by construction, a cone of the same length and mouth is 0.24, and the ordering across the brass family is the one players give when asked which of them can be made to blare.

The partials the tube makes itself

Eleven earlier essays compute a passive linear resonator, and none of them ever says so. At a real fortissimo the air in a brass instrument is not linear: a compression outruns a rarefaction, the wave leans forward as it travels, and the fourth partial of a loud trumpet note is seventy decibels louder than a scaled-up quiet one — generated in the tube rather than at the lips. How much of it happens is an integral over the bore, and it is why a flugelhorn cannot be blown into being a trumpet.

instruments · Air column A plucking point 50 millimetres from the nut, across a harpsichord's compass. The jacks stand in a rail and the rail is one object, so a register plucks at a fixed distance from the nut while the strings shorten by a factor of ten from the bass to the treble. At 50 millimetres that is one 36th of the string at C2 and one 3.5th at C6 — a plucking fraction that changes by a factor of 10.1 without the maker moving anything. The comb corner is one over that fraction and falls with it, from 36 to 3.5; the dispersion corner has the fraction under a cube root and falls only from 106 to 21. They never meet. Setting one over p equal to the cube root of two over three times the inharmonicity and the fraction gives a plucking point at p = √(1.5B), which on this instrument's iron wire is between 3.4 and 9.9 millimetres from the nut — nearer to it than any register a harpsichord has ever carried, the lute stop included. So the maker's one free choice is the binding term at every pitch and every register position, which is exactly what the piano's is not: on a piano the contact time takes the decision away above E3.

What the second register is for

Nine earlier essays take the plucking fraction as given — an eighth, a sixth, a seventh — and a harpsichord's jacks stand in a rail, so what is given is a distance and the fraction is a consequence: 50 millimetres is one thirty-sixth of the string in the bass and one three-and-a-halfth in the treble. Engage two registers on one string and the amplitudes add exactly, the near one fills the far one's missing partials, the pair digs holes of its own that neither has, and the combination comes out louder and rounder than either — including the bright one.

instruments · Excitation point Who owns clarinet, oboe, voice at every balance. The composite of three players at 392 hertz belongs to whichever of them it is nearest in log-spectral distance, and here that is drawn over the whole plane of balances a conductor could set — the second and third players from 24 decibels below the first to 24 above. voice owns 79 per cent of the square. The three regions meet where all three distances are equal, which is the only balance at which the composite belongs to nobody: it is at -0.3 and -19.1 decibels, inside the square and therefore a balance an ensemble could actually be asked for. A trio has a colour of its own at one point, not over a region.

A section has a loudest member, not a colour

Two players on one note have a balance at which the composite belongs to neither, and that is what blending means. Three should have three such balances and no reason for them to agree — a trio with a rock-paper-scissors ownership would have no strongest member at all. Twenty trios, sixty pairwise comparisons, and not one disagreement: the possibility is real, arbitrary spectra do it once in twenty, and instruments never do.

timbre · Spectrum A struck note's two ends are the same for every loss law. The partial levels of a string spectrum struck at 80 decibels on 130.8 hertz, and what is left of it when the fundamental itself falls under the threshold of hearing, for three laws relating a partial's decay rate to its number. The left panel is every one of them: a loss law cannot change the spectrum at the instant of the strike, because no time has passed. The other three are every one of them too: whatever the law, the note ends with nothing above the threshold. So both ends of the slide are shared, and everything that distinguishes an exponent of 0.5 from an exponent of 1 from an exponent of 2 is in the middle.

The middle nobody could have guessed

A struck note has no steady state, only a slide from one spectrum to another — so the question is what the middle carries that the ends do not. The answer is exact rather than statistical: every loss law in the family leaves the strike with the same spectrum and ends in the same silence, so both endpoints carry precisely nothing about which of them it is. The whole difference is 41.3 decibels, and it peaks 0.38 seconds in, seven per cent of the way through the note.

timbre · Envelope There is less to collapse the higher the note is. The same string model struck at 80 decibels on nine fundamentals, an octave apart. The heavy line is how far the spectral centroid falls between the strike and the note's death, in semitones — the whole of the colour drain, in the unit a musician has for pitch. It runs from 25.2 semitones at A0 to 6.6 at C8, a factor of 3.8, and it falls monotonically. The reason is the dashed line, which is how many partials exist at all: 727 under the audio ceiling at A0 and 4 at C8. From C7 upward a struck note has fewer partials than the ear could have resolved — 9 against 10 — so there is nothing left for a collapse to take away.

