Tuning & Temperament
How the same interval differs across tuning systems — equal temperament, just intonation and Pythagorean tuning — in cents, with the just-intonation ratios. Why 'in tune' is a choice.
| Interval | Equal (cents) | Just (cents) | Just ratio | Pythagorean (cents) |
|---|---|---|---|---|
| Perfect unison | 0 | 0.0 | 1/1 | 0.0 |
| Minor second | 100 | 111.7 | 16/15 | 90.2 |
| Major second | 200 | 203.9 | 9/8 | 203.9 |
| Minor third | 300 | 315.6 | 6/5 | 294.1 |
| Major third | 400 | 386.3 | 5/4 | 407.8 |
| Perfect fourth | 500 | 498.0 | 4/3 | 498.0 |
| Tritone | 600 | 590.2 | 45/32 | 611.7 |
| Perfect fifth | 700 | 702.0 | 3/2 | 702.0 |
| Minor sixth | 800 | 813.7 | 8/5 | 792.2 |
| Major sixth | 900 | 884.4 | 5/3 | 905.9 |
| Minor seventh | 1000 | 996.1 | 16/9 | 996.1 |
| Major seventh | 1100 | 1088.3 | 15/8 | 1109.8 |
| Perfect octave | 1200 | 1200.0 | 2/1 | 1200.0 |
Equal temperament splits the octave into 12 identical 100-cent steps; just intonation uses pure small-integer ratios; Pythagorean tuning stacks pure fifths. The small differences are why some intervals sound sweeter in one system than another. Hear cents deviation on the note/frequency converter.
How to read this
A cent is 1/100 of an equal-tempered semitone, so an octave is 1200 cents. Comparing tuning systems means comparing how far each interval sits from the pure whole-number ratio.
cents = 1200 × log2(ratio)
just perfect fifth 3/2 = 701.96 cents (ET: 700)
just major third 5/4 = 386.31 cents (ET: 400)
just minor third 6/5 = 315.64 cents (ET: 300)Notes & gotchas
- Equal temperament's major third is 13.7 cents sharp of pure — the largest compromise in the system, and clearly audible on sustained chords.
- Its fifth is only 2 cents flat, which is why fifths sound fine and thirds are the ones that beat.
- The Pythagorean comma (23.46 cents) is the gap left when you stack twelve pure fifths and land not quite seven octaves higher. Every tuning system is a different answer to what to do with it.
- Just intonation is perfect in one key and progressively worse as you move away — which is exactly the problem equal temperament was adopted to solve.
- 5–6 cents is roughly the threshold at which most listeners hear mistuning on a sustained tone.