Physical · your machine is the apparatus

Ten Millionths of a Second, Between Your Ears

A sound arriving from your left reaches your left ear first. The head is small and sound is fast, so the head start is tiny, and yet you can point at the thing. How tiny a head start is still enough is a measured quantity, it was measured in 1956 and again in 1958, and the answer, for trained listeners in a quiet laboratory, is about ten microseconds: shorter than the gap between two consecutive samples of a CD. This page builds that delay inside your browser, proves it survived your own machine's audio path before it asks you anything, measures your own threshold with a staircase built to refuse rather than to flatter, and then walks the test up in frequency until your hearing gives out. Where it gives out is also published, and it is not where you would guess.

Headphones, and this is not negotiable. Over speakers each ear hears both channels, the delay you are supposed to be judging arrives mixed with its own opposite, and the measurement becomes a measurement of your room. Wired is safer than wireless: some Bluetooth codecs give the two ears independent latency, which this page has no way to see. Section 2 runs a check that catches a swapped or a dead channel, and it will not let the test start until it passes. It cannot catch speakers. A tone in one channel comes out of one speaker, you will name the right side, and the check will pass. Nothing this page can do inside a browser tab, short of asking for your microphone, tells the difference, and section 8 is where that limit is written down rather than hinted at.

1Before anything is asked of your ears

The delay this page is about is smaller than the spacing of the numbers your machine plays. At the rate most sound hardware runs, consecutive samples are a little under twenty one microseconds apart, and the delay in question is around ten. So the obvious question comes first, and it comes before you are asked to judge anything: can this machine even make a delay that small, or is the whole exercise theatre?

It can, and the reason is that a delay does not have to be built by moving samples. For a tone, a delay of t is exactly a phase shift of 2πft, and phase is a number you can put into the formula before you evaluate it. The two ears get the same gate, the same amplitude, the same everything, and their carriers are computed from clocks offset by a fraction of a sample. Nothing is resampled and nothing is rounded.

Instrument 1: what your audio path did with ten microseconds

This makes no sound. It renders the stimulus through your browser's own audio engine, reads the samples back, and recovers the delay from them with two independent estimators, one that knows the carrier frequency and one that does not. It also renders the two ways this page decided not to build the delay, so you can see what they do instead.

not yet run

Here are the same two channels drawn. Move the slider and watch the pair separate. The dots are the actual samples: at ten microseconds not one of them has moved sideways, because there is nowhere sideways to move to. What changed is every sample's value, and that is the same thing.

The two ears, drawn

delay shown

A 700 Hz tone, one cycle and a bit, left in blue and right in pink. At the far left of the slider the two curves are the same curve.

2The channel check, which is allowed to stop the page

A tone will play in one channel only. Say which ear it arrived at. Three in a row, and if any is wrong the listening test does not open, because the likely explanations are that your channels are crossed, that one side is not working, that whatever you are on is mixing them, or that your two ears do not hear a 700 Hz tone equally well, and none of those can be measured through.

That last one is not a fault of yours and the page cannot tell it apart from the other three. This instrument needs two ears that hear this tone at about the same loudness, and asymmetric hearing, a single working ear, a blocked one or a passing cold all mean it has nothing to measure. If the gate closes on you, what has happened is that a test built for a particular pair of ears has met a different pair. There is no score here and no way to fail one.

Passing it does not prove you are wearing headphones, and the page does not pretend otherwise. This test asks which side a sound was on, and a pair of desk speakers answers that question correctly. What it can rule out is a crossed pair, a dead earcup and a channel collapsed to mono. The rest is on you, and the cost of getting it wrong is that your number in section 3 is a measurement of your room rather than of your hearing.

Instrument 2: channel identity

loudness

Comfortable is right. Thresholds do depend on level, which Zwislocki and Feldman measured directly in 1956, so a level you would not choose to listen at is not a level worth measuring at.

not yet run

3Your own number

Now the real task. A 700 Hz tone plays in both ears at once, with one ear a hair ahead. It will not sound like an echo. It will sound like one sound sitting slightly to one side of the middle of your head. Say which side. Get two right and the delay halves; get one wrong and it grows. That rule is called two-down one-up and it walks itself to the delay you get right about seventy per cent of the time.