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.

timbre · Envelope Two registers 50 microseconds apart on one string. The partials of a 355-millimetre string plucked at 32 and 50 millimetres from the nut, drawn twice: once with the two quills releasing together, once with the far one releasing 0.05 milliseconds later, which is 0.026 of this string's period. A delayed release turns partial n through 2·pi·f_n·dt, so the rotation is proportional to the partial number: the fundamental is turned 9 degrees and partial 24 is turned 226. Simultaneous, the pair is missing partials 17 and 20 — holes it digs for itself where the two combs are equal and opposite. Staggered, it is missing none of them: a rotation of anything at all takes two amplitudes out of opposition. The partials neither comb can excite at all — 7, 11, 22 — are filled either way, because where one comb is zero the sum is the other one whatever its phase. The fundamental's gain over the far register alone falls from 4.36 decibels to 4.34.

The interval between two quills

Two jacks on one key are voiced separately and do not let go at the same instant. That interval turns each partial of the later pluck through an angle proportional to its number — so it leaves the fundamental alone and inverts the twentieth partial, and the holes the pair digs for itself vanish at a hundredth of a period. What a regulator can tolerate turns out to be one fixed fraction of a period at every pitch, which on a five-octave instrument is a factor of sixteen in milliseconds.

instruments · Excitation point A bow shows the loss law a blow conceals. The same string on 130.8 hertz under three loss laws, drawn twice each: struck, and held by a continuous drive. The pale marks are the spectrum a blow produces, and they are identical in all three rows — a strike is the source spectrum and has no loss in it yet, which is why both endpoints of a struck note were found to carry nothing about the law. The solid marks are where each partial settles when a drive balances its own loss, at drive over loss, so the steady spectrum rolls off as the source's roll-off plus the exponent. At an exponent of 0.5 the held spectrum's centroid sits at 167 hertz, 5.7 semitones under the strike's 232; At an exponent of 1 the held spectrum's centroid sits at 144 hertz, 8.2 semitones under the strike's 232; At an exponent of 2 the held spectrum's centroid sits at 133 hertz, 9.6 semitones under the strike's 232. The quantity that is invisible at both ends of a struck note is the slope of a bowed one, for as long as the bow moves.

A bow holds the number a blow hides

Two struck notes with different loss laws are identical at the strike and identical at the end, which is why separating them at all meant looking in the middle. Drive the same two strings continuously and the loss law stops being a rate and becomes a slope: each partial settles at its drive over its own loss, so the exponent adds to the source's roll-off and sits in the spectrum for as long as the bow moves. It is 18.7 decibels of separation available from the first instant, against 41.3 that a blow delivers after four tenths of a second and then takes away.

timbre · Envelope From partial 3 the room is the slower of the two. Decay rates in nepers a second for each partial of a note on 130.8 hertz, in a concert hall. The rising curve is the string's own loss, which grows as the partial number to the power 1. The flat-ish curve is the room's, from its reverberation time at that partial's frequency. A reverberant field is the source convolved with the room, so a partial's tail falls at the SLOWER of the two — the heavy line — and the room keeps returning energy the string has stopped making. From partial 3, at 392 hertz, the room is in charge: 6 of the note's 8 partials are held up by the room rather than let go by the string. Those are exactly the partials the string was losing fastest, which is why the room does not merely lengthen the note.

