Why no piano is in tune
Stack twelve perfect fifths and you land 23.46 cents above seven octaves. No tuner has ever closed that gap, and none ever will.
Start at the A below the bass staff, 55 Hz. Go up a perfect fifth twelve times, multiplying by 3/2 each time, and you arrive at 7136.05 Hz. Start at the same 55 Hz and go up seven octaves, doubling each time, and you arrive at 7040 Hz.
On a keyboard those two journeys end on the same key. In arithmetic they end 96.05 Hz apart.
The fifth is not a convention
It would be convenient if the perfect fifth were an arbitrary choice that we could simply redefine. It is not. Open the Standing Waves tool and leave the string at its default metre. Set the harmonic to 2 and it reads 343.0 Hz. Set it to 3 and it reads 514.5 Hz. The ratio is exactly 1.5.
Those are not two notes chosen to sound nice together. They are the same string vibrating in two of the ways it is physically able to vibrate. A plucked string produces all of them at once, which is why a fifth sounds consonant: the two notes already share most of their harmonics. The 3:2 ratio is what a string does, not what a culture decided.
The octave is the same story with the ratio 2:1, one harmonic against the fundamental.
Why the gap cannot be closed
So we have two intervals that physics hands us, and we would like some whole number of one to equal some whole number of the other. We want (3/2)m = 2n.
Multiply out and that demands 3m = 2n+m. A power of three on the left, a power of two on the right. Every whole number factors into primes exactly one way, so this is only true when both sides are 1. There is no m, no n, no cleverness.
This is worth being precise about, because it is easy to file the problem under "instruments are imperfect". It is not a limitation of strings, or of ears, or of manufacturing. It is a statement about the integers 2 and 3, and it would hold in any universe.
Twelve fifths against seven octaves is simply the near miss that lands closest, and the size of the miss is 129.746338 / 128 = 1.013643. Expressed the way musicians express intervals, that is 23.46 cents, and it has a name: the Pythagorean comma.
Cents are logarithms
A cent is one hundredth of an equal-tempered semitone, and the conversion is 1200 × log₂(ratio). That logarithm is doing real work. Intervals multiply, but we hear them as though they add: two fifths stacked feel like twice one fifth, though the frequency ratio is 2.25 rather than 3. Taking the log turns the multiplying into adding, which is why a piano keyboard can be laid out with evenly spaced keys at all.
The Logarithm Explorer is the same operation with the music removed. Ratios that look wildly different in frequency sit close together once you take the log, which is exactly the compression that lets a 23.46 cent error hide inside a scale spanning seven octaves.
The settlement
Equal temperament spreads the damage evenly. Define the semitone as 21/12 = 1.059463 and twelve of them close the octave perfectly, by construction. Every other interval then inherits a small error.
The fifth comes out at exactly 700 cents against the true 701.955, so it is 1.96 cents flat. That is inaudible to almost everyone.
The major third is the one that hurts. The natural ratio 5/4 is 386.31 cents. Equal temperament gives it 400. Thirteen point seven cents sharp is not subtle, and it is why an equal-tempered major chord has a faint restlessness that a barbershop quartet, free to tune each chord as it comes, does not have.
Earlier tunings made a different bargain. Meantone temperament bought beautiful thirds in the common keys by making some fifths much worse, which worked until a piece needed a key the tuning had sacrificed. What equal temperament actually purchased was not accuracy but portability: every key is equally slightly wrong, so a piece can go anywhere.
When you hear that a piano is in tune, what is meant is that it is out of tune in the agreed way.
References (1)
- what each temperament bought and sold Barbour (1951). Tuning and Temperament: A Historical Survey. Michigan State College Press.