Acoustics & Music Theory
why it sounds like that
Concept

Brightness — where it appears

How many of a mode's degrees are raised relative to its neighbours, which orders the seven along the chain of fifths. It is a property of the rotation rather than of the set, so it distinguishes modes that every set-based measurement finds identical.

Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.

The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.

The same seven, started later

A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to.

scales · Modes The rotations of the diatonic set, brought to one tonic. The same rotations started on the same note rather than on their own, ordered by how many of their degrees are raised. Every step lowers exactly one note by a semitone, and the notes it lowers, in order, are F♯, B, E, A, D, G — which is the chain of fifths read backwards. A rotation is a rotation whatever the scale; the chain is a property of a set generated by a single interval, and it disappears the moment the set is not.

Parallel and relative are two different maps

Bring the seven modes to one tonic and order them by brightness, and each step lowers exactly one note by a semitone — and the notes it lowers, in order, are F♯, B, E, A, D, G, the chain of fifths read backwards. Do the same to harmonic minor's rotations and each step moves two, three or four notes at once. The famous chain belongs to the generator, not to rotation.

scales · Modes What gets out of an opening, for 4 openings. The fraction of the wave's energy radiated at an open end against frequency, in the baffled-piston model — the radiation resistance of a circular piston, normalised to the tube's own impedance. Each curve runs from nothing at the bottom, where the opening is far smaller than a wavelength and the wave simply turns round, to everything above ka ≈ 2. Half the energy leaves at 6364 Hz for a flute's embouchure end (radius 10 mm), 2015 Hz for a clarinet's bell (radius 30 mm), 975 Hz for a trumpet's bell (radius 62 mm), 403 Hz for a horn's bell (radius 150 mm). The crossover goes as one over the radius, so the widest and narrowest here are 15.8 times apart in frequency. The same number decides how strongly the tube resonates and how much sound it makes, which is why a bell cannot brighten an instrument without also weakening its own resonances.

The bell decides what gets out

A tube resonates because the wave turns round at the open end, and it is audible because some of the wave does not. Those are the same number with opposite signs. One quantity — the size of the opening against a wavelength — decides how loud an instrument is, how bright it is and how directional it is, and a bell moves the boundary rather than removing it.

timbre · Air column A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there.

The mark that is not a level

There are six of them, they carry no units, and a performer has to turn one into a number before it means anything. What they instruct is not loudness. On a struck string a harder blow shortens the hammer's contact from 2.26 milliseconds to 0.95, which moves the first null of its own pulse from the third partial to the sixth: the partials between those are not quieter at pianissimo, they are gone. A fortissimo is a different sound, and the page has one word for both things it changes.

timbre · Notation C4, in every place it can be played. A guitar neck with the 4 places C4 can be stopped, drawn at the fret spacing a 64.8-centimetre scale actually has. The stave writes one note and the tablature writes one of these; each notation says exactly what the other leaves out. The speaking lengths run from 61.2 down to 27.2 centimetres, so a hand plucking 12 centimetres from the bridge meets between 20 and 44 per cent of the string.

What a tablature keeps

Middle C can be stopped in four places on a guitar. The speaking lengths run from 61 to 27 centimetres, so a hand plucking twelve centimetres from the bridge meets between a fifth and nearly a half of the string, and the comb of missing partials is different at every one: the second partial is thirteen decibels stronger in the best position than in the worst. A stave writes one note for all four. A tablature writes four different things and cannot say which note any of them is.

instruments · Notation 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.

The note that gets duller as it dies

Every envelope drawn so far is one curve applied to a whole sound, and no struck string behaves that way. A string loses energy to air, to internal friction and to the bridge, and all three losses rise with frequency — so a note with a six-second fundamental has a sixteenth partial that is gone in under half a second, and the sound moving toward the listener is a spectrum collapsing toward its own fundamental. Which means an instrument is identified twice: once by the fifty milliseconds of its attack, which the earlier essays measured, and again by how fast its colour drains, which they did not.

timbre · Envelope One instrument, five spectra. The first 12 partials of a bowed string at 5 pitches, each passed through the same fixed body response and normalised to its own loudest partial. The resonances stay where they are and the partials slide under them, so the pattern is different at every note: the power-weighted centroid runs from 1.02 to 1.71 across the compass and is not monotonic in pitch. A source with no body at all would give 2.35 at every pitch, which is the single number every roughness figure here uses for a string.

An instrument is not one timbre

Every roughness verdict so far uses one partial list per instrument, and a violin does not have one. Its body's resonances stay where they are while the fundamental moves, so the radiated spectrum is different at every pitch: the power-weighted centroid runs from 1.02 to 1.64 across the compass, and it is not monotonic. Which means the result that a spectrum chooses its own scale gives a different scale at every note on the same instrument — three minima in the dissonance curve at the bottom of a violin's range, two in the middle and four at the top.

timbre · Spectrum What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series.

What the climb was a search for

The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose.

tuning · Pitch standard Trumpet at three dynamics, as a spectrum rather than a level. The radiated partials of a trumpet at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 2.19 to 6.41, a factor of 2.92, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 3 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument.

A dynamic mark changes what a note is

Every spectrum until now is a shape with a level in front of it, so that playing ten decibels louder raises every partial by ten. That is true of exactly one instrument in an orchestra. Everybody else steepens their own spectrum as they lean on it, and a trumpet's centre of gravity moves from the second partial to the sixth across a dynamic range while an organ flue pipe's does not move at all.

timbre · Orchestration Which of the four terms is doing the removing, note by note. Each of three terms has a corner — a partial number above which it is the one taking the energy away — and the lowest corner at each pitch is the term that binds. The comb is flat at 8, because where the hammer lands is a fraction of the length and does not change with the note. The contact time's corner falls from 34 at A0 to 0.5 at A7. They cross at about E3, 165 hertz: below it the strike point decides the spectrum and above it the contact time does. The dispersion's corner is above both at every pitch on the instrument — 5.8 at the top against a contact corner of 0.5 — so that earlier term is real and is nowhere the binding one. That is a statement no one of them could make on its own.

