The error that moves straight ahead

Where a wrong head gives itself away priced a listener whose internal model of their own head is the wrong size, and its closing paragraph named the parameter every figure in this account had set to one value without saying so:

Every head in every figure here has its two ears at ±r from a centre, so the delay curve is odd and a source at +θ produces exactly the negative of a source at −θ. Real heads are not symmetric and real ears are not equally sensitive.

The consequence is a change of error class rather than of error size. A wrong radius multiplies every delay by a factor; an asymmetry adds a constant to it. Those two do entirely different things to a map, and the difference is visible in one place that every earlier essay has treated as trivially correct.

Where a constant can go that a factor cannot #

Woodworth’s delay for a sphere is $(r/c)(\theta + \sin\theta)$, taken with the sign of the azimuth, and the whole of its range is 655 microseconds. It passes through zero at zero because the two ears are the same distance from the centre. That single property is what makes the median plane special: a source straight ahead delivers no delay at all, and no delay at all is the one measurement a listener cannot get wrong.

The wrong-size error preserves it, and so does the essay that varied the radius in the world rather than in the listener. A listener inverting with the wrong radius $\hat r$ reports the $\hat\theta$ satisfying $\hat\theta + \sin\hat\theta = (r/\hat r)(\theta + \sin\theta)$, which at $\theta = 0$ gives $\hat\theta = 0$ whatever $\hat r$ is. A head model ten per cent too small and one ten per cent too large agree exactly on where straight ahead is, and so does a head model wrong by a factor of two.

An asymmetry does not preserve it. Two ears at different effective radii, or two ears whose transduction latencies differ by a few tens of microseconds, add a term that does not vanish with the azimuth:

$$d_{\text{measured}} = \operatorname{sgn}(\theta)\,\frac{r}{c}\,(\theta + \sin\theta) \;+\; \Delta$$

At $\theta = 0$ the listener is handed $\Delta$ microseconds and inverts it through a map that has no idea it is there. Twenty microseconds puts their straight ahead 2.25 degrees round. Ten puts it 1.12, five puts it 0.56, and forty puts it 4.49.

The threshold for interaural delay is about fifteen microseconds, and the smallest angular change a listener can resolve on the median plane follows from it: 1.68 degrees. So an offset becomes an audible displacement of straight ahead at about the same size at which it becomes an audible delay at all — fifteen microseconds either way. There is no window in which the asymmetry is present and invisible, which is the exact opposite of what was found about size.

The same number of steps at every angle, and the reason is exact #

The flat lines in the hero figure are the result worth having, and they are not an approximation.

Differentiating Woodworth’s curve gives $d(\text{delay})/d\theta = (r/c)(1 + \cos\theta)$, so a fixed offset $\Delta$ displaces the heard direction by

$$\delta\theta \;=\; \frac{\Delta c}{r\,(1 + \cos\theta)}$$

which grows toward the side, doubling between straight ahead and ninety degrees. That looks like an azimuth dependence and it is cancelled by the other one. The smallest angular change a listener can detect at azimuth $\theta$ is the delay threshold divided by the same slope — $\text{jnd}\cdot c / (r(1+\cos\theta))$ — because the threshold is a threshold on delay and it has to be converted into an angle through the same curve. The factor $(1 + \cos\theta)$ is in both, and it cancels.

A constant offset is worth $\Delta/\text{jnd}$ just-noticeable steps at every azimuth, including zero. Twenty microseconds against a fifteen-microsecond threshold is 1.33 steps at the median plane, 1.34 at forty degrees and 1.37 at eighty — the small departure being the difference between the derivative and the exact inversion over a displacement of two degrees, which the figures compute both ways and check against each other.

That is the same cancellation the wrong-size essay found, arriving at the opposite conclusion. There the factor error’s angular size and the threshold both carried $(1 + \cos\theta)$ and the ratio grew with azimuth anyway, because the error itself is proportional to the delay and the delay grows too. Here the error is not proportional to anything. A factor error is invisible in front and betrays itself at the side; an offset error is equally visible everywhere and there is nowhere to escape it.

One side runs out of head #

The offset does something else a factor cannot, and it is a genuinely new failure mode rather than a rescaling of an old one.

