Acoustics & Music Theory
why it sounds like that
Concept

Piano — where it appears

The struck-string keyboard instrument, and this collection's standing example of an instrument whose timbre is a function of how hard it is played. Its felt hammers shorten their contact under a harder blow, so louder is brighter, which a plucked or bowed instrument does not do.

Named by 21 essays across 5 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 C4: the pulse computed and the pulse assumed. Above, the force the hammer delivers to the string at C4, integrated forward against the felt's nonlinear force and the string's returning corner, drawn against the half-sine of 1.60 milliseconds that every earlier figure assumed. The computed contact lasts 2.21 milliseconds and the corner comes home 4.6 times inside it. Below, the excitation each pulse gives to each partial. They agree at the bottom and part company higher up — worst at partial 6, by 34 decibels — because the assumed pulse has nulls the computed one does not.

The pulse that was assumed

Every figure until now low-passes the string's excitation with the spectrum of a half-sine, which is what a hammer would deliver against a rigid wall. An earlier essay said so and declined to do better. Doing better takes forty lines and refuses the prediction that came with it: the corner's round trips govern the spectrum as expected, and the contact time is governed by something else entirely — the mass ratio discovered one essay earlier.

timbre · Excitation point The partial above which the returning corner has lost its phase. Up a piano, the lowest partial whose extra travel after one round trip to the near end exceeds a quarter of a cycle — above it, the component comes home somewhere other than where a corner needs it. The curve follows the inharmonicity coefficient, which is smallest in the tenor and rises both ways, so the corner survives best at A2 — to the 24th partial — and worst at A7, where only 5 partials are still in step. The earlier coupled model assumes a clean corner at every partial it uses.

The corner does not come back a corner

Stepping a hammer forward against the string's own returning wave showed that the coupling matters most in the bass. It assumed the wave that came back was the shape that left. On a stiff string it is not: partials travel at different speeds, and by the top octave of a piano the sixth partial is the first to arrive out of place. But a periodic wave can only see phase modulo a cycle, and counted that way twenty-one of forty-eight partials are still in step — which is why the returning wave is degraded rather than destroyed.

timbre · Excitation point The heard moment against the pitch, on an instrument whose own attack is 8 ms. A note cannot establish an amplitude in less than 4 of its own cycles, so the attack has a floor of 4 periods — 145 milliseconds at A0 and 1.9 at C7. Below A4 the floor is longer than the instrument's own attack and the pitch decides the heard moment; above it the instrument does. The lag runs from 46.0 milliseconds at the bottom to 2.5 at the top, a spread of 44 milliseconds that no player can play their way out of.

A low note cannot start on time

Three earlier essays have held the pitch at one value. A note cannot establish an amplitude in less than a few of its own cycles, so the attack has a floor that rises as the pitch falls — 146 milliseconds at the bottom of a piano and three at the top. On an instrument whose action takes eight milliseconds everywhere, that is a forty-three millisecond spread across the keyboard from the period alone, and no player can do anything about it.

rhythm · Perceptual-centre Re-gauging at a fixed tension: how close sheep gut comes to breaking. Holding the tension at 700 newtons and re-gauging every string to suit the standard, the diameter each note needs goes as one over its frequency — so the stress goes as the frequency squared, and the margin against breaking falls the same way. At A = 392 the worst note has 1.79 times the stress sheep gut will take; at A = 466 it has 1.27. The margin reaches one at A = 525 hertz, which is far above anything the four hundred years of climb reached.

It was never the strings that stopped the climb

Every figure so far holds the instrument still and moves the standard. History did the opposite — instruments were rebuilt to suit the pitch, string by string. Hold the tension instead and each string's gauge is forced: the diameter goes as one over the frequency and the stress as its square. Gut breaks at A = 525 hertz and steel at 604, and the climb stopped at 466. The ceiling was somewhere else entirely.

tuning · Pitch standard 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 Three instruments, three wires, and an order of magnitude between them. The inharmonicity coefficient of each instrument's own string design, across its own compass. At middle-of-the-keyboard C a piano's wire is 315 millimetres long and 1.02 thick and gives B = 1.5e-3; a harpsichord's at the same pitch is 355 millimetres and 0.30, and gives 8.1e-5 — a factor of 19. The harpsichord's line is nearly flat across four octaves because its scaling is Pythagorean: the length doubles for every octave down, exactly, until the case runs out. The piano's is a V, because its case runs out two octaves earlier and everything below the break is a compromise. Every column before now used the top line for all three instruments.

