Reference
Tuning Systems & Temperaments
Cents deviations from equal temperament for six historical tuning systems, with what each sounds like and when to use it — plus an interactive comparison.
| Temperament | C | C♯/D♭ | D | D♯/E♭ | E | F | F♯/G♭ | G | G♯/A♭ | A | A♯/B♭ | B | Hear |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 12-tone equal temperament | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| Pythagorean (D♭–F♯) | 0 | -9.8 | +3.9 | -5.9 | +7.8 | -2.0 | +11.7 | +2.0 | -7.8 | +5.9 | -3.9 | +9.8 | |
| Quarter-comma meantone | 0 | -23.9 | -6.8 | +10.3 | -13.7 | +3.4 | -20.5 | -3.4 | -27.4 | -10.3 | +6.8 | -17.1 | |
| Werckmeister III | 0 | -9.8 | -7.8 | -5.9 | -9.8 | -2.0 | -11.7 | -3.9 | -7.8 | -11.7 | -3.9 | -7.8 | |
| Vallotti | 0 | -5.9 | -3.9 | -2.0 | -7.8 | +2.0 | -7.8 | -2.0 | -3.9 | -5.9 | 0 | -9.8 | |
| 5-limit just intonation (on C) | 0 | +11.7 | +3.9 | +15.6 | -13.7 | -2.0 | -9.8 | +2.0 | +13.7 | -15.6 | +17.6 | -11.7 |
Values are cents relative to equal temperament with C as the reference. Press ▶ to hear C major then E major chords in each system — notice how meantone's pure C major turns sour in E.
12-tone equal temperament
Every semitone is exactly 100 cents (a frequency ratio of 2^(1/12)). The universal modern standard.
Pythagorean (D♭–F♯)
Built from pure 3:2 fifths. Fifths are beatless, major thirds are 8 cents wide of pure and harsh.
Quarter-comma meantone
Fifths narrowed by ¼ syntonic comma so that major thirds are pure (5:4). Renaissance keyboard standard; one 'wolf' fifth.
Werckmeister III
A 1691 well temperament: four fifths tempered by ¼ Pythagorean comma, all keys usable with distinct colours.
Vallotti
Six fifths (F–C–G–D–A–E–B) narrowed by ⅙ Pythagorean comma; the others pure. Popular for baroque ensembles.
5-limit just intonation (on C)
Simple whole-number ratios (5:4 thirds, 3:2 fifths) relative to C. Perfect in one key, unusable in remote ones.