Twelve perfect fifths overshoot the octave. Every piano pays for it.

What you're seeing

The top picture stacks fifths around a circle where one full turn is one octave. Start anywhere, go up a fifth, again, again — twelve times — and you should have travelled exactly seven turns and be home. With pure 3:2 fifths you overshoot by a sliver: 23.46 cents, about a quarter of the gap between two adjacent piano keys. That sliver is the Pythagorean comma, and it is not a measurement error. Twelve pure fifths multiply out to 531441⁄4096; seven octaves are 128. Those are different numbers, and they always will be — 3 raised to any power is odd, and 2 raised to any power is even, so the two can never meet. Stack more fifths and it still never closes: 41 gets closer, 53 closer still, 665 closer than that, and none of them land.

So the gap can't be removed. It can only be moved. Drag the dial and you tune every fifth a touch flat, which shrinks the overshoot; the bottom chart shows what that costs, key by key. At the far left (Pythagorean) every fifth is perfect and the whole error piles onto one interval — the wolf, so called because it howls — while eight of the major thirds sit 21.5 cents sharp, sour enough that medieval music mostly avoided them. Nudge to 1.955 cents and the circle closes exactly: that's equal temperament, the tuning of essentially every keyboard you have ever touched. No wolf, no unusable key, and nothing pure either — every fifth is imperceptibly flat and every third is 13.7 cents sharp, forever.

Push further, to 5.38 cents, and something lovely happens: eight major thirds become exactly pure — that's quarter-comma meantone, and it is why Renaissance music in a handful of keys sounds so much sweeter than a modern piano. The bill arrives at the wolf, now 35.7 cents sharp and genuinely unplayable, so four keys are simply off-limits. That was the real bargain for two centuries: gorgeous in the keys you used, unusable in the ones you didn't. Watch the two error bars as you drag — they never both sit inside the shaded "nobody can hear this" band. Not at any setting. Equal temperament is exactly the tuning that makes the worst fifth as small as possible; meantone is exactly the one that makes the thirds perfect; and because those two answers are different numbers, the choice is forced.

This exhibit is silent on purpose. Every drawer in this cabinet has to work with the sound off, on a phone, in a library, for someone who can't hear the difference anyway — so the beating is drawn rather than played. The number under the verdict is the honest translation: a tempered A–E fifth wobbles about 0.37 times a second, slow enough to sound like warmth. The Pythagorean wolf wobbles 4.4 times a second, which is the growl the name is about. If you have a piano nearby, that's the experiment.

One quiet detail worth hunting for: at the far left, four of the twelve "major thirds" — the ones whose path crosses the wolf — come out almost perfectly pure, off by under 2 cents, while their eight neighbours are badly sharp. Nothing was tuned differently. It is the same accident of arithmetic, landing somewhere else.