Four tunings · one chord
Meantone buys a pure third with a 5-cent fifth. Equal temperament pays 14 on the third.
Tuning is a budget. The octave is fixed, so every tuning decides which of the fifth and the third to pay for. Below, the same major triad on 220 Hz, billed against pure 5:4 and pure 3:2.
| Tuning | Third | Third charge | Fifth | Fifth charge |
|---|---|---|---|---|
| 12-tone equal | 400.00 | +13.69 | 700.00 | −1.96 |
| Pythagorean | 407.82 | +21.51 | 701.96 | +0.00 |
| Quarter-comma meantone | 386.31 | +0.00 | 696.58 | −5.38 |
| 5-limit just | 386.31 | +0.00 | 701.96 | +0.00 |
Plays root 220 Hz, third and fifth together for four seconds.
What each one pays
12-tone equal. Every semitone 100 cents. The keyboard.
Pythagorean. Built from pure 3:2 fifths only. The third is four fifths up, two octaves down.
Quarter-comma meantone. Fifths narrowed by a quarter of the syntonic comma so that the third comes out pure.
5-limit just. Pure 5:4 and pure 3:2 for this chord. Pays for it elsewhere: no fixed twelve-key keyboard can do this in every key.
Where the bill goes
Twelve pure fifths overshoot seven octaves by 23.46 cents, the Pythagorean comma. Something must absorb it. Equal temperament spreads it over all twelve fifths. Pythagorean tuning and quarter-comma meantone put it all in one fifth, the wolf: eleven good fifths and a twelfth of 678.49 cents in Pythagorean tuning, 737.64 cents in meantone. A pure fifth is 701.96; the meantone wolf is 35.68 cents sharp of that, and you can hear it. Meantone keeps the thirds pure in the keys it was built for by narrowing eleven fifths by 5.38 cents each, and that one is the price.
That is the ledger's claim restated: equal temperament pays a little on every interval so that nothing is ruinous. Glossary: the wolf is the one fifth left over when the circle of twelve fifths is closed, so called for its howl; the syntonic comma (about 21.5 cents) is the gap between four pure fifths and a pure major third two octaves up; 5-limit means every ratio is built from the primes 2, 3 and 5. Sources: Pythagorean tuning, Quarter-comma meantone, Just intonation (Wikipedia, CC BY-SA). Every figure here is computed in the build.