Three exciters and three wires

Every column drawn so far has run a plectrum and a beater on a piano's string, because a piano's was the only scaling to hand. A harpsichord's wire is an order of magnitude less stiff, for a reason that turns out to be the tensile strength of iron — and with all nine combinations computed, the crossing found before stays exactly where it was, on every wire.

instruments · Excitation point A mistuned 2:1 makes 8 beats, not one. Every pair of partials that coincides in a 2:1 mistuned by 6 cents, with the rate each one beats at. On a perfectly flexible string the rates are 1.5, 3.1, 4.6, 6.1 and so on — an exact harmonic series of the slowest, because partial 2k is exactly twice partial k. On a stiff one they are 1.3, 0.9, 2.7, 11.1, and the 8th member is at 122 hertz. The family grows as the cube of the member number rather than linearly, so 4 of the 8 are slow enough to be beats at all and the rest are roughness.

A beat is never one beat

Every beat counted until now has come from one pair of partials. A real spectrum has many, so a mistuned octave produces eight beats at once — and on a perfectly flexible string those eight are an exact harmonic series of the slowest. On a real piano string they are a cubic, half of them are not beats at all, and no single width of octave silences more than one. That is where the tuner's five octaves come from.

intervals · Beating Of 8 beats at A3, 3 can be attended to. Every member of the beat family a 2:1 mistuned by 6.0 cents makes on a real string at A3, placed by how separable its coincidence is from the partials beside it — in auditory filter widths, across — and by how fast it beats, up. The shaded region is the set a listener can receive: wider than one filter, and between 0.4 and 15 fluctuations a second. 4 of 8 members fall inside it, and once rates within a factor of 2 are counted as one modulation channel there are 3. The members that fail do so for two different reasons: the low ones sit under the roughness ceiling but their coincidences are buried in a filter that holds three partials, and the high ones are resolved and far too fast.

Three beats at most, and only in the middle of the keyboard

A mistuned octave on a real piano makes eight beats at once, and a listener attending to one of them is doing something that has a threshold. Two thresholds, in fact — a rate and a place — and once both are applied the eight become four at A3, one at A1 and one at A5. Every interval a tuner sets goes to zero at both ends of the compass and peaks at eight countable beats in the octave the bearing is laid in.

intervals · Beating 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 Every beat in a family is the same depth, and none is near the threshold. The 8 members of the beat family a octave mistuned by 6.0 cents makes at A3 on a real string, each placed at the rate it beats and at the modulation index a listener's filter delivers there. The rising curve is the published detection threshold for amplitude modulation, which is flat at 0.03 below about fifty fluctuations a second and rises above it. The flat dashed line at 0.667 is the depth the pair has in isolation, and it is the same for every member: on a spectrum falling as one over n, the k-th member pairs partials 2k and 1k, whose ratio is 2.00 whatever k is. What the filled points show is the smaller effect that does depend on the member — the partials on either side of the coincidence leak into the same filter, add level without adding fluctuation, and dilute the index from 0.662 to 0.405. Inside the countable rate window the narrowest margin over the threshold is a factor of 16.7, on member 4. The depth criterion removes nothing.

Every member of a beat family is the same depth

A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.

intervals · Beating 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 harder strike buys beats only if the drop does not deepen with it. The beats a mistuned octave on A3 delivers on a string that decays in two stages, against how hard both notes are struck, with the aftersound's drop below the strike 20 dB at 80 dB and changing by a stated number of decibels for each decibel of strike. -0.5 dB per dB: 70 dB 2.5, 75 dB 5.7, 80 dB 8.9, 85 dB 11.9, 90 dB 12.2, 95 dB 11.6, 100 dB 10.9; 0 dB per dB: 70 dB 4.7, 75 dB 6.9, 80 dB 8.9, 85 dB 11.0, 90 dB 12.3, 95 dB 12.2, 100 dB 11.8; +0.5 dB per dB: 70 dB 7.2, 75 dB 8.1, 80 dB 8.9, 85 dB 9.8, 90 dB 10.7, 95 dB 11.2, 100 dB 11.9; +1 dB per dB: 70 dB 9.6, 75 dB 9.2, 80 dB 8.9, 85 dB 8.4, 90 dB 8.1, 95 dB 7.8, 100 dB 7.5. Where the drop stays put, a harder strike keeps buying beats to about 90 dB; where it deepens by a decibel per decibel, every strike past 70 dB costs beats.

