A quiet bass is further off-centre than a loud violin

The error that moves straight ahead added a constant to every interaural delay and found it doing the one thing a wrong head size cannot: displacing the median plane, by 2.25 degrees for twenty microseconds, and by the same number of detectable steps at every other azimuth. Its conclusion was reassuring. A constant displaces an orchestra rigidly — every desk moved by the same number of steps in the same direction — and a rigid displacement is the one distortion a listener has almost no way to notice, because nothing in the scene changes shape.

It then named the mechanism most likely to produce such a constant, and in doing so undermined the reassurance. Two ears rarely have the same threshold. An ear that is a few decibels less sensitive hears a tone as the better ear would hear it a few decibels quieter, and a quieter tone reaches the brainstem later. That makes the offset a latency difference — and latency depends on level, so the offset would not be a constant at all.

The closing paragraph predicted the shape: a curve that starts a few degrees off at a whisper and closes toward zero at a forte. It also said the curve was one the essays on loudness already had. They do not. Nothing in this collection has a transduction latency in it, and the curve has to be brought in from outside and stated. Once it is, both halves of the prediction turn out to be wrong, in opposite directions.

The curve that was missing #

The standard measurement of how long a sound takes to reach the brainstem is the fifth wave of the auditory brainstem response, a voltage recorded from the scalp a few milliseconds after a tone burst. Its latency is well documented against both level and frequency, and the fit this essay uses is the one Neely, Norton, Gorga and Jesteadt published in 1988 for tone bursts:

$$\tau \;=\; a \;+\; b\,c^{-L/100}\,f^{-d}$$

with $a$ = 5.0 milliseconds, $b$ = 12.9 milliseconds, $c$ = 5.0, $d$ = 0.413, the level $L$ in decibels and the frequency $f$ in kilohertz. The constant $a$ is the part of the journey that does not depend on the sound; the second term is the part that does, and it is what makes a quiet tone slow.

The worse ear receives the same pressure and responds as though it had received less, so its latency is the law’s latency at $L - \Delta T$, where $\Delta T$ is the difference in threshold. Subtract and the constant cancels:

$$\delta\tau \;=\; b\,f^{-d}\,c^{-L/100}\,\bigl(c^{\Delta T/100} - 1\bigr)$$

Three things can be read straight off that expression before any figure is drawn. The gap is proportional to the asymmetry, for asymmetries of a few decibels, since $c^{\Delta T/100}-1$ is nearly linear there. It falls exponentially with level, halving every 43 decibels — $100\ln 2/\ln 5$ — exactly, whatever the asymmetry or the frequency. And it grows toward the bass as the frequency to the power $-0.413$, so the gap for a note two octaves down is 1.77 times as large.

A latency gap is an interaural delay that no source put there. At three decibels of asymmetry it is 323 microseconds at a moderate level — half the whole range of delays a head produces, which two ears and one difference put at 655.

Neither half of the prediction survives #

The hero figure feeds that gap to the reader the preceding essay built, exactly as it fed a constant, and asks where the median plane goes.

The prediction said a few degrees at a whisper. The law says nineteen and a half degrees at 30 decibels for one decibel of asymmetry, forty for two and sixty-six for three. Those are not small displacements of the kind a lifetime of vision might gently correct. They are the difference between a singer at the front of the stage and one at its edge.

The prediction said closing toward zero at a forte. The law says it cannot close: an exponential halving every 43 decibels takes 43 decibels to lose half and another 43 to lose half of what is left, and the whole dynamic range of an orchestra is about 60. From a whisper at 30 decibels to a forte at 90 the displacement falls by a factor of 2.6, not to zero. At 100 decibels a one-decibel asymmetry still displaces straight ahead by 6.3 degrees, nearly four detectable steps.

So the curve is real and its size is alarming, and the alarm is itself the finding. Something is badly wrong with taking the law at face value, and the ear’s own thresholds are enough to show what.

The ear already says what a decibel is worth #

A listener has two detectable steps that every essay on localisation here has used. The smallest interaural delay they can detect is about fifteen microseconds; the smallest interaural level difference, which where a wrong head gives itself away took to be about one decibel when it asked the level cue to check the delay cue. Each is one step of lateral displacement, so between them they fix an exchange rate: fifteen microseconds of delay and one decibel of level are worth the same, to the listener, as a cue to where a sound is.

A threshold asymmetry is also a level difference. An ear three decibels worse hears everything three decibels quieter, and the binaural comparison of level reads that as three decibels of interaural level difference — three steps, at every level. So the latency law is making a claim about the exchange rate between the two cues: how many microseconds of delay one decibel of asymmetry produces alongside the decibel itself.

