The beat that is not in the air

Two tones a few hertz apart swell and fade at their difference, and the swelling is in the air. Add two sinusoids and the sum has an envelope; a microphone records it, an oscilloscope draws it, and any detector at all responds to it. That is why beating is the most direct evidence available that the ear does sums, and it is also why it is not evidence that the ear does sums: the sum has already happened before the ear is involved.

There is a version of the experiment in which the sum cannot happen, and it is a one-word change to the apparatus. Put one tone in each ear.

Nothing sums, and there is still a beat #

With 500 hertz going to the left ear and 504 to the right, the pressure at each eardrum is a steady sinusoid of constant amplitude. No modulation anywhere. Two microphones in the two ear canals would record two flat envelopes.

Listeners report a beat at four per second. Not the same beat — it is usually described as a rotation or a fluttering inside the head rather than as a loudness that swells — but a periodicity at exactly the frequency difference, which is not in either signal.

That the percept is reported as inside the head rather than as a source going round the room is usually treated as a detail of vocabulary. It is not: the section on the ceiling below shows it is a consequence of the arithmetic, because at the frequencies where the effect is clearest most of the phase differences the beat sweeps through are ones no real source could produce. A percept with no possible external cause is heard as having no external location, which is the ordinary thing for the auditory system to do and is worth noticing as a prediction rather than as an idiom.

It is worth noticing what has to be true for that report to be possible at all. The two ears’ signals do not meet anywhere in the head that is made of air: the skull between them attenuates by tens of decibels at these frequencies, and the head’s own shadow is the mechanism the localisation ladder’s other cue depends on. Whatever combines them combines representations rather than pressures.

So there is arithmetic being done on the two ears’ inputs, and it is being done centrally. That is a much stronger claim than ordinary beating supports, and it is the reason the dichotic version is the interesting one.

What is actually varying #

The quantity that changes is the phase between the ears, and it changes at a rate the difference sets: a whole cycle of relative phase every quarter of a second at four hertz.

Interaural phase is what the ear’s timing mechanism uses to place a sound. So a slowly rotating interaural phase is a slowly rotating direction, and the percept follows it.

That reframes the phenomenon rather than explaining it away. A beat in the air is an amplitude modulation; a beat between the ears is a spatial modulation, and the two share a rate and nothing else.

It is worth adding what the phase rate is, since the essay’s own quantity is a rotation and a rotation has a speed. At four hertz the interaural phase advances a full cycle every 250 milliseconds, which on a 500-hertz carrier is a delay running from one extreme to the other and back four times a second — a rate of change of implied direction of about five thousand degrees a second if it were a source. Nothing physical moves that fast, which is a second reason the percept has no external location and a first reason it is described as a flutter rather than as a sweep.

There is one more difference between the two, and it is the one that decides which mechanism is which. An acoustic beat’s depth depends on the amplitudes: two tones of equal size give a full swing to silence, and a ten-to-one imbalance gives an envelope that varies by under two decibels. A dichotic beat has no depth at all to depend on anything, because there is no envelope. Turning one ear’s tone down changes how far the image swings and not how strongly it beats.

Which limit stops it, and it is the wrong one #

Binaural beats are reported for low tones and are gone by about a kilohertz to fifteen hundred hertz. This site has two candidate explanations for that already, and they are far apart.

The first is phase locking: an auditory nerve fibre fires in step with the waveform up to about five kilohertz, above which the timing information is not there to be compared. The second is phase ambiguity: a head is 17.5 centimetres across, which gives at most 656 microseconds of interaural delay, and a phase difference only names a direction while half a period exceeds that. Half a period equals 656 microseconds at 762 hertz.

The measured upper limit is at 1,000 to 1,500. That is within a factor of two of the head’s own number and nearly a factor of five from the nerve’s.

So the ceiling on a binaural beat is the width of a head, not the speed of a nerve. That is a satisfying answer because it is the same number the localisation ladder started from — 655 microseconds is the whole of the difference — arriving in a place nobody put it.

The 762 hertz is worth deriving once more, though, because there is a more exact statement of it and it changes what the frequency means. The quantity to compare a period against is not the maximum delay but the whole left-to-right range, which is twice it: 1,312 microseconds. As a fraction of a period that range is 2f × 656 microseconds, and it equals exactly one cycle at 762 hertz.

