Just intonation
Just intonation fixes the intervals of a single key exactly according to the simple frequency ratios: third 5:4, fifth 3:2. In this key everything sounds flawless. If you modulate, it becomes unbearable, because individual intervals go wrong by more than 20 cents.
- Principle
- simple frequency ratios
- Examples
- octave 2:1, fifth 3:2, fourth 4:3, major third 5:4, minor third 6:5
- Strength
- flawless in one key, no beats
- Limit
- modulation into other keys
- Counterpart
- equal temperament (twelve equal semitones)
Flawless in one key
The article Equal, meantone, just: why your piano is slightly out of tune places just intonation among three answers to the Pythagorean comma. It fixes the intervals of a key exactly by simple ratios and therefore works in this one key only. For choirs singing a cappella it is nevertheless the normal case, because singers can adjust the pitch continually.
The physical background is on the page Interval Ear Training: the octave is 2:1, the perfect fifth 3:2, the perfect fourth 4:3, the major third in just intonation 5:4. The simpler the ratio, the calmer and more fused the sound together.
| Interval | Just ratio | Equal-tempered versus just |
|---|---|---|
| Fifth | 3:2 | 2 cents too low |
| Major third | 5:4 | 14 cents too high |
| Minor third | 6:5 | 16 cents too low |
The table summarises figures from the articles Equal, meantone, just: why your piano is slightly out of tune and Major and minor: the third as a switch: in equal temperament the fifth lies 2 cents too low, which nobody hears, and the major third 14 cents too high, which you do hear. Conversely, in just intonation the major third lies 14 cents lower than on the piano, the minor third 16 cents higher.
In an ensemble
The article Major and minor: the third as a switch gives a practical rule for wind players and singers: anyone who plays the third of a chord in an ensemble tunes it by ear instead of following the tuner. When the beating disappears, it is right, and then the needle simply does not point to zero. According to The leading tone the leading tone lies somewhat higher in Pythagorean tuning than on the piano, while in just intonation (15:8) it lies about 12 cents lower; the choral habit of singing it high follows the Pythagorean leading tone.
The page Intervals shows how just ratio and guitar fret are connected: the frets follow equal temperament; the just semitone 16:15 is the counterpart to the equal-tempered fret spacing.
Common mistakes
- Taking just intonation for “better”: It applies to one key, not to all.
- Following the tuner on the third: The tuner measures against the equal-tempered grid.
- Taking the fifth for the problem: In equal temperament it deviates by only 2 cents, the third by 14.
Continue on Maikstrings
Related terms
Pythagorean commaWell-temperedCentBeatingIntonationHarmonic seriesOctavePerfect fifthMajor third
Editor: Maik Krohm · reviewed on 3 October 2026
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