Roughly one trial in six is a deliberately obvious one, far larger than any threshold. Those do not move the staircase. They exist so that the reduction can tell a listener from a coin, and if you miss too many of them this page will not print a number for you.

Instrument 3: the staircase, at 700 Hz

Locked until the channel check in section 2 passes. Left arrow and right arrow work as well as the buttons. About forty trials, two or three minutes. Guessing is fine and expected near the end; that is what a threshold is.

not yet run

every trial, as numbers

Nothing is sent anywhere by this page. If you want it in the repo, it has to travel by hand, and that is the honest state of the human half of this instrument.

4How many degrees is that?

A delay on its own is hard to feel the size of. Turn it into an angle. Woodworth's model of a rigid spherical head, tested and corrected by Aaronson and Hartmann in 2014, says the delay between the ears for a source at azimuth θ is (r/c)(θ + sin θ). Near straight ahead that is 2r/c per radian, which is the steepest the function ever is: straight ahead is where a given turn buys the most delay, which is why straight ahead is where you are most precise.

Instrument 4: your head, in microseconds

head circumference, cm

air temperature, C

A tape measure round the widest part. The model wants the radius of a sphere with that circumference, which is a real simplification, and the apparatus says what it costs.

That formula is the high-frequency limit, and 700 Hz is not high. Aaronson and Hartmann's abstract says so of the model in their own title: it is a frequency-independent, ray-tracing model of a rigid spherical head that is expected to agree with the high-frequency limit of an exact diffraction model, and their paper exists to quantify the discrepancy when the frequency is not high. Down where this page measures, the wave bends around the head rather than cutting across it and the same head makes about half again as much delay for the same angle, so the degrees printed above are an over-estimate by something in that neighbourhood. Which way the error runs is worth knowing; its exact size is in a paper behind a 403, and the apparatus says so rather than guessing.

Mills reached the same place from the opposite direction in 1958, by putting people in an anechoic chamber and moving a real loudspeaker until they noticed:

5Where it stops working

In 1907 Rayleigh set out why a head has two cues rather than one. At low frequencies the wavelength is long compared with the head, so the head casts no shadow worth having and the only difference between the ears is timing. At high frequencies the head shadows, so there is a level difference to use, and the timing cue develops a problem: once half a wavelength is shorter than the path difference, a phase lead and a phase lag stop being distinguishable. So a phase-based time cue has an arithmetic ceiling, and it is worth working out where that ceiling is before reading anybody's measurement of where the cue actually failed.

Work it out, and the head gives the wrong answer by a factor of two. This is the part the obvious story gets wrong. Section 4's model, at the default 57 cm head and 20 degrees, says the largest delay a head that size can make is . Half a period is that long at , and that is Rayleigh's ambiguity frequency for this head. The low-frequency form of the same physics, where the wave bends round the head instead of cutting across it, puts it , which is lower still. Both sit below of the 1400 Hz the measurements below are about. To put the ambiguity at 1400 Hz you would need a head about half the width of yours. Head size does not set this ceiling. What it does line up with is the other end of the story: Zwislocki and Feldman put the frequency of best time sensitivity near 800 cps and Brughera and colleagues put it between 700 and 1000, which is where the head arithmetic lands. The head predicts where this works best. It does not predict where it stops, and a page that told you it did would be telling you something none of its sources say.

Three published measurements put a number on where it does fail, spread over fifty five years, and they do not all agree. Mills, from a loudspeaker in an anechoic chamber, put the crossover below about 1400 cps. Brughera, Dunai and Hartmann, from headphones and four listeners, put it here:

And Zwislocki and Feldman, from headphones in the same volume of the same journal as Klumpp and Eady, put it a hundred cycles lower, and in the same breath say where the cue is at its best:

Two of those three are not independent of each other. Mills did not read a crossover off his own free-field data alone. His abstract says the frequency comes from a comparison of these thresholds with those reported for dichotic stimulation, which is to say from comparing his loudspeaker thresholds against headphone measurements of the era, and Zwislocki and Feldman's is one of the headphone measurements of the era. So the agreement between two studies fifty five years apart is worth something, and it is worth less than two independent methods would be. The honest summary is that the limit is somewhere between about 1300 and 1400 cycles, that three groups using two kinds of apparatus over fifty five years all landed in that hundred-cycle band, and that this page draws its dashed line at Brughera's 1400 because that is the modern redo of exactly the headphone experiment.

