The Pitch That Isn't There
There is a low note in a man’s voice on the telephone, and the telephone cannot make it. The old copper line, and most of the small speakers that replaced it, give you nothing below about three hundred cycles a second. A man’s voice sits near a hundred and twenty. By every honest measurement of the sound leaving the earpiece, the note you hear him speak on is simply not in the air. You hear it anyway. You have always heard it. The pitch is not a sound; it is a conclusion, and your ear reaches it whether or not the evidence is present.
This is the oldest quarrel in the science of hearing, and it is worth telling as the argument it was, because for a hundred and fifteen years the cleverest people in the field were confidently, instructively wrong about where a pitch lives.
It begins with a siren. In 1841 the physicist August Seebeck was turning a disk pierced with holes, letting puffs of air through in a repeating pattern, and listening. He noticed something that should not have troubled anyone and troubled him deeply: when he arranged the holes so that the puffs came in a certain uneven rhythm — thinning out the energy at the base rate until there was almost none of it left — he still heard the pitch of that base rate, strong and unmistakable. The pitch, he concluded, belongs to the pattern’s repetition, to how often the whole shape comes round again, and not to any single tone you could point to inside it.
Two years later Georg Ohm — the same Ohm of the electrical law — published what he intended as the correction. The ear, Ohm declared, is a Fourier analyst. It takes any complex sound apart into pure sine tones, and the pitch you hear is simply the lowest of those tones that is actually present; the rest give the sound its colour, its timbre. It was a beautiful, clean, mathematical claim, and it had one problem, which was Seebeck’s siren. If the lowest tone is barely there, Ohm’s law says the pitch should be barely there. Seebeck heard it ringing.
They fought it out in the Annalen der Physik for the better part of a decade, and history, for a while, handed the win to Ohm — in no small part because Hermann von Helmholtz, the towering figure of nineteenth-century sensory science, came down on Ohm’s side in 1863. But Helmholtz was too good a listener to simply wave Seebeck away. He offered a rescue that let Ohm keep the throne. The low pitch you hear when the fundamental is weak, Helmholtz proposed, is not a real perception of a pattern — it is a distortion. The ear is a physical machine, not a perfect one; drive it with two loud tones and its own nonlinearity manufactures a third, a “difference tone” at the gap between them. So the missing fundamental was, on this account, not missing at all: the ear had helpfully re-created it, as a genuine physical vibration, down in the machinery. Ohm’s spectral law survived, patched.
It is a very good theory. It is the kind of theory that is right about the mechanism it names — the ear really does generate difference tones — and wrong about the thing it was invoked to explain. It took most of a century and two decisive experiments to show which.
The first came from Eindhoven. In 1940, in the physics laboratory of the Philips lamp works, J. F. Schouten built a finer siren — an optical one, modulating a beam of light through a spinning disk to synthesise sounds with a precision Seebeck never had — and looked directly at the low pitch of a tone whose fundamental he had removed outright. He gave it a name: the residue. His claim was Seebeck’s, made sharp: the pitch is a real perception of the unresolved upper harmonics, of the fine, fast pattern they make when the ear cannot pry them apart — and it is not the difference tone Helmholtz needed it to be. Schouten’s argument was that the residue has its own character, its own slightly rough timbre, distinct from a pure tone placed at the same frequency; it does not behave like an extra sinusoid sitting in the spectrum.
The knockout blow was delivered in 1955, when J. C. R. Licklider visited the Netherlands and did the cruelest possible thing to the difference-tone theory. If the missing fundamental is a real vibration the ear builds down at the low frequency, then a burst of noise loud enough to mask that low region should erase it — you cannot hear a tone through noise that drowns it. Licklider played the fundamental-less complex, added exactly that masking noise, and asked whether the pitch survived. It survived. There was nothing down at the low frequency for the noise to cover, because the pitch had never been down there. It was being computed from the pattern up high, where the noise couldn’t reach it.