So 762 hertz is the one frequency at which interaural phase and direction are in one-to-one correspondence, and the failures either side of it are of two different kinds. Above it every phase is produced by more than one direction, so a rotating phase is ambiguous, and the ambiguity is complete when the range covers two cycles at 1,525 — which is the top of the measured 1,000-to-1,500 ceiling. The measurement sits between the onset of ambiguity and its completion, which is what a gradual failure looks like and is a better fit than either endpoint alone.

Below it the head cannot produce most of the phases the beat sweeps through, and that is the more interesting half, because it explains the percept rather than the ceiling. At 250 hertz a rotating interaural phase spends two thirds of every cycle at a phase difference no real source could create. There is no external direction to hear it at, and what listeners report at those frequencies is exactly that: a rotation inside the head rather than a source moving round the room.

So the two halves of the phenomenon come from one number read in two directions. Below 762 the beat is unlocatable because the phase is impossible; above it the beat fades because the phase is ambiguous; and at 762 itself the head is, for one frequency, a perfect instrument.

Read as a delay it is a fact about localisation; read as a period it is a ceiling on this beat, and the second reading is what happens when the two tones are slow enough for every phase the rotation asks for to be a phase the head can actually produce.

The same head geometry that produces this beat produces the reason it can only happen low down.

Two cues, and each one fails where the other works. The binaural beat needs the delay cue, so it lives in the band below the ambiguity — and above that band the ear is using level differences instead, which carry no phase and therefore cannot beat. The frequency limit on this phenomenon is not a limit of the effect; it is the edge of the cue it is made of.

The demonstrations on this page are the wrong way round #

A caution about the buttons, because it is unavoidable and worth being explicit about. This site’s synthesiser is monophonic, so nothing here can play a dichotic stimulus: both tones go to both ears, they sum in the air, and what the buttons produce is the ordinary acoustic beat.

That is not useless — it is the contrast. Pressing them gives the sound the figure is about the absence of, and the difference between what is heard and what the figure draws is the whole phenomenon. A reader with headphones and a tone generator can do the real experiment in a minute; a reader with a browser cannot, and a page that claimed otherwise would be lying about its own evidence.

It is also the sharpest instance of a rule this site keeps: the sound corroborates and never carries. If the argument needed the button, the argument would be unavailable here.

Why this is the strongest evidence in the ladder #

It is worth saying plainly what the dichotic experiment buys, because the ordinary beat has been carrying the argument up to now and it cannot.

An acoustic beat is compatible with an ear that does nothing at all. The sum happens in the air; a passive detector with a slow response reports the envelope; nothing needs to be inferred about the listener. That is why the beat rung’s claim — that it is evidence the ear does sums — is weaker than it sounds, and the rung says so.

The dichotic version removes every passive explanation at once. There is no summation in the air, none within a cochlea, none within a critical band, and none available through the ear’s own nonlinearity. Whatever produces the four-per-second percept has to combine two signals that never meet before the auditory nerve, and there is exactly one place in the anatomy where that happens.

So this rung is not a curiosity attached to the localisation ladder. It is the experiment that makes the localisation ladder’s premise checkable — that the two ears’ timing is compared, centrally, at a resolution finer than either ear’s own response.

Where else the two ears are one detector #

Once the ear is doing central arithmetic on two inputs, other things follow, and one of them is a demonstration the site has drawn from another direction.

There is a second one, and it is the reason the dichotic experiment matters beyond a curiosity. A binaural comparator that can hear a four-hertz rotation can also hear a coherence — whether the two ears’ signals are versions of one thing. That is what makes a reverberant room sound spacious rather than merely loud, and it is what a pair of loudspeakers exploits.

There is a third mechanism to rule out. Two tones close enough together to beat are inside one critical band, so an ordinary beat is a within-channel event — one filter carrying two components and rectifying their sum. The beat this essay is about is not: its two components are in different ears and therefore in different channels, and no filter anywhere is carrying both.

There is one further consequence, and it is why the effect has a following outside acoustics. Because the percept is generated centrally, it is available at rates far below anything the ear can otherwise resolve — a one-hertz difference gives a one-hertz rotation, where a one-hertz acoustic beat is at the edge of being a rhythm rather than a beat. The order threshold and the integration window are properties of a channel, and a comparison between two channels is not bound by either.

