Tool
Temperament comparator
The same note in different tunings, in cents and in sound: what equal temperament costs and what it buys.
A pure major third, at the ratio 5:4, and the piano's major third, at a hundred cents per semitone, are not the same thing: the piano's is fourteen cents wider, and the difference is heard as a beat. Choose two temperaments and a sound, and hear one after the other from the same C. The table shows where each note falls in each tuning, and how far it sits from equal temperament.
Equal temperament. Twelve equal semitones of a hundred cents. No fifth is pure, no third is pure, and everything works in every key. It is the modern piano's.
Just intonation. Every note in a simple ratio with the tonic. The major third is pure, 5:4, and the tonic chord sounds without beating; other degrees pay for it.
| Semitones | Equal temperament | Just intonation |
|---|---|---|
| 0 | C 0.0 (0) | C 0.0 (0) |
| 1 | C♯/D♭ 100.0 (0) | D♭ 111.7 (+11.7) |
| 2 | D 200.0 (0) | D 203.9 (+3.9) |
| 3 | D♯/E♭ 300.0 (0) | E♭ 315.6 (+15.6) |
| 4 | E 400.0 (0) | E 386.3 (−13.7) |
| 5 | F 500.0 (0) | F 498.0 (−2.0) |
| 6 | F♯/G♭ 600.0 (0) | F♯ 590.2 (−9.8) |
| 7 | G 700.0 (0) | G 702.0 (+2.0) |
| 8 | G♯/A♭ 800.0 (0) | A♭ 813.7 (+13.7) |
| 9 | A 900.0 (0) | A 884.4 (−15.6) |
| 10 | A♯/B♭ 1000.0 (0) | B♭ 996.1 (−3.9) |
| 11 | B 1100.0 (0) | B 1088.3 (−11.7) |
Temperament
A temperament is a way of tuning the twelve notes of the octave. Pure intervals come from simple ratios between frequencies: 2:1 for the octave, 3:2 for the perfect fifth, 5:4 for the major third. Twelve pure fifths stacked up overshoot seven octaves by about 23.5 cents, the Pythagorean comma, so no twelve-note tuning keeps every interval pure. Each temperament spreads that difference in its own way.
Cents
A cent is a hundredth of an equal-tempered semitone; the octave has 1,200 cents. With this measure tunings can be compared note by note: a pure major third is about 386 cents, and the equal-tempered one is 400.
The four temperaments
In equal temperament, the tuning of the modern piano, the octave is divided into twelve equal semitones of 100 cents. Every key sounds the same, and no interval but the octave is pure: the fifth is about 2 cents narrow and the major third about 14 cents wide.
Pythagorean tuning builds the notes from pure fifths. The fifths sound clean, and the major thirds are about 22 cents wider than pure ones; the fifth that closes the circle is narrower than the others by the whole comma and is called the wolf fifth.
Just intonation uses simple ratios for the intervals from one tonic, C here: the thirds and fifths of C major are pure. In keys far from C some of those intervals are out of tune.
Meantone narrows each fifth by a quarter of the syntonic comma, about 5.4 cents, so that four fifths make a pure major third. It was the usual tuning of keyboard instruments in the sixteenth and seventeenth centuries; keys with many accidentals sound out of tune in it.