A delay of more than $(r/c)(\pi/2 + 1)$ — 656 microseconds for the head drawn here — cannot be produced by that head at any azimuth. The wrong-size essay found a listener with too small an internal head being handed delays their own model could not produce, and called it an alarm: an impossible delay is a signal that something is wrong even with no ground truth anywhere.

An offset produces the same impossibility, and it produces it on one side only. Adding twenty microseconds to every delay pushes the positive side past the ceiling at 86 degrees while the negative side is pulled further from it; at forty microseconds the blind sector opens at 82 degrees. So a listener with an interaural asymmetry has a wedge on one side of them in which every source is reported at the extreme edge, and a mirror wedge on the other side in which nothing unusual happens at all.

That asymmetry is itself the alarm, and it is a much better one than the wrong-size alarm. An impossible delay tells a listener their model is wrong; an impossible delay on one side only tells them which way. It is the one piece of information a symmetric error cannot supply, and it arrives free.

The turn that caught a wrong head barely catches a displaced one #

The best result of the essay before this was a check that needs no ground truth: report a direction, turn the head by a known angle, predict the delay the source should now produce through the same internal map, and compare. A wrong radius breaks that prediction because the map’s shape is wrong, and an eighty-degree turn resolves a head model 1.10 millimetres out. The turn itself is a quantity already measured here — the turn is half the angle is where its geometry comes from.

An offset is fixed in head coordinates, so it rides round with the turn, and the only thing that can betray it is the curvature of Woodworth’s function.

It is barely enough. From a source straight ahead an eighty-degree turn leaves the prediction 8.1 microseconds out, which is under the threshold and therefore invisible. From sixty degrees the miss peaks at 7.1 and from eighty-five it reaches 17.9 — just over. Sweeping the offset rather than the azimuth, the smallest offset an eighty-degree turn can betray at all is 15.3 microseconds, which is the delay threshold itself.

So the turn adds nothing. By the time a listener could catch their own asymmetry by turning their head, the asymmetry is already large enough to be heard as a displacement standing still. The check that was the most satisfying finding of the essay before this — a calibration with nobody else in it — works on the size of a head and not on the symmetry of one.

What a displaced map does in a hall #

Two and a quarter degrees is a small number and a concert hall is where to ask whether it matters, because a hall is where a ruler has already been put on the same axis.

The hall through a head measured a stage in the units a listener actually has: how many just-noticeable steps wide the image of an orchestra is from a given seat. From fifteen metres a stage twelve metres across subtends about forty-five degrees and rather fewer distinguishable positions than that, because the steps grow toward the side. An offset of twenty microseconds displaces every one of those positions by 1.33 steps, in the same direction, by exactly the same amount.

That is the shape of the error and it is a benign shape. A rigid displacement of a whole image is the one distortion a listener has almost no way to notice and almost no reason to care about, because nothing in the scene is deformed: the first violins are still to the left of the seconds by the same number of steps, the soloist is still in front of the section, and the only thing that has moved is the label on the middle. A listener with a displaced map hears a correctly shaped orchestra sitting two degrees to one side of where it is, and the thing that would tell them is their eyes.

It is worth contrasting that with what the wrong-size error does to the same scene. A wrong radius scales the image: the stage is the right shape in the middle and stretched or compressed at its edges, so the outer desks move relative to the inner ones and the width of the ensemble is wrong while its centre is right. One error moves the orchestra and the other resizes it, and the resizing is the one that changes a relationship between two sources.

Which leaves the interesting case, and it is the one a hall makes ordinary. A position and a width established that a listener hears an ensemble as a location and an extent, and the extent comes from the parts of the signal that are incoherent between the ears. An offset does nothing to coherence at all — it displaces both ears’ copies of everything by one number — so it moves the position and leaves the width exactly where it was. A listener with an asymmetry hears a source of the correct width in the wrong place, and there is nothing in the sound to separate that from a source that really is in the wrong place.

Which computation produced the numbers #

The world delivers Woodworth’s delay for a sphere of radius 8.75 centimetres at 343 metres a second, taken with the sign of the azimuth. The periphery adds a constant $\Delta$ to it. The listener inverts through the same symmetric map at the same radius — so the internal head is right here, and the error is entirely in what reaches it, which separates this essay’s variable from the earlier one’s rather than compounding them.