So here is a prediction, fixed before you run anything, and it is not a monotone one. Your threshold in microseconds should be worse at 500 Hz than at 700, because in that range what stays constant is the threshold in phase rather than in time. It should be smallest somewhere between 700 and 1000. Above 1000 it should climb, faster than exponentially, and somewhere near 1400 the staircase should stop converging at all.

Part of that last sentence is arithmetic, not hearing, and here is the size of it. No delay past 0.45 of a carrier period is ever presented, because beyond half a period a lead and a lag are the same stimulus. That ceiling falls as the frequency rises, and the largest threshold this page will report falls with it: So a listener with a perfectly flat threshold, identical at every frequency and with no high-frequency limit at all, is refused more often at the top of the sweep than at the bottom, purely because of the rule. The check panel measures that on simulated flat listeners at three different thresholds and prints all three sets of rates, because how much of it bites depends entirely on where your own threshold sits: near that falling ceiling it dominates, and a listener at 15 us is untouched by it. That is the only honest way to read your own curve: part of the shape is the rule and not the ear, and the part that is yours is whatever sits on top of it.

Instrument 5: the same test, walked up in frequency

Each is its own staircase and takes about as long as section 3. You do not have to run them all, but the shape needs at least three. A refusal at the top is not a failed run; read it against the note above, which says how much of it the rules can produce on their own.

nothing run yet

6The same delay, moved somewhere else

Whatever ends the cue above 1400 Hz, what ended was not your ability to use a delay. It was your ability to use the delay of a carrier. Anything else about the sound that moves slowly should still carry a delay perfectly well, and unlike the mechanism, that part is testable on you in two minutes.

This page does not know why the carrier cue stops, and it will not borrow an explanation it cannot support. The tempting one is that the auditory nerve stops firing in step with the waveform around there. Two things on this page argue against reaching for it. The first is what just happened to the other tidy explanation: section 5 put the head through one line of arithmetic and it came out wrong by a factor of two, and a page that has watched one obvious mechanism fail a test it could run should be slow about adopting the next one it cannot. The second is that the one statement of mechanism anybody working on this page has actually read does not say nerve. Brughera and colleagues' own deposited abstract says where their model starts:

The medial superior olive is in the brainstem, two synapses past the auditory nerve. That is a claim about their model rather than a settled fact about you, they are one group, and this page has read one paragraph of them. What is measured here is that the cue stops near 1400 Hz. Why it stops is not measured here, and this page is not the place to find out.

Henning tested exactly that in 1974, with a 3900 Hz tone whose loudness was wobbling 300 times a second. His finding, from the abstract, is that delaying the whole stimulus and delaying only its wobble are equally detectable, and both are as good as a 300 Hz pure tone. The step from that to "at 3900 Hz the carrier's delay contributes nothing" is an inference, not a measurement of his: if delaying everything is no better than delaying only the wobble, then the carrier's delay added nothing on top of it. This page puts that inference to you directly, which he did not.

So: same tone, same delay, twice. Once the delay is in the carrier and the wobble is identical in both ears. Once the wobble is delayed and the carrier is identical in both ears. Twenty four trials, the two conditions shuffled together so you cannot tell which is which.

Instrument 6: carrier against envelope, at 3900 Hz

locked until the channel check in section 2 passes

Both conditions carry the identical delay, verified in the check panel below: in one the carrier lag is the delay and the envelope lag is zero, in the other it is the other way round, and the estimators are told nothing about which is which.

7Now let the machine take it

Your number is a number about you, and it is worth nothing unless the path underneath it can carry something much smaller. There is one honest way to find that out: give the same task to a machine, over the same rendered audio, through the same staircase and the same reduction, and see where the machine stops.

It runs four ways and three of them are supposed to fail. One uses the page's own synthesis. One rounds the delay to whole samples, the way you get if you forget that a sub-sample delay has to be built rather than requested. One throws the delay away and renders zero. One renders unrelated noise into each ear. A reduction that returns a confident number from the last two is not an instrument.