The room is the slower of the two

A reverberant field is the source convolved with the room, so a partial's tail falls at the slower of the two rates rather than at their sum — and the room is slower for exactly the partials the string is losing fastest. Half a note's colour is gone in 0.163 seconds in no room at all, 0.313 in a concert hall and 1.441 in a stone church. The destination is identical in all three, because a room cannot hold a partial up above the fundamental it is also holding. What a hall takes away is the rate, and the rate was the whole of the identity cue.

timbre · Envelope The drain does not stop when the key does. The note does.. Semitones of colour gone, against time, for a note on 130.8 hertz left to ring and for the same note released after 0.4 seconds onto a damper of 0.15 seconds. The two curves lie on each other until the key comes up, and the damped one then ends: the note is inaudible at 0.52 seconds with 8.7 of the free note's 9.9 semitones delivered. A damper adds one loss to every partial alike, so it adds the same number to every decay rate and leaves every DIFFERENCE between rates exactly as it was — the spectrum at each instant is the ringing spectrum shifted bodily down by 400 decibels a second. The colour goes on draining at its own rate the whole time. What the damper takes away is not the drain but the seconds.

A damper changes the clock, not the colour

A damper is an extra loss on the string rather than a second decay, so it adds the same number of nepers a second to every partial — and adding a constant to every rate leaves every difference between rates exactly where it was. The damped spectrum at any instant is the ringing spectrum at that instant shifted bodily down, to machine precision. The colour goes on draining at its own rate; the note simply runs out of seconds, and how many it gets is written on the page as a note value and a tempo.

timbre · Envelope A damper is a loss on the string, so a room can overrule it. Decay rate in nepers a second against partial number, for a note on 130.8 hertz in a room of 2 seconds. The rising line is the string's own loss, 1.15 nepers a second at the fundamental and growing as the partial number to the power 1. The line above it is that plus the damper's 46.1, which is what the string does once the key comes up. The flat line is the room. What a listener receives is the SLOWER of the damped string and the room, because a hall goes on radiating what the string has already given it — and here the room is slower on 8 of 8 partials, from the fundamental upward. The composition proposed earlier — take the slower of the string and the room, then add the damper to whichever won — would put the damper outside the minimum, where nothing can overrule it, and would predict a note 2.54 seconds shorter than ringing where the arithmetic here predicts 0.90.

A damper cannot reach into the room

The essay before this one proposed the arithmetic for a damped note in a hall: take the slower of the string's rate and the room's, then add the damper's to whichever won. The composition is wrong, and it is wrong in the one place that decides the answer. A damper is a loss on the string, so it belongs inside the minimum where a room can overrule it — and past about three seconds of reverberation it is overruled on every partial, so the damper removes no audible seconds of note at all.

timbre · Envelope A louder final chord stands higher and still stands for a fraction of a second. How far a final chord stands above the listener's running impression at the instant it is released, against how long it lasts. The chord is a struck six-note tonic; "a step louder" is the hammer velocity doubled, which raises its loudness 7.12 phons above the tutti's. At the tutti's level, after 1.2 s of silence, the impression is 8.09 dB below the chord and the chord stands highest, 5.14 dB, at 54 ms; a step louder, straight out of the tutti, the impression is 7.12 dB below the chord and the chord stands highest, 4.51 dB, at 36 ms; a step louder, after 1.2 s of silence, the impression is 15.21 dB below the chord and the chord stands highest, 9.46 dB, at 38 ms. Every curve returns to zero by half a second: the impression climbs to whatever level the chord is played at, so a louder mark is not a stand that lasts but a deeper fall to climb out of, and a silence and a mark add as depths.

A louder final chord is a deeper silence and a brighter sound

A final chord marked a step louder than the passage was supposed to stand above a listener's running impression for as long as it sounded, since the impression can climb no higher than the chord. It climbs exactly that high, and the stand closes in half a second as it always did. What a louder mark actually buys is depth — about seven phons, the same depth a second of silence buys — and a spectrum whose balance point sits most of a whole tone higher, which, unlike the stand, lasts for the whole chord.

form · Closure

Named alongside it

The objects these essays reach for when they reach for this one.

BrightnessDecayEnvelopeIdentificationPianoRegisterSpectrumDampingDynamicsExcitation pointHarpsichordInharmonicity

All concepts