Acoustics & Music Theory
why it sounds like that
Concept

End correction — where it appears

The extra length a pipe behaves as though it has, because the air just beyond an open end moves along with the column inside it. It is a fixed physical length, so it is a larger fraction of a short tube than a long one and stretches the upper modes.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

The end correction, for a bore of radius 7.5 mm. How flat a tube sounds against what its physical length alone would predict, because the wave carries on past the opening before it turns round. The correction is 4.6 mm at every note — Levine and Schwinger's 0.6133 times the radius for an unflanged end — and the error it causes is 13 cents on a 60 cm sounding length and 52 cents on 15 cm. It is the same millimetres in both cases.

The tube ends after it ends

A wave does not turn round at the opening. It carries on into the room for about six-tenths of the bore radius and reflects there, so every tube is acoustically longer than it is. The correction is a fixed number of millimetres against a wavelength that halves every octave — a rounding error at the bottom of an instrument's range and most of a semitone at the top.

instruments · Air column What 2 partials imply, and how many answers there are. The partials are at 32.7, 49.0 Hz. Each row is a harmonic series they are consistent with to within 30 cents: the filled dots are the partials in their assigned slots and the open dots are slots the series predicts that nothing occupies. There are 5 such series with harmonic numbers up to 16, the best fitting them to 1.4 cents with a fundamental of 16.34 Hz and no unoccupied slot. The rest sit at one half, one third, one quarter, one fifth of it and their arithmetic is exactly as exact; what separates them is the count of empty slots, which is why a residue pitch built from few partials is reported an octave out by a minority of listeners and not by the rest.

Two pipes for a note neither makes

The resultant stop sounds a sixteen-foot pipe and a ten-and-two-thirds-foot pipe and asks the listener for a thirty-two-foot note. It is the missing fundamental built on purpose from the fewest partials that can imply anything — and counting how many fundamentals two partials actually imply is the arithmetic behind three centuries of builders disagreeing about whether it works.

instruments · Missing fundamental A wind instrument is a thermometer, and a string is not. Cents from the pitch at 20 degrees, against the temperature of the air inside a wind instrument and of a steel string, computed from the speed of sound as 343.2 metres a second times the square root of absolute temperature, and from a string's tension falling by Young's modulus times the expansion coefficient per degree. The wind slope is 2.95 cents a degree at 20 degrees, so 0.0 cents at 20, 11.7 cents at 24, 23.3 cents at 28, 34.7 cents at 32. The Pythagorean comma is reached at 28.1 degrees — 8.1 degrees of warming, which a wind instrument does from breath alone within a few minutes. The string goes the other way, 36 cents flat at 34 degrees, so the gap between the two sections opens at 5.4 cents a degree.

A wind instrument is a thermometer

Pitch goes as the square root of absolute temperature, so a warming clarinet sharpens by about three cents a degree and passes a Pythagorean comma after eight. The strings beside it go flat as they warm. Nobody chose either number, no temperament addresses either of them, and together they are twice the size of the discrepancy this whole field is named after.

tuning · Air column 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 What a hand in the bell buys, and what it costs. How far the instrument flattens and how much radiation it loses, against the share of the bell's mouth the hand blocks, for an F horn's 15 centimetre mouth on a 3.7 metre acoustic length. The two curves are the same aperture radius read twice: a narrower mouth adds inertance, which lengthens the tube, and is acoustically smaller, which stops radiating. The mark is the 25.6 cents left owed earlier on the eleventh partial — it needs 62 per cent of the mouth blocked and costs 3.8 decibels of radiated power. Fully stopping is a semitone and nine decibels.

The hand that changes the bore

A conjecture refuted here left a question with a number on it: the natural trumpet's eleventh partial is 48.7 cents flat, the lips can move it 23.1, and 25.6 cents are owed by something that is not an embouchure. There is exactly one thing a player can change about the bore while playing, and putting the hand in the bell buys those 25.6 cents at a cost of 3.8 decibels — because the aperture that tunes the instrument is the aperture that radiates it.

tuning · Air column cylinder and Bessel flare: the length each mode behaves as though it has. Each mode's own acoustic length, m·c over twice its frequency, for a bore 148 cm long. A cylinder would give one number repeated. This gives 164 cm at the second mode and 154 at the 8th — a spread of 10.1 centimetres, or 110 cents, because a flare's end correction is a length that shrinks as the note rises. The first mode is off the top of this axis and is not a mode a player uses.

A horn has one length per partial

Every tube until now has had an acoustic length: its physical length plus a correction for the wave carrying on past the opening. A flaring bore does not have one. Its second mode behaves as though the tube were 164 centimetres long and its eighth as though it were 154, and the ten centimetres between them are the same physical fact — a fixed correction against a shrinking wavelength — arriving as a hundred cents.

tuning · Bore profile The hand closing, and the note it is holding. The fourth resonance of a 370 cm horn against how much of the bore the hand occludes, at 97 per cent of the way along where the radius is 36 mm. Nothing much happens for the first ninety per cent. The note then falls to -401 cents — most of a fourth — over the next few, and between 99.5 and 99.90 per cent it jumps to 96 cents SHARP, because the lowest resonance has left the bottom of the range and every mode has taken the place of the one below it. The series' ratios are unchanged across the jump.