Instrument 7: the machine listener, on this machine

Every trial is rendered through a real offline audio context in this browser before the machine is allowed to hear it, and it hears through sixteen-bit quantisation, because that is what the output path does. A few seconds.

not yet run

Here is the same battery as it came out in three different browser engines on the machine this page was built on, replayed through the identical reduction that just ran on yours. Two of them run at 48 kHz and one at 44.1 kHz, so the whole-sample control lands in a different place in each, which is exactly the tell: a real threshold does not care what your sample rate is, and that one is nothing but your sample rate.

8What this page will not tell you

A page that always produces a figure is a page that will eventually produce a wrong one. Every place this instrument declined to answer during your visit is listed here, with the reason.

The refusals, this session

The standing ones, which no reader can talk it out of, are listed in the apparatus and are computed from the same constants the staircase runs on. The one worth naming here is the last: this page cannot see your headphones. It can prove that the numbers it handed to your audio system carried the delay, and it does. It cannot prove that what left your earcups did. Nothing running in a browser tab can.

What the check panel checks

Everything above is recomputed below, in your browser, from the shipped module and the committed captures. Lines marked OFFLINE are results your browser cannot redo in the time you would wait. The rest it just did.

Show the check

Running...

Apparatus

What is being run

A two-alternative lateralisation task under a two-down one-up adaptive rule. The stimulus is a gated sinusoid whose gate is computed once and applied identically to both ears, so the envelope carries exactly zero interaural delay and the only cue is the carrier's phase. The delay is put in by evaluating the carrier at two clocks offset by half the delay each way, which is exact for any delay, including delays far shorter than the sample spacing. Section 6 inverts that: the carrier is identical in both ears and the modulation is what gets offset.

All of it lives in one file, public/strata/ten-millionths-of-a-second/itd.mjs, which holds no DOM, no Web Audio, no clock and no unseeded randomness. That is how the same code synthesises in your browser, in node during the lab captures, inside both verifiers, and over audio recorded from engines none of us is running. There is exactly one copy. The verifier asserts there is exactly one copy.

The page publishes the trial you are being asked, and here is why. An instrument that measures a person does nothing until somebody answers a trial, so a checker that only loads the page and reads the result can never touch the part that matters. The browser verifier therefore takes this test three times, as a simulated listener with a known threshold, as a listener answering by coin flip, and as a listener who is right every single time, and asserts that the page returns a number for the first and refuses the other two. To answer a trial it has to be able to see which side that trial actually was, so the page exposes the waiting trial at window.__aw. It changes nothing for a reader, and a reader who opens it is only cheating themselves out of a measurement.

Sources

Every field below (title, authors, journal, volume, issue, pages, year) was read from the publisher's own Crossref deposit, and those deposits are committed verbatim in research/ten-millionths-of-a-second/data/sources/. Quotations are from the abstract as deposited. The publisher's article pages for every one of these papers return HTTP 403 to us, so nobody working on this page has read any of these primary articles. Six are at pubs.aip.org and one at tandfonline.com, and all of them refuse, with a browser user agent and a search engine referer. What we have instead is the deposit, the abstract inside it, and the DOI resolver's own redirect, whose URL independently carries the journal, volume, issue, first page and title. The statuses were measured, not assumed, and they are committed in data/sources/access.json:

Those seven are not equally well supported, and singling out the 1956 pair would be the wrong ranking. The load-bearing one of the single-sourced group is Mills, not either 1956 paper: he supplies both the about-ten-microseconds figure the whole page is named after and the below about 1400 cps that section 5 is built on. That is the weakest link in the citation chain here.

Lord Rayleigh, XII. On our perception of sound direction. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, volume 13, issue 74, 1907, pages 214-232. doi:10.1080/14786440709463595
R. G. Klumpp, H. R. Eady, Some Measurements of Interaural Time Difference Thresholds. The Journal of the Acoustical Society of America, volume 28, issue 5, 1956, pages 859-860. doi:10.1121/1.1908493
J. Zwislocki, R. S. Feldman, Just Noticeable Differences in Dichotic Phase. The Journal of the Acoustical Society of America, volume 28, issue 5, 1956, pages 860-864. doi:10.1121/1.1908495
A. W. Mills, On the Minimum Audible Angle. The Journal of the Acoustical Society of America, volume 30, issue 4, 1958, pages 237-246. doi:10.1121/1.1909553
G. Bruce Henning, Detectability of interaural delay in high-frequency complex waveforms. The Journal of the Acoustical Society of America, volume 55, issue 1, 1974, pages 84-90. doi:10.1121/1.1928135
Andrew Brughera, Larisa Dunai, William M. Hartmann, Human interaural time difference thresholds for sine tones: The high-frequency limit. The Journal of the Acoustical Society of America, volume 133, issue 5, 2013, pages 2839-2855. doi:10.1121/1.4795778
Neil L. Aaronson, William M. Hartmann, Testing, correcting, and extending the Woodworth model for interaural time difference. The Journal of the Acoustical Society of America, volume 135, issue 2, 2014, pages 817-823. doi:10.1121/1.4861243

