Acoustics & Music Theory
why it sounds like that
Concept

Head shadow — where it appears

The attenuation a listener's own head applies to sound arriving at the far ear. It grows with frequency, because a head is small compared with a long wavelength and large compared with a short one, which is what makes a level difference between the two ears a usable cue to direction.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

A source 45° off centre, and the path difference it makes. A head from above with a source to one side. The near ear is reached first; the far ear's path runs round the head, and the difference between the two is 13.1 centimetres, which at 343 metres a second is 381 microseconds. That number, and the level difference the head's shadow produces, are the whole of what the ear has to work with.

Two ears, and the whole of the difference is 655 microseconds

Direction is computed from two numbers — when a sound reaches each ear and how loud it is at each — and which of the two is usable is decided by the wavelength against the width of a head. The changeover frequency is not a design choice. It falls out of 343 metres a second and 17.5 centimetres, and it is why the mechanism of hearing where something is changes halfway up the piano.

perception · Localisation Where a room stops sending the two ears the same sound. The correlation between the two ears' signals against frequency, for a seat 15 metres from the source in a 15,000 cubic metre room with a 2 second reverberation time. The pale curve is the diffuse field alone — sin(kd)/(kd) for an ear separation of 17.5 centimetres, which first crosses zero at 980 hertz. The heavy curve adds the direct sound, which is coherent and lifts the whole thing by an amount the direct-to-reverberant ratio sets. At 125 hertz the coherence is 0.98 and at 1000 it is 0.16.

Where the two ears stop agreeing

A room sends both ears versions of the same sound, alike at low frequencies and increasingly unlike at high ones. Where they stop resembling each other is 980 hertz, and it is set by the 17.5 centimetres between the ears rather than by anything about the room — which is within a quarter of a frequency found earlier for a completely different reason. One minus that correlation is spaciousness, and it is computable from a room's own reverberation.

perception · Localisation A turn of 0.99 degrees tells front from back. A source 45 degrees off centre and its mirror image 135 degrees off, which produce the same interaural delay and are therefore the same signal to a listener who does not move. As the head turns the two predictions separate: the front source's delay falls and the rear source's rises, because the fold at ninety degrees puts them on opposite branches of the same curve. They differ by the 15-microsecond threshold after 0.99 degrees of turn — which is exactly half the 1.97 degrees a source would have to move for the same listener to notice it moving, and it is half for a reason: a turn displaces the two hypotheses from each other by twice what it displaces either of them from where it started.

The turn is half the angle

A stationary head cannot tell a sound in front from the same sound behind, and an earlier essay said so at length. The turn that breaks the confusion is 0.84 degrees — exactly half the angle a source would have to move for the same listener to notice it moving, and half for a reason. In a hall the same turn does something else: the source swings at 8.9 microseconds a degree and the room swings at 2.7, so a listener who moves is separating the soloist from the reverberation as well as the front from the back.

perception · Localisation 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 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.

Interaural time differenceLocalisationJust-noticeable differenceSpeed of soundCalibrationImage-sourcePrecedence effectReverberationCone of confusionCritical distanceDiffuse fieldDuplex theory

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