The hand goes in, and the note jumps

A horn player's hand closes the bell and the pitch falls — 19 cents, then 55, then 132, then four hundred, accelerating the whole way. Then, in the last half per cent of closure, it stops falling and lands a semitone above where it started. An earlier essay guessed the mechanism was the boundary condition changing kind and the series going odd-only. It is not. The series never changes at all.

tuning · Air column A woodwind cannot be pulled to a new standard. Every earlier figure computes strings, because a string's tension and gauge give a closed-form scaling law. A wind instrument does not have one, and this is why. To move from 440 to 415 hertz the player pulls out 11.7 millimetres at the joint, which lengthens the sounding tube of every fingering by the same absolute amount — so the interval each note drops is a fixed length against a shrinking one, exactly the shape of an end correction. The tuning note lands where it should and nothing else does: D3 is 67 cents sharp of where it belongs and G5 is 75 flat, a spread of 142 cents across the compass. A rebuilt instrument has no such problem — scale every length by one ratio and every mode moves by the same interval, and the tone-hole lattice cutoff moves with it, from 1766 hertz to 1666, which is 101 cents and therefore the same instrument transposed. That is the difference between an afternoon and a year.

A woodwind cannot be pulled to a new standard

Every earlier essay computes strings, because a string has a closed-form scaling law. A wind instrument does not. Pulling out at the joint lengthens every fingering's sounding tube by the same number of millimetres, which is a fixed length against a shrinking one — so the tuning note lands and the compass spreads by 142 cents. The strings could be regauged in an afternoon; the winds had to be rebuilt.

tuning · Pitch standard Seven holes that all sound 196 hertz, and none of them agrees about the twelfth. Each dot is a hole radius, placed at the station that makes the first resonance 196 hertz. The stations run from 411 millimetres for a 7.5-millimetre hole to 307 for a 1.4-millimetre one, which is a fifth of the tube. Up the axis is what the second resonance does: a cylinder's should be three times the first, and it is -2 cents from it for the widest hole and -453 for the narrowest. The hole's inertance rises with frequency, so a narrow hole lengthens the tube more for the twelfth than for the fundamental — and two holes that are interchangeable in the first register are a fourth apart in the second.

A hole is a short tube

Four earlier essays have treated an open tone hole as a point where the pressure is released. It is not: the air in a hole has mass, and a hole with mass does not end the bore, it loads it. Seven holes drilled at seven stations all sound the same G — and their twelfths are spread over a fourth. Cross-fingering falls out of the same arithmetic, and it is not made of what everybody says it is.

instruments · Tone holes Four of the five are a whole number of semitones, and one is exactly half of one. Each mismatch in cents, against the ticks at whole semitones — which are the only places a transposing keyboard can put a player. 4 of the 5 land within six cents of a tick: the Chorton–Kammerton gap is 197 cents against a whole tone's 200, and Chorton against French pitch is 296 against a minor third's 300. The exception is an English organ against Handel's fork, at 50 cents — 50 cents from the nearest tick, which is as far as it is possible to be. So the small mismatches are the unsolvable ones, and the large ones were solved by shifting the keys.

The instrument that cannot be moved

A string is regauged and a woodwind is scaled. An organ's pitch is the length of its pipes, and metal can be cut off and cannot be put back — so an organ is a ratchet that only goes sharp. The mechanical answer was to shift the keyboard against the pipes, and its cost is not the transposition. It is that the temperament's key colours rotate out from under the notation, by an amount measured in fifths rather than in semitones.

tuning · Pitch standard A woodwind with holes graduated 12 mm to 6 mm, drilled so that every fingering is in tune. A cylindrical bore 567 millimetres of acoustic length and 15 across, with twelve tone holes through a 4-millimetre wall. Opening them one at a time from the far end takes it up a chromatic scale from D3 to D4. The stations are not copied from a maker's drawing: each was solved so that its own fingering sounds its equal-tempered note in this model, one hole at a time down the tube with every hole below it already open, which is what a reamer and a tuning fork do. The worst fingering is 11.9 cents out. The diameters run 12.0 millimetres at the bell end to 6.0 at the top, and the spacings close from 30 millimetres to 19.

The cutoff that is a list

Five earlier essays have quoted one number for a woodwind's cutoff — 1,824 hertz for a clarinet — from a formula written for an infinite lattice of identical holes. Solve a whole twelve-hole chart instead and the number is eleven different numbers, running from 2,193 hertz down to 1,574, which is 574 cents. The lowest fingering has no cutoff at all, and which way the list runs turns out to be a design decision rather than a fact about woodwinds.

instruments · Tone holes Every pitch standard, given the width 8 degrees gives it. Each documented standard drawn not as a point but as the band an ensemble occupies while the room warms by 8 degrees: the air columns sharpen by 23.3 cents, the steel strings flatten by 20.1, and 13.8 cents of spread inside each wind instrument's own register cannot be pulled out because it is a gradient along the bore rather than an offset. The band is 57 cents wide, and 5 of the 6 adjacent steps in the whole record are narrower than it — which is to say that 5 of the distinctions four centuries of committees argued about are smaller than the pitch spread inside one orchestra on one evening.

A standard is a point, and a performance is a band

Nine earlier essays draw every pitch standard as a single number, because none of them has a temperature in it. An air column sharpens as the room warms and a steel string flattens, at 2.95 and 2.49 cents a degree; add the 13.8 cents of spread inside one wind instrument's own register and eight degrees makes an orchestra 57 cents wide. Five of the six steps in four hundred years of pitch standards are narrower than that.

tuning · Pitch standard

Named alongside it

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

BoreCutoffIntonationResonanceBoundary conditionPitch standardRegisterStanding waveTemperamentTone holeBrassCents

All concepts