The rule fixed before measuring

Two rules were changed after seeing data, and this is the disclosure. Both are in the section below, both made the reduction stricter rather than more permissive, and both were changed because a negative control got through, and both are described under An error this page made and kept. The measured cost of the second one is that runs whose true threshold is near the ceiling are now refused far more often, and that number is in the check panel.

What this page does not claim

It does not claim your threshold is comparable to a published one. The published figures come from trained listeners at a hand-picked level in a quiet room with calibrated earphones. Yours comes from whatever you have on, wherever you are, first time. The criteria do not match either: a two-down one-up staircase settles where you are right about 70.7 per cent of the time, and Klumpp and Eady's figures are at 75 per cent, so even a careful comparison is not like for like. Treat the published number as a floor that has been reached by people trained to reach it, not a target you have failed to hit.
It does not claim to measure what your headphones delivered. It measures what the browser rendered. Everything after the render is invisible to it, and if your headphones are wireless the two ears may not even be receiving in step.
It does not claim to reproduce anybody's protocol. Klumpp and Eady used noise, a tone and a click; Brughera and colleagues used their own levels, durations and listeners. This page fixed its own parameters and they are listed above.
It does not claim that a refusal at 1400 Hz proves the published limit. One listener, one staircase, in a browser, is weak evidence about a boundary that took four trained listeners and a modelling paper to place. What it can honestly show you is the direction and the steepness, on you.
It does not claim to know why the cue stops near 1400 Hz. It rules one candidate out with arithmetic anybody can redo, and the check panel redoes it: the head is the wrong size by about a factor of two, so Rayleigh's ambiguity is not what ends this. Beyond that the page has one deposited sentence about mechanism, from one group, and it says medial superior olive rather than auditory nerve. Naming a cause on that much evidence would be borrowing an authority nobody here has.
It does not claim the angle in section 4 is exact at 700 Hz. Woodworth's formula is the high-frequency limit of an exact diffraction calculation, which is what the paper it comes from says about it, and at these frequencies the real head makes more delay per degree than the formula does, so the printed angle runs high.
It does not claim the sweep's shape is all yours. The largest threshold this page will report falls with frequency, because the largest delay it may present falls with the carrier period, so a listener with no high-frequency limit at all is still refused more often at 1400 Hz than at 500. The size of that is measured on a flat simulated listener and printed in the check panel, and it is named in section 5 rather than left in the code.
It does not claim the channel check in section 2 proves you are on headphones. It proves the channels are not crossed, not dead, and not collapsed to one. A pair of desk speakers passes it, and there is no test in a browser tab that would not.
It does not claim the head model is your head. A head is not a sphere and the ears are not at its equator, which is the error Aaronson and Hartmann quantify.
It does not have a single human staircase behind it. The signal half of this instrument is checked against real audio captured out of three different browser audio engines at two different sample rates. The listener half is checked only against simulated observers whose thresholds are true by construction, because nobody with ears ran this before it shipped. The staircase you just took may be the first. That asymmetry is the honest shape of the checking here, and it is why section 3 offers you your own trace as a file.
It does not claim anything about hearing that is not this task. Asymmetric hearing, tinnitus, a blocked ear or a cold all change this, and a reader who cannot do the task has not failed a test. There is nothing here to be good at.

