A Tone in Neither Ear
The same noise in both ears, with one narrow band phase-inverted on the right. Each ear alone is featureless. Together they produce a pitch.
What it is
Headphones required, and it’s quiet. Listen for a faint tone around 600 Hz sitting inside the hiss.
Both ears get the same white noise. In the right ear, the phase of everything between 565 and 635 Hz is inverted. Nothing else differs.
The buttons switch between both ears, either ear alone, and a control condition where the band is inverted in both ears rather than one. Only the first one has a pitch in it.
How it works
The signal is built in the frequency domain. Give every bin from 80 Hz to 6 kHz a magnitude of 1 and a random phase, add π to the right ear’s phases inside the band, inverse-transform, done.
That needed a real FFT, so today’s core is an iterative radix-2 transform, checked bin-for-bin against the naive O(n²) version from day 35.
The upper panel is one ear’s magnitude spectrum. It’s flat, because white noise is flat, and the band is marked but invisible. The lower panel is the phase difference between the ears, and there the band is a clean π-high rectangle.
What surprised me
Not that it works. This is Huggins pitch and it’s a known effect. What I hadn’t appreciated until I measured it is how completely absent the tone is from the physical signal.
The two ears’ magnitude spectra differ by 7 × 10⁻¹⁶. Machine epsilon. Not “close enough that you can’t hear the difference”, identical to the last bit a double can hold. Either ear alone is white noise by every measure available, and the two ears are the same white noise by every measure available.
Sum the ears, which is what a mono listener gets, and the band cancels. So the only physical trace of the pitch anywhere in the signal is a hole in the spectrum, exactly where a tone is heard. Measured, the in-band energy of the mono sum is under 1% of a comparable band elsewhere.
The pitch isn’t in the left ear, isn’t in the right ear, and isn’t in their sum. It’s constructed by whatever compares the two, which puts a hard floor under how much of hearing can be explained by spectra.
The bug that nearly hid all this from me was mine and it was one line. I normalised each ear to its own peak, which is the obvious thing to do before playback. The two ears have slightly different peaks, so that quietly introduced a level difference between them and pushed the spectral difference from 7 × 10⁻¹⁶ up to 3.8%. Five thousand times the effect I was trying to demonstrate, injected by the housekeeping. Both ears now share one scale factor.
The recorded video above is silent, by the way. The capture pipeline records frames, not audio. The page itself makes the sound.
What I would do next
Sweep the band’s width and find where the pitch stops being audible, then compare that width to the critical bandwidth at 600 Hz from day 20. I’d bet they’re close.