Play a violin note for two seconds and you can name the pitch exactly. Clip it to a
millisecond and it becomes a tick with no pitch at all. That is not your ear failing. The short
sound genuinely does not have one pitch, and the amount it loses is fixed in advance.
“Pure” here means narrow in pitch — close to a single
frequency. Not clean, not beautiful, not free of noise.
One note, twice● the sound in time● the same sound in pitch
spread in time—spread in pitch—the two multiplied—above the floor by—
—
—
1 ms a tick8 ms — default40 ms a held note
110 low A440 — concert A1760 high A
↳ your turn — drag
How long the note lasts to the left and watch the pitch hump balloon by exactly
as much as the note narrowed
What you're seeing
Both pictures are the same sound. The top one plots it against time; the bottom one plots how
much of each pitch it contains.
Shorten the note. The wave squeezes into a narrow burst — and the pitch hump spreads out.
Lengthen it and the hump collapses towards a single line.
Now watch the two multiplied while you drag. It does not move. Every millisecond you
take out of the note is paid for in hertz, at a fixed exchange rate.
The floor is 1⁄4π, about 0.0796, and a note that fades in and out along a bell curve sits
exactly on it. Nothing sits below it.
Change the pitch. The hump slides along the axis and keeps its width. High
notes are not blurrier than low ones. The trade is about the length of the sound, not where it
sits.
Then press Switch it on hard. Cutting the note on and off with a sharp edge does not
just cost you a little accuracy — it costs an unlimited amount. Look at the ripples running
off both sides of the hump: they never stop. In the ideal signal drawn here,
a perfectly sharp edge has no finite width in pitch at all: measure more finely and the number
keeps climbing. A real loudspeaker, microphone and ear all round the edge off, so a real click is
very wide rather than infinitely wide — which is still why it sounds like every pitch at
once, and why audio engineers fade edges rather than cut them.
Dennis Gabor wrote this down for sound in 1946. It is the same inequality as Heisenberg's, and
it applies for the same reason. Heisenberg's is often told as a story about
measurement disturbing a particle. It isn't. It is this: a wavepacket short in one variable is
wide in the other, whether the wave is a violin note or an electron.
The rule, exactly. Write the spread of a pulse in time and in frequency as the
standard deviations of its energy, σt and
σf. Then for every pulse
σt · σf≥ 1 ⁄ 4π ≈ 0.0796
with equality only for a Gaussian. This page measures both spreads from the signal it actually
draws, using its own fast Fourier transform — the numbers are not read off the formula.
Verified in node (improve/verify/114-gabor-limit.js, 17 checks):
The transform is self-tested first, on a pure tone and on Parseval's theorem, before any
physical claim rests on it.
A Gaussian pulse lands on 1⁄4π at four widths spanning 30×, and its
bandwidth is 1⁄(4πσt) each time.
Halving the length exactly doubles the bandwidth across a 32× range, with zero
mismatch to the precision measured.
Six window shapes are tested and none beats the floor; the Gaussian is strictly best, and
a deliberately lopsided pulse is handled correctly, so the moment code assumes no
symmetry.
Negative control. A hard gate has no finite bandwidth at all. Its
σf grows like the square root of the sampling rate —
measured ratios 1.415, 1.415, 1.414, 1.414 against √2 — so it diverges rather
than settling on some larger number.
Negative control. Eleven chirped Gaussians, the classic attempt to cheat
the bound, all fail; the best of them is the unchirped one.
Negative control. Sampling four times more finely leaves the measured
bandwidth unchanged, which refutes the reading that this is a limit on the ruler rather
than on the note.
Negative control. A pulse built as two separated blips lands ten times
above the floor, proving the measure does not simply return the floor for everything.
What this leaves out: real instruments have overtones, and a real ear does
not compute a Fourier transform over the whole note. The trade-off is the same shape, but a
cochlea makes it in narrow bands rather than all at once.