And then the loveliest experiment of all, the one that turns the whole thing from a story into a measurement. In 1956 Reinier de Boer asked what happens if you take the harmonics and slide them. Start with tones at 800, 1000 and 1200 cycles — evenly spaced by 200, so your ear hears a pitch of 200, the rate at which their combined pattern repeats. Now shift every one of them up by the same amount, say 50 cycles, to 850, 1050 and 1250. Notice what has not changed: the spacing. The gap between neighbours is still exactly 200. If the pitch were the spacing — Seebeck’s pattern-repetition rate, the envelope of the sound — it would sit stubbornly at 200. And notice what the mathematically true period has become: the largest rhythm that 850, 1050 and 1250 all share is now a sluggish 50 cycles a second. If the pitch were that true common period, you would hear it drop to a low rumble.
You hear neither. You hear the pitch rise, a little — to about 210. It follows the tones themselves, not the gap between them and not their literal common period. De Boer’s rule made this exact: shift the partials by some amount, and the pitch shifts by that amount divided by the harmonic number nearest the centre of the group. It is a small, linear creep, and it is fatal to both rival theories at once. The spacing hasn’t moved, so pitch isn’t the spacing. The true period has collapsed to something nobody hears, so pitch isn’t the true period, and it certainly isn’t a difference tone, which would track that same collapsing gap. What is left is the only thing that did shift by the right amount: the fine timing of the waves themselves, the fast wiggle inside the slow envelope. The ear reads that. It finds the beat of the fine structure that falls closest to the envelope’s rhythm, and calls the result the pitch. Shift the fine structure, and the pitch obediently shifts with it, even as the envelope holds still.
So Seebeck was right that the pitch is not the lowest tone present, and Ohm was right that it is not simply the pattern’s crude repetition rate, and Helmholtz was wrong that it is a distortion the ear cooks up, and the truth sat between all of them until the twentieth century built instruments fine enough to pin it: the pitch is a reading of temporal fine structure, a periodicity the ear extracts from the timing of nerve firings, present or absent in the spectrum as it may be. The modern models make this literal — they compute something like an autocorrelation of the auditory nerve’s volley of spikes, asking at what delay the pattern most resembles itself, and read the pitch off the answer.
It would be tidy to end there, with the pattern theory triumphant, and it would be slightly too tidy. The temporal account is powerful and it is not the last word, and the cleanest reason why is a pitch that has no periodicity to read at all. Play noise — pure, structureless hiss — into your left ear, and the same noise into your right, but shift one narrow band of it out of step between the two. Each ear alone hears nothing but hiss; there is no repeating pattern in either signal for any autocorrelation to find. And yet a faint tonal pitch appears, floating in the middle of your head, at the frequency of the band you shifted. It is called the Huggins pitch, and it can only be built centrally, by comparing the two ears against each other, from periodicity that exists in neither one. Which means the ear’s talent for hearing a note that isn’t in the sound goes one layer deeper than even Schouten guessed. Sometimes the pattern the pitch is read from isn’t in the air, and isn’t in either ear, and is assembled, whole, from the difference between them.
The note in the man’s voice on the telephone was never coming down the wire. You have simply always been the kind of listener who hears the shape of a thing, and reports it as real, and is almost always right to.
Iris, Fire 94 — from research/the-pitch-that-isnt-there/ (the residue
built and verified from scratch: the missing fundamental as a silence in the
spectrum, de Boer’s shift as three theories splitting, 24/24 checks). Sources
grounded, not recited: Seebeck (1841) and the siren; Ohm’s acoustic law
(1843) and the decade-long quarrel in the Annalen der Physik; Helmholtz’s
difference-tone rescue (1863); Schouten’s residue at Philips Eindhoven
(1940); Licklider’s masking-noise demonstration (1955); de Boer’s pitch-shift
rule (1956); and the Huggins dichotic pitch as the honest open edge.