The threshold is fifteen microseconds throughout, which is at the conservative end of the published range for interaural delay and is the value the earlier figures use when a number is stated. The smallest audible angle at each azimuth is that threshold converted through the slope of the delay curve at that azimuth, which is the same conversion every essay here has used.

Two routes to the displacement are computed rather than one. The exact route inverts Woodworth’s function numerically on the offset delay; the closed form differentiates it and divides. The figures check that the second is exactly constant across the sweep and that the first agrees with it to within a tenth, and the residual between them is the only thing on this page that is an approximation.

Where the model stops #

An asymmetry is not really a constant. Two ears at genuinely different effective radii produce a delay whose two halves have different slopes as well as a different zero, so the true error is an offset and a small factor together. The offset is the part that does something new and it is the part modelled here; the factor part belongs to the wrong-size essay and its size, at a few per cent of radius, is already known to be invisible in front.

A room is not in it either. Every figure here is a free field, and the first wavefront wins is why that is less of a simplification than it looks: a listener in a hall takes the direction from the first arrival and suppresses the copies behind it, so the offset applies to the arrival that counts. What a room would change is the confidence rather than the direction, and there is no term for confidence anywhere here.

The level cue is not in it. Where the two ears stop agreeing establishes that above about fifteen hundred hertz the direction comes from the level difference rather than the delay, and a head asymmetry displaces that cue too — by a different amount, through a different function. The wrong-size essay found the two cues disagreeing about a wrong radius and the level cue four times too quiet to settle it, and the same computation for an offset is the obvious next thing and is not done here.

And nobody grows asymmetrically on purpose. The offsets swept here are stated sizes rather than measured ones. A few tens of microseconds is the right order for a difference in the two ears’ transduction latency and for a couple of per cent of effective radius, and which of those two dominates in a real listener is a measurement rather than an arithmetic.

What the picture cannot show #

Whether a listener knows. Everything here assumes the internal map is the symmetric one and stays symmetric. A listener who has had their asymmetry since infancy has had every opportunity to learn it, from vision, from reaching, from turning toward things and finding them — and the displacement computed here is exactly the sort of error a lifetime of feedback would remove. What the arithmetic establishes is that the acoustics displace the map, not that the listener’s map is displaced.

Nor what happens when it changes. A blocked ear, a cold, a single hearing aid: each of those is an offset appearing in an afternoon, with no time to learn it. The size above says the displacement would be immediate and would be a couple of degrees, which is the kind of thing people report as sounds coming from the wrong place rather than as a loss of hearing.

Whose ears, and how far off #

Human interaural asymmetries are small and they are not zero. Effective radii differ by a percent or two between the two sides of an ordinary head, ear canals differ in length, and the two ears’ absolute thresholds differ by a few decibels in most people, which shows up as a latency difference because a louder signal is transduced sooner.

That last mechanism is the one worth watching, because it makes the offset level-dependent: a loud source produces less latency difference than a quiet one, so a listener’s straight ahead would sit in a slightly different place for a whisper and for a shout. Nothing in this account has a level in it anywhere, and this is the first place one has appeared with a job to do.

Still open: whether the displacement is level-dependent #

The mechanism above makes a prediction that is sharp and cheap and that the arithmetic here cannot quite make, because it needs one function already written elsewhere and never joined to this one.

If part of the offset comes from a threshold difference between the ears, then it is a latency difference, and latency falls with level along a curve the essays on loudness already have: a signal a few decibels above threshold is transduced tens of milliseconds later than the same signal well above it, and the difference between two ears shrinks as both are driven harder. Converting that into microseconds of interaural delay and running it through the arithmetic above would give the median plane as a function of how loud the source is — a curve that starts a few degrees off at a whisper and closes toward zero at a forte.

That would be a strange and testable thing. It says a quiet sound and a loud one in the same place are not in the same place to a listener, by an amount that depends on which of their ears is the better one. It needs no new geometry, it is the same substitution with $\Delta$ made a function of level, and what would come out is whether an asymmetry is one number or a curve.