Four terms, and only one of them binds

Six earlier essays each computed one term of a piano's excitation spectrum and handed it to the next, and nobody had multiplied them. Multiplied, three of the four have a corner — a partial above which they are the term doing the removing — and putting the three corners on one axis says which is in charge at each pitch. Below E3 the strike point decides; above it the contact time does; and the dispersion, which is the newest and most laborious term, is nowhere the binding one.

timbre · Excitation point 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 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 Four players on three notes, every arrangement. The 36 ways of putting 4 players on a 3-note chord so that every note is covered, ranked by roughness, all at one total loudness of 26.9 sones. Each row is shaded by which note carries the pair. The best is flue | clarinet+violin | oboe and the worst is clarinet | oboe+flue | violin, a factor of 2.21. Every earlier essay puts exactly one player on each note, which is a permutation; a doubling makes the arrangement a surjection instead, and the doubled note sounds neither of its two players but the composite they make. Which note gets the pair explains 7 per cent of the spread here and which players sit on the lowest note explains 89: the fourth player is a much smaller decision than the three that were already there.

The fourth player is a spectrum, not a decision

Six earlier essays put exactly one instrument on each note, which makes an arrangement a permutation — and the commonest operation in orchestration is a doubling, which does not. Four players on three notes give thirty-six arrangements instead of six, and the extra choice turns out to be the smallest thing on the page: which note carries the pair explains three per cent of the spread and which players sit on the bass explains eighty-nine. A doubled note can be priced as one player, and which one is not the one a spectral account would have named.

instruments · Orchestration What each strike point on a marimba bar excites. The amplitude each of the first 4 partials receives from a strike at each point along the bar, each partial drawn to its own maximum. On a string this comb is |sin nπx| and is periodic, so one strike point silences a whole family; on a bar it is the mode shape itself, whose zeroes are not at rational fractions, so no strike point silences a family. The exception is the middle: every antisymmetric mode is zero there by symmetry, so a centre strike drives partial 1 at 100 per cent of its maximum and partial 2 at nothing at all. The cord's hole at 0.2031 is the mirror case — the fundamental is the partial a strike there cannot reach.

The four ways a marimba loses what its arch placed

The centre of a bar is a node of every even mode by symmetry, so a stroke there gives the second partial exactly nothing and a mallet's width cannot recover it; five millimetres off centre returns 14 per cent. Add the yarn mallet's 3-millisecond contact, the tube's rigid lid and a decay four times faster than the fundamental's, and four independent mechanisms take the tuned partial 80 decibels below it. On a xylophone the same four come to 24.

instruments · Struck bar 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 The partials the air makes, on the series the bore actually has. The input impedance of a trumpet, with the partials wave steepening manufactures from its A4 at 428 hertz drawn on it as vertical marks. The steepening is a distortion of one periodic waveform, so it makes its partials at exact integer multiples — 856, 1285, 1713, 2141, 2569 hertz. The bore's own resonances are not at integer multiples of anything: the peaks the manufactured partials aim at sit at 886, 1346, 1812, 2278, 2741. So every manufactured partial lands flat of the peak it might have used, by 59, 81, 98, 108, 112 cents — 2.6, 2.7, 3.0, 4.0, 4.9 half-widths of the peaks in question, which is outside the half-power point of every one of them. The dashed line is where the bell stops reflecting, at 2945 hertz; above it there is no peak to land on or miss.

Which notes go brassy first

Wave steepening manufactures partials at exact integer multiples of the note being played. A brass instrument's resonances are not at integer multiples of anything, and twelve earlier essays have measured how far off they are without ever putting the two on one axis. Put them there and a manufactured partial never lands on a peak — never once, at any note, by a miss that is the same number of hertz every time. So the alignment cannot be what decides which notes blare, and the thing that does turns out to be the bell.

instruments · Air column 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 Where a woodwind's A♭3 leaves it, below its corner and above it. The same fingering — 6 holes open on a 15-millimetre bore 567 millimetres long, sounding A♭3 at 207 hertz — drawn twice, with each opening's circle scaled by the share of the radiated power that leaves through it. At 400 hertz 78 per cent of it leaves through the first open hole, the bell takes 0 per cent, and the number of apertures really doing the radiating is 1.6; At 2600 hertz 6 per cent of it leaves through the first open hole, the bell takes 49 per cent, and the number of apertures really doing the radiating is 3.3. The lower frequency is below this fingering's corner and the higher one above it: below the corner the instrument is a short tube with one opening at the end of it, and above the corner it is the whole lattice at once. The power-weighted station — where a listener would say the sound is coming from — moves from 392 millimetres to 515.

Where a woodwind actually sounds from

Every number so far is read at the mouthpiece, and the corner's whole musical meaning is at the other end. Run the same solver forwards and it gives the flow leaving every hole — from which a clarinet's radiating aperture turns out to be a function of fingering and of frequency, but not the way it was predicted to: the fingering sets how far the aperture opens, almost exactly to the number of open holes, and barely moves the frequency at which it does.

instruments · Tone holes 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.

Spectral centroidSpectrumDecayEnvelopePartialTimbreRegisterExcitation pointIdentificationBoreDampingDynamics

All concepts