A firm touch buys beats until the aftersound sinks with it

A piano string decays twice, and a mistuned octave's countable beats were found to rise with how hard the note is struck — on the assumption that the aftersound always starts twenty decibels below the strike. It need not. The moment the count shuts is set by the level the aftersound starts at and nothing else, so a harder strike buys beats only if its aftersound does not sink with it. If the drop deepens by more than 0.85 decibels for each decibel of strike, the firm touch costs beats instead.

intervals · Beating How wide the unison is left decides how far down the aftersound starts. A two-string unison on A3, each string decaying in 20 seconds alone and the pair's bridge-driving mode in 1.5, struck and heard at the bridge, against how many cents apart its strings are tuned: the drop from the prompt sound to the aftersound, for a hammer that strikes both strings alike and for one that strikes the second 20 and 50 per cent softer. Even strike: 0.25 cents 65.2 dB, aftersound 19.9 s; 0.5 cents 53.1 dB, aftersound 19.5 s; 1 cent 41.0 dB, aftersound 18.0 s; 1.5 cents 33.7 dB, aftersound 16.0 s; 2 cents 28.4 dB, aftersound 13.8 s; 2.5 cents 24.2 dB, aftersound 11.6 s; 3 cents 20.5 dB, aftersound 9.7 s; 4 cents 13.8 dB, aftersound 6.5 s; 5 cents 6.3 dB, aftersound 4.0 s. The earlier count assumed 20 dB; the unison gives that at about three cents. At a perfect unison the aftersound is not excited at all, and past 5.33 cents the two modes split and the unison beats instead. A strike's force scales both modes alike and appears nowhere in the drop.

The octave is counted best over a unison left two and a half cents wide

A firm touch buys a mistuned octave's beats only if the aftersound's drop does not deepen with the strike, and the drop was left as a number to be measured. It has a mechanism, and the mechanism has been known since the essay on three strings: the unison's two coupled modes. Computed from them, the drop does not depend on the strike at all. It depends on how wide the tuner left the unison — sixty-five decibels at a quarter of a cent, twenty at three — and the octave's beats peak, at 6.8, over the same two-and-a-half-cent unison that makes the longest note.

intervals · Beating Where the tilt of the blow matters, and where it does not. The octave's beats at 80 dB against the tilt of the blow from the vertical: one string, others muted, 0° 2.81, 1° 2.81, 2° 2.81, 3° 2.81, 5° 2.81, 8° 2.87, 12° 3.19, 16° 3.62, 20° 4.27; a unison half a cent wide, 0° 1.51, 1° 1.51, 2° 1.51, 3° 1.51, 5° 1.57, 8° 1.95, 12° 2.60, 16° 4.00, 20° 4.87; a unison 2.5 cents wide, 0° 6.81, 1° 6.81, 2° 6.81, 3° 6.81, 5° 6.87, 8° 6.87, 12° 6.70, 16° 6.97, 20° 7.25. One string alone and a clean unison both get their aftersound from the second plane and both gain beats as the blow tilts. The wide unison already has an aftersound twenty-odd decibels down from its own detuning, and the second plane, at forty, adds nothing it can hear.

The plane of the blow matters only where the unison is clean

A piano string vibrates in two planes, and the one parallel to the soundboard gives even a single string a second, slower decay. A hammer tilted off the vertical feeds it. That was the last way left for a firm touch to cost a tuner beats: if harder blows came down straighter. They would have to straighten by half for every seven decibels, and even then only on a single string or a unison tuned clean. Over a unison left two and a half cents wide, no tilt of the blow reaches the count.

intervals · Beating A stiff string quickens the narrow intervals' beats and slows the wide ones'. Each equal-tempered interval above A3 on a piano whose strings are those of the grand modelled here, each note with its own string's inharmonicity: the beat rate of its coincident partials as a share of the rate a bearing plan prints, which assumes whole-number partials. fifth (narrow): printed 0.74, true 0.96, a share of 1.29; fourth (wide): printed 0.99, true 0.63, a share of 0.64; major third (wide): printed 8.73, true 8.27, a share of 0.95; minor third (narrow): printed 11.87, true 12.88, a share of 1.08; major sixth (wide): printed 9.98, true 8.86, a share of 0.89. In every coincidence the lower note supplies the higher-numbered partial and stiffness raises it more, so a narrow interval, whose lower partial is already on top, is pushed further apart and a wide one is pulled together.

A stiff string slows the fourths and quickens the fifths

Every rate in a bearing plan is computed from whole-number partials, and a piano string's partials are not whole numbers. The correction is usually summarised as 'every real beat is faster than the table'. It is not: stiffness quickens the beat of every narrow interval and slows the beat of every wide one, and on a small upright the tempered fourth stops beating altogether. Both errors push the same way, so a tuner who counts the printed table leaves every interval wide, and a fifth and a fourth counted that way make an octave stretched to the 4:2 type or a little past it.

intervals · Beating The fourth below a note and the fifth above it differ by the same amount wherever the note is put. The fourth A3–D4 and the fifth D4–A4 on the grand modelled here, each note with its own string's stiffness, with the octave 1.9 cents wider than 2:1 in its fundamentals: the beat rate of each, signed so that a wide fourth and a narrow fifth are both positive, as the middle note is moved three cents either side of its tempered place. Both lines move together, and the fourth always beats 0.15 a second faster than the fifth — the difference between the upper note's second partial and the lower note's fourth, which the middle note does not touch. The two beat alike only when the octave is 1.60 cents wide, the octave whose 4:2 member is silent.

