Build a tone out of nothing but the upper harmonics of a note — and delete the note itself, every scrap of energy at its frequency. You still hear it. The pitch you hear corresponds to no frequency anywhere in the sound. It is the missing fundamental, and your ear computes it from a fact of arithmetic: the rate at which the whole waveform repeats, which is the greatest common divisor of the parts that are present.
verify-audio.mjs) confirms the cut section carries no energy at 200 Hz to −102 dB while still repeating at 200 Hz. Every word on screen is verbatim from this page's verifier — 12/12; the Schouten 1940 / de Boer 1956 result is cited, not measured. Source: research/the-pitch-that-isnt-there/film.
Here is a tone with a fundamental at 200 Hz and its harmonics stacked above it. Press play. Now switch off the green key — the fundamental, the 200 Hz itself — and listen to what does not happen to the pitch.
Each key is one harmonic: a pure sine at a whole-number multiple of 200 Hz. The bars below are the spectrum — exactly what is in the air. When the fundamental is off, its bar goes ghostly: the ear keeps reporting a pitch there, but the microphone would find nothing. The pitch you hear is the greatest common divisor of the frequencies that remain — the slowest beat that all of them share, the rate the summed waveform repeats. Switch harmonics on and off and watch that number; it is what your ear is tracking.
A/B test: tap play a real 200 Hz and compare. The reconstructed pitch and the genuine fundamental sit at the same place — yet in the tone above, 200 Hz is nowhere. · Try the even keys only (2,4,6): the GCD jumps to 400 and you hear the octave. Pitch follows the divisor, not a wish for the bottom note.
This is old, and it was a fight. August Seebeck heard it in 1841 from a siren — a disc of evenly spaced holes — whose tone sat an octave below anything the disc was physically making. Seebeck said the pitch lived in the periodicity. Georg Ohm, and then Helmholtz, said the ear is a bank of resonators that hears each pure component separately, so a pitch with no component can't be real — it must be an artifact, a faint distortion product the ear manufactures. The argument ran for a century.
You have heard the missing fundamental ten thousand times without noticing, because the telephone is built on it. A landline passes only 300 – 3400 Hz. A man's voice has its fundamental near 120 Hz, with the second harmonic at 240 — both below the band, both thrown away. What arrives is harmonics 3, 4, 5, 6… and the greatest common divisor of those is still 120. The low pitch of the voice is not transmitted down the wire; it is reconstructed in your head from the harmonics that survived. The same trick lets a laptop or phone speaker — physically unable to move air at 120 Hz — still sound like it has bass.
Switch to telephone: the fundamental (120) and the second harmonic (240) vanish from the air, the lowest surviving partial jumps to 360 Hz — and the pitch your ear assigns does not move. The GCD of {360, 480, 600, …} is still 120.
Then in 1940 J. F. Schouten ended the old dispute with an experiment. If the low pitch were a distortion product the ear cooks up — Helmholtz's combination tone — then masking that exact region with noise should erase it. He masked it. The pitch stayed. The missing fundamental is not a frequency hiding in the signal or fabricated at the eardrum; it is a residue, a pitch the brain reads off the timing of the upper partials directly. Which raises the obvious test: if pitch comes from timing, can you make the timing lie?
Take three neighbouring harmonics — the 9th, 10th, 11th of 200 Hz, i.e. 1800, 2000, 2200 — and shift all three upward by the same amount Δ. Watch one thing: the spacing between them never changes. It is 200 Hz no matter what Δ is.
So a "pitch = the spacing" rule predicts the pitch can't move. The old difference-tone rule predicts the same: f₂ − f₁ = 200, fixed. But the partials themselves have slid, and the brain doesn't read the spacing — it finds the simplest set of harmonics the actual frequencies could be, and reports that fundamental. After a shift of Δ, the best fit is (2000 + Δ)/10, which climbs. Drag Δ and hear it: a pitch rising while the spacing sits perfectly still.
de Boer's rule: the pitch shifts by Δ ÷ n, where n = 10 is the central harmonic — slope 0.1. Slide to the far end (Δ = +100): the central partial lands exactly halfway between two harmonics, the fit becomes two-valued, and the pitch is genuinely ambiguous — a real bistability, not a rounding glitch.
The two pointers part company, and the question of which one is the pitch is settled by the ear, not the arithmetic. The experiments are clear: listeners follow the residue — the moving pointer (Schouten 1940; de Boer 1956, "the pitch shift of the residue," with exactly the slope ≈ 1/n this instrument computes). That is the verdict against the resonance picture and for periodicity: pitch is read from the fine timing of the partials, not from their spacing and not from any single resolved frequency. What this page can prove on its own is only the arithmetic the experiment then judges — that the two models give different numbers, and by how much. That an ear chooses the residue is reported by the people who measured it, and named as theirs.
Every number above is recomputed by a small deterministic verifier on the exact signals this page synthesises — no listening panel, just arithmetic and one autocorrelation. It is committed beside the page.
Honesty note: this page computes the GCD, the waveform period, and the two rival predictions in movement III. The claim that a human ear, hearing movement III, follows the residue rather than the spacing is an experimental result (Seebeck 1841; Schouten 1940; de Boer 1956) — cited, not measured here. Your own ear, with movement III in front of it, is the one experiment this page can't run for you.