Just Intonation / Cents Deviation Calculator
In just intonation intervals are tuned to small whole-number ratios (the octave 2:1, perfect fifth 3:2, major third 5:4), giving pure, beat-free harmony. In 12-tone equal temperament (the piano's tuning) every semitone is exactly 100 cents, so intervals are powers of 21/12. The two systems disagree: the equal-tempered major third is ~13.7 cents sharp of the pure 5:4, and the equal-tempered fifth is ~2 cents flat of the pure 3:2. One cent is 1/1200 of an octave, so cents = 1200·log2(ratio). See a full interval table (ratio → cents → frequency → nearest equal-tempered interval and its deviation), plus converters between ratios, cents, and frequencies. Everything runs locally in your browser.
Just intervals vs equal temperament (base = 440 Hz)
Ratio ⇄ cents
Cents → frequency
Cents between two frequencies
The math. A musical interval as a frequency ratio r is measured in cents as c = 1200·log2(r); one cent is one hundredth of a semitone and 1200 cents make an octave. Invert with r = 2c/1200. The cents between two frequencies is 1200·log2(f2/f1). Just intonation builds intervals from small integer ratios drawn from the harmonic series — the 5-limit ratios used here: unison 1:1 (0¢), minor second 16:15, major second 9:8, minor third 6:5, major third 5:4 (~386.3¢), perfect fourth 4:3 (~498.0¢), tritone 45:32, perfect fifth 3:2 (~702.0¢), minor sixth 8:5, major sixth 5:3, minor seventh 16:9, major seventh 15:8, octave 2:1 (1200¢), plus the 7-limit harmonic seventh 7:4. Equal temperament instead divides the octave into 12 equal semitones of 100¢, so an interval of n semitones is n·100 cents with ratio 2n/12. The deviation column shows justCents − equalCents for the nearest equal-tempered interval: a positive value means the just interval is sharper than the piano, negative means flatter (e.g. the pure 5:4 major third at 386.3¢ sits 13.7¢ below the equal-tempered 400¢, so the piano's major third sounds sharp). Pairs with the Note Frequency Calculator (which handles equal-tempered note↔frequency) and the Wavelength / Frequency Calculator. Everything runs locally — nothing leaves your browser.