At 60 decibels and 500 hertz the law prices a decibel at 105 microseconds — seven times what the ear’s thresholds say it is worth. At a whisper it is eleven times, in the bass it is twelve, and even at the top of the figure, a 1 kHz tone at 100 decibels, it is nearly three. Taken whole, the law says that a threshold asymmetry is overwhelmingly a timing error and only incidentally a level one, and that a listener with a one-decibel asymmetry should hear a tone seven steps off-centre by its timing and one step off by its level, in the same direction.

That is not how the trading of the two cues is measured. The published trading ratios for low tones — how much delay offsets how much level, when a listener centres an image with both — run from a few microseconds per decibel to a few tens, which is the range the two thresholds imply and not the range the brainstem law implies. And a listener with an ordinary one- or two-decibel difference between the ears does not report the middle of the stage twelve or twenty-four degrees off.

So most of the latency the brainstem response shows cannot be reaching the comparator that reads interaural delay. How much can? The share at which the law agrees with the ear’s own exchange rate, at a moderate level and a middle frequency, is fifteen over 105: about one seventh. Why only a seventh is a question about the physiology — about where along the pathway the level-dependent part of the latency accumulates, and whether the comparison of the two ears is made before or after it — and the arithmetic here cannot answer it. What the arithmetic can do is say that a seventh is the most that is consistent with the listener’s own thresholds, and then ask what that seventh still does.

What survives the scaling #

Any share, a seventh or a twentieth, multiplies the whole curve by one number. It changes the size of every displacement and it changes none of their ratios — and the ratios are the part of the law that the preceding essay’s conclusion depended on.

Scaled to a seventh, a two-decibel asymmetry is worth 2.8 steps for a 500 Hz tone at 40 decibels — a displacement of nearly five degrees, which is the size of thing the preceding essay was discussing. But it is 5.0 steps for a 125 Hz tone at the same level and 0.95 for a 1 kHz tone at 90. Those numbers are not rescalings of one another: the law separates level from frequency, so the ratio between a quiet and a loud note is 2.24 in every register and the ratio between a low and a high note is 2.36 at every level, and both hold for any share at all.

That is the conclusion to carry away from the size argument. The asymmetry’s displacement is not one number but a surface over level and pitch, and the only thing the ear’s thresholds constrain is its height. Its slope in both directions is the latency law’s and cannot be scaled out.

An orchestra deformed rather than moved #

The hall through a head measured a stage in the units a listener has — detectable steps, fewer toward the sides — and the preceding essay’s reassurance rested on every one of those steps moving together. A displacement that moves every source by the same number of steps preserves every relation between them — the first violins still to the left of the seconds by the same margin, the soloist still in front of the section — and a listener has nothing inside the sound to compare against. A displacement that depends on pitch and level does not preserve those relations.

At 40 decibels — a quiet passage heard from the stalls — the double basses on their G move 5.5 detectable steps toward the better ear, the cellos 4.9, the violas 3.7, the second violins 3.1 and the first violins 2.8. In degrees the basses move 10.5 and the first violins 4.9. The stage has not been shifted; it has been stretched on one side and compressed on the other, and the stretch is concentrated in the bass.

Then the dynamic changes, and the stretch changes with it. At 90 decibels the basses move 2.5 steps and the first violins 1.2, and every section has come back toward where it is. A listener with a small threshold asymmetry hears the lower half of an orchestra drift toward their better ear in a pianissimo and return in a fortissimo, while the violins barely move — and that is a change inside the scene, a relation between two sources altering with the music, which is precisely what a rigid displacement could not produce.

Whether anybody notices is a separate matter. A crescendo changes a great deal at once — loudness, brightness, and the balance of early reflections that a position and a width found moving, widening and colouring an apparent source — and a three-step drift of the basses may be lost among those changes or mistaken for one of them. The arithmetic does not say it is heard. It says it is there, in any listener whose ears differ at all, at a size set by a share nobody has measured.

A crescendo is a calibration the head turn never was #

The preceding essay’s most disappointing result was about self-checking. Where a wrong head gives itself away had found a test that needs no ground truth: report a direction, turn the head by a known angle — the geometry the turn is half the angle works out — predict what the delay should become, and compare. It catches a wrong head size easily. Against a constant offset it catches almost nothing, because a constant rides round with the head and the only thing left to betray it is the curvature of the delay function; the smallest offset an eighty-degree turn could expose was 15.3 microseconds, which is a displacement already audible standing still.

