Acoustics & Music Theory
why it sounds like that
Concept

Speed of sound — where it appears

How fast a pressure disturbance travels, about 343 metres a second in air at twenty degrees. It rises with temperature by about a sixth of a per cent per degree, which is why a wind instrument's pitch is a reading of the room's temperature.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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 How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string.

A standard is a specification

Choosing where to put A looks like a convention and is a mechanical decision. Tension goes as the square of frequency, so a piano built at 440 and tuned to 466 carries twelve per cent more load — a tonne and a half in this model's arithmetic. And there is a hard ceiling nobody can engineer round: frequency times length is capped by half the square root of a material's specific strength, which for gut is 240 hertz-metres. A violin E at A440 runs at 89 per cent of that. At A493 it is at a hundred, and every complaint in the historical record about rising pitch is about that one string.

tuning · Pitch standard 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 The same geometry at four sizes of head. Woodworth's interaural delay against direction, for 4 head radii from 5.8 to 9.8 centimetres. The whole range runs from 431 microseconds for a newborn to 735 for a large adult, and it scales exactly with the radius because the delay is (r/c)(θ + sin θ) and r is a multiplier. The detection threshold does not scale with the listener, so the number of distinguishable delays across the whole range falls from 98 to 57: a smaller head has the same directions in front of it and a shorter ruler to measure them with.

A smaller head in the same hall

Ten earlier essays draw one head. Every parameter belonging to the room has been varied by some figure and the one belonging to the listener never has, and it is the only one whose change the detection threshold does not follow: a six-year-old in the same seat receives the same fifty-four reflections at the same instants and reads them onto an axis with seventy distinguishable positions instead of eighty-seven. The speed of sound, swept over every temperature a hall is ever at, changes nothing at all — and the reason it cannot is the reason head size can.

perception · Localisation 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 A head model a few millimetres out reads every azimuth but the front. The azimuth a listener reports against the azimuth a source is at, for internal head radii from 8.22 to 9.28 centimetres against a true radius of 8.75. The delay a source produces is (r/c)(θ + sin θ) and the listener inverts it with the radius they believe they have, so their answer solves θ̂ + sin θ̂ = (r/r̂)(θ + sin θ). Every curve passes exactly through the origin: on the median plane there is no delay and therefore no error, whatever the head model is. The error grows with azimuth and is largest at the side. An internal head 5.3 millimetres too small runs out of azimuth at 82 degrees: beyond that the world is delivering a delay larger than any its owner's model can produce, and every source out there collapses onto the side.

Where a wrong head gives itself away

Every claim so far maps a delay to a direction through one fixed geometry, and the listener acquires that map while the geometry grows under them by seventy per cent. So the map can be wrong — and the essay before this one said the error would be largest on the median plane, where the delay curve is steepest. It is exactly zero there. The steepness is in the error and in the threshold and cancels between them, which leaves a listener whose internal head is 1.3 millimetres out with one place to catch it: hard to the side, where nobody localises well.

perception · Localisation A displaced map is displaced by the same amount everywhere. How far a listener's heard direction is displaced, in units of the smallest angular change they could detect at that azimuth, for four constant offsets added to every interaural delay. Each curve is flat. an offset of 5 microseconds is worth 0.33 just-noticeable steps at every azimuth; an offset of 10 microseconds is worth 0.67 just-noticeable steps at every azimuth, and past 88° hands the listener a delay their own head cannot produce; an offset of 20 microseconds is worth 1.33 just-noticeable steps at every azimuth, and past 86° hands the listener a delay their own head cannot produce; an offset of 40 microseconds is worth 2.67 just-noticeable steps at every azimuth, and past 82° hands the listener a delay their own head cannot produce. The reason is exact: differentiating Woodworth's curve gives a slope proportional to (1 + cos θ), so the angular displacement a fixed offset produces carries a factor of 1/(1 + cos θ) — and so does the smallest detectable angle, so the ratio has no azimuth in it. That is the opposite of a wrong head radius, whose displacement is zero on the median plane and grows toward the side.

The error that moves straight ahead

The essay before this one found that a listener whose internal head is the wrong size makes no error at all on the median plane, and has to look hard to the side to catch it. Every head drawn here has its ears at equal radii, which makes the delay curve odd and every error a factor — and a factor cannot move a zero. Real heads are not symmetric. A constant offset of twenty microseconds displaces a listener's straight ahead by two and a quarter degrees, and it displaces every other direction by the same number of just-noticeable steps, exactly.

perception · Localisation

Named alongside it

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

BoreHead shadowInteraural time differenceIntonationJust-noticeable differenceLocalisationString tensionCalibrationCentsEnd correctionPitch standardTemperament

All concepts