Just Intonation Interval Calculator
In just intonation, musical intervals are expressed as simple integer ratios derived from the harmonic series. The frequency of the upper note is the base frequency multiplied by the ratio: f_interval = f_base x (numerator / denominator). A perfect fifth above A4 (440 Hz) in just intonation is 440 x 3/2 = 660 Hz. This calculator lets you apply any rational ratio to any base frequency and also reports the interval in cents relative to the base for comparison with equal temperament values.
Just intonation formula
finterval = fbase × (numerator / denominator)
Cents = 1200 × log2(numerator / denominator)
The interval in cents is the logarithmic measure of the frequency ratio. 100 cents = 1 equal-temperament semitone; 1200 cents = 1 octave.
Common just intonation intervals
- Perfect fifth: 3:2 = 702.0 cents (ET = 700 cents)
- Perfect fourth: 4:3 = 498.0 cents (ET = 500 cents)
- Major third: 5:4 = 386.3 cents (ET = 400 cents)
- Minor third: 6:5 = 315.6 cents (ET = 300 cents)
- Octave: 2:1 = 1200.0 cents (ET = 1200 cents)
Frequently asked questions
What is just intonation?
Just intonation is a tuning system in which musical intervals are based on simple integer ratios of frequencies. A perfect fifth is 3:2 (1.5x the base frequency), a perfect fourth is 4:3, a major third is 5:4. These ratios are derived from the harmonic series and produce beatless, pure intervals.
How does just intonation differ from equal temperament?
Equal temperament divides the octave into 12 equal semitones of 2^(1/12) each. Just intonation uses pure integer ratios. A just perfect fifth (3:2 = 702.0 cents) is slightly wider than the equal temperament fifth (700 cents). The difference (2 cents) is the Pythagorean comma remnant.
What are common just intonation ratios?
Unison 1:1; octave 2:1; perfect fifth 3:2; perfect fourth 4:3; major third 5:4; minor third 6:5; major sixth 5:3; minor seventh 7:4; major seventh 15:8.
Why is just intonation not used for pianos?
Just intonation produces beatless intervals in a single key, but transposing to other keys changes the frequency ratios and the instrument sounds out of tune. Equal temperament is a compromise that allows playing in all keys with acceptable intonation everywhere.
How do I enter a ratio in this calculator?
Enter the numerator and denominator separately (e.g. numerator = 3, denominator = 2 for a perfect fifth). The resulting frequency is base x numerator / denominator.
Official sources
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.