Predictions, and how they did

P1 held. Predicted that a delay put in as a phase offset would survive a real browser audio path unchanged. It does, and better than expected: the windows that came back out of chromium, firefox and webkit are bit for bit what node put in, at both sample rates.
P2 held, and it was the one worth being nervous about. Predicted that the whole-sample control would stall the machine listener near half a sample. It does, and half a sample sits on top of the published human floor, which is the coincidence the whole apparatus is built around. A page that got this wrong would have agreed with the literature.
P3 died, and the design survived it. Predicted that Web Audio's own DelayNode would round a ten microsecond delay away to nothing, which was the stated reason for not using it. It does not: all three engines interpolate, and the delay comes back close to right. What is actually wrong with that route is a different thing entirely: it does not deliver a pure delay, it delivers a delay plus an interaural level difference that grows with frequency, and a level difference is a localisation cue in its own right. Its size, measured in all three engines on the committed captures, is in a 4 kHz stimulus; at the 3900 Hz carrier of section 6 the same predictor, which those captures pin to within two thousandths of a decibel, says . Note what that is not. Mills's level threshold is half a decibel, so this artefact sits below it everywhere this page measures: it is a contaminant that has not been shown to be audible, not a demonstrated confound, and saying otherwise would be reading the numbers harder than they read. It keeps growing with frequency and crosses that threshold above 5 kHz. The decision not to use the route was right and the reason originally given for it was wrong.
P4 died. Predicted that a buffer built at one sample rate and played into a context running at another would lose a sub-sample delay in the resampler. It survives, to within a fiftieth of a microsecond, in every engine tested.
P5 died, and it is the reason the machine listener has reference trials. Predicted that a machine does not need catching out, because a machine does not lose attention. The null control then walked through the reduction and printed a threshold from a stimulus with no delay in it. Reference trials are not about attention; they are about demonstrating you can do the easy version of the task.
P6 held, with the number printed rather than claimed. Predicted that the finished battery of refusal rules would keep a coin-flipping listener from getting a number. It mostly does. The rate is measured, it is not zero, and it is in the check panel above.

An error this page made and kept

The first one is the one the whole wave is about. The machine listener's rules originally set the reference-trial interval to zero, on the reasoning that a machine does not lose attention and so does not need catching out. The null control, which throws the requested delay away and renders silence into the difference between the ears, then walked through the reduction and came out the other side with a threshold of a number, from a stimulus containing no interaural delay of any kind. It passed because a staircase driven by coin flips reverses often, hit its reversal quota in twenty five trials, and cleared the binomial test by luck.

The trace that did it is committed at research/ten-millionths-of-a-second/data/incident-null-passed.json, and the check panel replays it twice on every page load: once under the rule that let it through, where it must still produce that number, and once under the shipped rule, where it must be refused. A fix with no failing case behind it is a story about a fix.

The second is smaller and is in the estimator. The general cross-correlation estimator searched two samples wider than its own stated range, so that the parabola at the edge would have a neighbour to lean on. On the 3900 Hz stimulus in section 6, at 44 100 Hz, that let the peak one whole carrier cycle away into the search, and a pair built with a delay of one value came back as another, about one period low, with a clean fit. The search range is now exactly the stated range and the correlation is merely computed one sample wider.

Both of those numbers are estimator output, recomputed on this page load, not remembered. The retired version of the estimator is kept in the shipped module beside the current one, marked as retired and used by nothing that measures anything, and the check panel runs the two of them side by side on the same freshly synthesised pair and prints both answers. That also pins down something the first telling of this story left out: the alias only bites at 44.1 kHz. The two extra samples reach 158.7 microseconds at 44.1 kHz and only 145.8 at 48 kHz, and the false peak sits one carrier period away at 256.4, so at 48 kHz the retired estimator gets the right answer and looks fine. Two of the three engines here run at 48 kHz. It took the third one to see this at all, and the check panel asserts both halves of that.

Reproducing

node research/ten-millionths-of-a-second/01-fetch-sources.mjs
node research/ten-millionths-of-a-second/02-capture-engines.mjs
node research/ten-millionths-of-a-second/03-machine-staircases.mjs
node research/ten-millionths-of-a-second/04-simulated-observers.mjs
node research/ten-millionths-of-a-second/05-build-page-data.mjs
node research/ten-millionths-of-a-second/verify.mjs
node verify-ten-millionths-of-a-second.mjs

The captures in research/ten-millionths-of-a-second/data/engines/ are the raw float samples that came back out of chromium, firefox and webkit on the build machine, committed as base64 of the bytes, with the delay that was asked for stored beside them. The staircases in data/machine/ were run inside those same browsers. The simulated observers in data/simulated/ are not people and the file says so in its first line.