Just Intonation Ratio Calculator
Just intonation builds intervals from simple whole-number frequency ratios, giving the pure, beat-free sound of a 3:2 fifth or a 5:4 major third. This calculator takes a base frequency and a ratio expressed as a numerator and denominator, and returns the resulting frequency, the size of the interval in cents, and how far that pure interval sits from the nearest equal-tempered note. Use it to explore tuning systems, set up a synthesizer for just intonation, or understand why pure intervals sound different from the piano.
Just intonation formula
resulting frequency = base * (numerator / denominator)
cents = 1200 * log2(numerator / denominator)
nearest ET semitone = round(cents / 100) * 100
deviation = cents - nearest ET semitone
The ratio numerator over denominator is multiplied by the base pitch. Converting to cents lets you compare any pure ratio with the 100-cent grid of equal temperament.
Common just intervals
- Octave 2:1, exactly 1,200 cents.
- Perfect fifth 3:2, 701.96 cents.
- Perfect fourth 4:3, 498.04 cents.
- Major third 5:4, 386.31 cents.
- Minor third 6:5, 315.64 cents.
Just intonation: frequently asked questions
What is just intonation?
Just intonation tunes intervals to small whole-number frequency ratios such as 2:1 (octave), 3:2 (perfect fifth), and 5:4 (major third). These pure ratios produce beat-free, consonant intervals, unlike equal temperament, which slightly detunes every interval except the octave so that all keys are equally usable.
How do I get a frequency from a ratio?
Multiply the base frequency by the ratio. For a 3:2 fifth above 220 Hz, the result is 220 times 3 / 2 = 330 Hz. To find the interval in cents, take 1200 times the base-2 logarithm of the ratio.
What are the cents values of common just intervals?
The just perfect fifth (3:2) is 701.96 cents, the just major third (5:4) is 386.31 cents, the just minor third (6:5) is 315.64 cents, and the just major sixth (5:3) is 884.36 cents. Compare these with their equal-tempered neighbours (700, 400, 300, and 900 cents) to see how far each is detuned.
What does the equal-temperament deviation tell me?
It shows how many cents the pure ratio differs from the nearest equal-tempered semitone. A positive value means the just interval is sharper than the equal-tempered note; a negative value means it is flatter. The just major third, for example, is about 13.69 cents flat of the equal-tempered major third.
Sources and definitions
- Just intonation uses whole-number frequency ratios; the cents conversion (1200 * log2 of the ratio) is the standard definition of the cent. These are standard mathematical definitions.
- National Institute of Standards and Technology: SI units reference (frequency in hertz).
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.