Which computation produced the numbers #

The interaural delay is Woodworth’s formula for a sphere, which this ladder has used throughout: the delay is (r/c)(θ + sin θ) for a source at azimuth θ, giving 656 microseconds at 90 degrees for a head of radius 8.75 centimetres in air at 343 metres a second.

The rotation rate is the frequency difference by definition: the relative phase advances by Δf cycles every second.

The azimuth curve is that formula inverted numerically — bisection on θ, forty steps, which is exact to more digits than the model deserves — applied to the delay the running phase implies, wrapped into ±half a period. Where the implied delay leaves the range a head can produce, the curve is drawn at the extreme, because that is what the auditory system has to do with it.

The ambiguity frequency is 1/(2·ITDmax), which is 762 hertz for this head and is arithmetic rather than a fit.

Whose heads, and what is quoted #

The head radius is the standard 8.75 centimetres and every number scales with it: a child’s head gives a higher ambiguity frequency and an elephant’s a much lower one, which is a testable prediction nobody appears to have tested on binaural beats.

Three things here are quoted rather than computed. That a dichotic pair produces a beat percept at all; that the percept is a rotation rather than a loudness; and that it disappears somewhere between one and one and a half kilohertz. All three are from the psychoacoustic literature, this site has no listeners, and the argument uses all three.

What is computed is the consequence: if the percept follows interaural phase, then it must fail at 762 hertz for geometric reasons, and the reported failure at 1,000 to 1,500 is close enough to that to be an explanation and too far to be a confirmation.

What the picture cannot show #

The clipping is a caricature. Drawing the image pinned at ±90 degrees above the ambiguity frequency is the simplest thing that could happen; what a real auditory system does with an ambiguous phase is to weigh it against every other cue, and the result is a percept that becomes vague rather than one that snaps.

Only one mechanism is drawn. Interaural level is the other half of localisation and it is untouched by a frequency difference, so a real dichotic pair has one cue rotating and one cue saying “centre” throughout — a conflict the figure does not represent and which is probably why the percept is described as being inside the head rather than out in the world.

The 762 hertz figure is a hard edge on a soft phenomenon. Phase ambiguity does not begin at a frequency; it becomes progressively less resolvable as the accessible range of the cycle shrinks, and the localisation ladder’s own changeover between timing and level cues is a band rather than a line for the same reason.

A monaural distortion product would spoil the experiment and does not arise here. The ear generates frequencies of its own at high levels, and a difference tone at 4 Hz would be an acoustic beat by the back door. It is not available: the two tones never share a cochlea, so there is no nonlinearity for them both to pass through. That is a third thing the dichotic arrangement rules out by construction rather than by argument.

And nothing here is about music. A binaural beat is a laboratory stimulus. The nearest musical relative is the width and stability of a chorused or reverberant sound, which depends on interaural coherence and is a different measurement.

What it says about the ear’s clock #

The site’s two timing limits have been used for different jobs and never put on one axis. It is worth doing here, because this rung is the one place they compete.

5 kHz is where an auditory nerve fibre stops firing in step with the waveform. It is the number behind which harmonics carry the pitch, behind the residue mechanisms, and behind the claim that pitch above a few kilohertz is a different faculty.

762 Hz is where a phase difference stops naming a direction, and nothing but the width of a head sets it.

Both are timing limits and they answer different questions: one is about how finely the nervous system can mark an event, the other about how much delay the geometry can produce. Almost every phenomenon in this collection that has a timing limit has been attributed to the first, because it is the famous one. The dichotic beat is a case where the second is the binding constraint and the numbers say so.

That suggests a check worth making elsewhere. Wherever this site has a timing threshold, the question is this a nerve number or a geometry number is available and has usually not been asked.

Where this ladder goes next #

Two rungs of this ladder have asked how two ears place a sound and how they cope with a room full of copies. This one takes the two inputs apart and finds a periodicity that exists in neither of them, and that the frequency at which it stops is set by the size of the listener rather than by the speed of anything.

The rung after it is the one the coherence question opens. A room sends both ears versions of the same sound that are alike at low frequencies and increasingly unlike at high ones, and the frequency at which they stop resembling each other is computable from the room’s own reverberation — which would put a number on spaciousness using machinery the room ladder already has, and which this collection has never joined up.