The fourth–fifth check cannot hear the note it checks

A tuner tests a temperament by sounding the fourth below a note and the fifth above it, inside one octave, and comparing their beats. Both beats run to the same partial of the middle note, so moving that note speeds or slows both together and leaves their difference untouched. The check is a test of the octave, and of one particular octave: the 4:2. Measured with each note's own stiffness, the printed table laid fourth first builds exactly that octave on any piano; laid fifth first it overshoots, and the check pulls it back by a third of a cent.

intervals · Beating Each family of checks builds its own octave, and the thirds build one twice as wide as the fourths and fifths. On each lower note from F3 to F4, on the grand modelled here: how many cents wider than 2:1 the octave lands when it is filled by a chain of intervals each counted to the rate a bearing plan prints — a fourth then a fifth, three major thirds, a major sixth then a minor third, four minor thirds — against the 4:2, 6:3 and 8:4 octave types, each note with its own stiffness. At A3: fourth, then fifth 1.60, three major thirds 3.20, minor third, then major sixth 3.57, four minor thirds 8.09; types 1.60, 3.56 and 6.22. The fourth and fifth build the 4:2 octave exactly and the minor third and major sixth the 6:3 within a few hundredths of a cent; three major thirds build an octave 2.00 times as wide as the fourth and fifth's, just short of the 6:3.

The thirds and the fourths check different octaves

Every interval counted to its printed beat rate on a stiff string is left wide, by an amount set by the squares of the partial numbers that meet in it. Add the links of a chain and the sum lands on an octave type: a fourth and a fifth make the 4:2, a minor third and a major sixth make the 6:3, and three major thirds come just short of the 6:3. So the two most familiar checks in a bearing are exact tests of two different octaves, and on a stiff piano no octave passes both. Each fails on the other's octave by a beat a second at A3.

intervals · Beating Weighed as a share of the beats they compare, the two checks fail equally a ninth of the way from the 4:2 to the 6:3. On A3, on the grand modelled here, each note with its own stiffness: how far apart the fourth and fifth of the fourth–fifth check beat, and the minor third and major sixth of the third–sixth check, each as a share of the rates the table prints for them (0.99 and 11.87 a second), against how many cents wider than 2:1 the octave is. The 4:2 octave is 1.60 cents and the 6:3 3.56. The failures are equal at 1.82 cents, 11.2 per cent of the way from the 4:2 to the 6:3, where both checks fail by 11.3 per cent; balanced in beats a second instead, the octave is 2.78 cents and the fourth–fifth check fails by 60.7 per cent. Both checks are within the 15.0 per cent line on octaves from 1.31 to 1.89 cents.

The octave that fails both checks least

A tuner who cannot pass both the fourth–fifth check and the third–sixth check settles on an octave that fails them equally, and a listener compares beat rates by ratio. Weighed that way, the compromise sits a ninth of the way from the 4:2 octave to the 6:3 on every note of every piano, because the fourth beats twelve times slower than the minor third and that ratio belongs to equal temperament, not to the strings. On a grand both checks then pass to a fifteen-per-cent ear up to middle C, and above it no octave passes both.

intervals · Beating In an equal bearing the progression of thirds pulls the octave back onto the 4:2. On A3 on the grand modelled here, with the bearing set equal: the failures of the fourth–fifth check, the minor third–major sixth check and the progression of thirds across the octave's edge, each as a share of what it compares, against how many cents wider than 2:1 the octave is. The progression passes exactly on an octave 0.13 cents wide; the 2:1 type is 0.40, the 4:2 1.60, the 6:3 3.56. The two checks alone balance at 1.82 cents, failing by 11.3 per cent; with the progression as a third vote the octave that fails all three least is 1.63 cents, where the worst failure is 12.6 per cent (fourth–fifth 1.6 per cent, third–sixth 12.6 per cent, progression 12.6 per cent). All three pass at fifteen per cent on octaves from 1.31 to 1.88 cents.

The progression of thirds votes for the bearing, not the octave

Added as a third vote beside the two contiguous checks, the progression of thirds was expected to push a stiff piano's octave wider. It pulls it narrower, because the octave it asks for is not set by the strings at all: it is whatever stretch the bearing already carries, plus an eighth of a cent. In an equal bearing that drags the compromise back onto the 4:2 and costs the top of the bearing its pass; give the bearing a fifth of the octave's own stretch and the third vote falls silent.

intervals · Beating

Named alongside it

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

BeatingInharmonicityTuning by earPartialHammerDecayOctave stretchEqual temperamentSpectral centroidTemperamentBrightnessContact time

All concepts