A level-dependent offset offers a test the constant never could, and it needs no movement at all. A source that does not move, getting louder, should stay where it is. For a listener whose offset falls with level it does not: between 40 and 90 decibels a two-decibel asymmetry at a seventh of the law’s size moves a 500 Hz source back toward the centre by 1.56 detectable steps. That is a displacement within one listener’s hearing of one source, and it is above threshold. A crescendo from a still source is therefore a calibration signal — the ear hears a stationary instrument drift, the eye says it did not — and it is available every time an orchestra swells.

How small an asymmetry the crescendo can betray sets the practical reach of the test. Over that fifty-decibel range the drift reaches one step at an asymmetry of 1.3 decibels, and one decibel gives 0.77 steps. So a listener whose ears differ by more than about a decibel and a third has, in ordinary listening, a cue that says so and says which way — which is more than any test in the preceding essays offered for an error of this kind, and more than a smaller head in the same hall could offer a child whose map was being outgrown. Whether the auditory system uses such a signal to recalibrate is a question for an experiment. That the signal exists at all follows from the shape of the latency law, and would not exist if the offset were the constant the earlier arithmetic assumed.

Which quantities are stated and which are computed #

The world here is the free-field head of every earlier essay: Woodworth’s delay for a sphere of radius 8.75 centimetres at 343 metres a second, the median plane read by inverting that curve, and the threshold for interaural delay at fifteen microseconds. The new term is the latency law, stated with its published constants rather than derived, and applied as a difference between the two ears at the same sound pressure.

The share is a free parameter and the figures say so every time it appears. The one-seventh value is not a measurement: it is the single number at which the law’s exchange rate matches the ratio of the collection’s two thresholds, at 60 decibels and 500 hertz, and it is used to show a displacement of plausible size rather than to claim one. Everything this essay claims about shape — the halving every 43 decibels, the power of frequency, the constant ratios between registers and dynamics — holds at every share, and the figures check that directly rather than relying on the algebra.

The stage places each section’s cue at its fundamental, which is where the delay cue for a low note lives; a violin’s upper partials above about 1.5 kHz are localised by level rather than delay and would not move with the law at all, which makes the true deformation larger rather than smaller.

Where the argument is weakest #

The latency law describes wave V, not the binaural comparator. Wave V is generated high in the brainstem, after the stations where the two ears’ timing is first compared, and most of the level-dependent part of the latency may accumulate in the cochlea before that comparison or in the pathway after it. The seven-to-one mismatch is evidence that much of it does not reach the comparator; it is not evidence about where it goes.

Tone bursts are not notes. The law was fitted to short bursts presented alone; a sustained note has an onset, where the latency matters, and a steady state, where the comparator reads fine structure and the latency of the onset may no longer matter at all. For a sustained string section the whole effect may be confined to the attacks.

A threshold difference is not a gain difference. An ear that is less sensitive near threshold may not be less sensitive at 80 decibels — recruitment in damaged ears is exactly the opposite — so the level at which the worse ear “hears” a loud tone may be closer to the better ear’s than $L - \Delta T$ suggests. That would make the loud end of every curve here smaller still, and steepen them.

And a listener adapts. The preceding essay made the point and it binds harder here: an asymmetry present since childhood has had a lifetime of vision to calibrate against. What calibration can remove is an average displacement. A displacement that changes with the dynamic within a single phrase is a moving target, and whether a lifetime of calibration flattens a curve or only centres it is a question about learning, not about acoustics.

Still open: the attack and the steady state #

The largest caveat is the one about notes, and it is also the one that could be settled with arithmetic already set out in the essays on the first wavefront. A latency is a property of an onset. Once a tone is sounding, the two ears’ neural responses are locked to its waveform and the comparator reads the phase difference between them, which a level-dependent delay at the onset need not survive into.

If the displacement lives only in the attack, then the right question is not where a pianissimo bass is heard but where its entrance is heard — and the first wavefront wins already establishes that a listener takes a source’s direction from its first arrival and holds it. A displaced attack followed by an undisplaced steady state may therefore be heard entirely at the attack’s position, which would make the effect larger than this essay’s curves for struck and plucked notes and smaller for bowed ones that begin softly. The computation would give each note an attack, apply the latency difference only while the envelope is rising, and ask which position the precedence rule assigns to the note as a whole. It would say whether an asymmetric listener’s orchestra is deformed by its register, as here, or by its articulation.