Just Intonation Cents Calculator
Convert between frequency ratios and cents to compare just intonation, equal temperament, and historical tunings.
Find pitch difference for any interval.
Cents and Frequency Ratios
Music intervals are perceived logarithmically — each octave doubles the frequency, but our ears hear it as the same “step.” Cents are a fine-grained logarithmic unit that makes intervals comparable across tuning systems.
Formula
cents = 1200 × log₂(f₂ / f₁)
Inversely:
f₂ / f₁ = 2^(cents / 1200)
One octave = 1200 cents. One equal-tempered semitone = 100 cents. A skilled musician can hear differences as small as 5–10 cents.
Just Intonation vs Equal Temperament
Just intonation tunes intervals to small whole-number frequency ratios — the ratios that produce the cleanest, beatless harmonies. Equal temperament divides the octave into 12 equal steps, slightly detuning every interval (except the octave) so that any key sounds equally good.
| Interval | Just Ratio | Just Cents | ET Cents | Difference |
|---|---|---|---|---|
| Unison | 1:1 | 0 | 0 | 0 |
| Minor 2nd | 16:15 | 111.7 | 100 | +11.7 |
| Major 2nd | 9:8 | 203.9 | 200 | +3.9 |
| Minor 3rd | 6:5 | 315.6 | 300 | +15.6 |
| Major 3rd | 5:4 | 386.3 | 400 | −13.7 |
| Perfect 4th | 4:3 | 498.0 | 500 | −2.0 |
| Tritone | 45:32 | 590.2 | 600 | −9.8 |
| Perfect 5th | 3:2 | 702.0 | 700 | +2.0 |
| Minor 6th | 8:5 | 813.7 | 800 | +13.7 |
| Major 6th | 5:3 | 884.4 | 900 | −15.6 |
| Minor 7th | 9:5 | 1017.6 | 1000 | +17.6 |
| Major 7th | 15:8 | 1088.3 | 1100 | −11.7 |
| Octave | 2:1 | 1200 | 1200 | 0 |
Why Equal Temperament Won
A keyboard with just intonation sounds beautiful in one key but increasingly out-of-tune as you modulate. Bach’s Well-Tempered Clavier showcased the early compromises that led to today’s 12-tone equal temperament — one slightly detuned scale that works in every key.
Worked Examples
- A 3:2 perfect 5th (440 Hz → 660 Hz): cents = 1200 × log₂(660/440) = 1200 × log₂(1.5) = 701.96 cents.
- A 5:4 major 3rd (C → E in just intonation): 386.3 cents — about 14 cents flatter than the equal-tempered E.
Cents in Practice
| Use | Typical Range |
|---|---|
| Detuning two unison strings | < 5 cents |
| Schism / comma adjustments | 5–25 cents |
| Pythagorean comma | 23.5 cents |
| Syntonic comma | 21.5 cents |
| 19-TET vs 12-TET 5th | About 5 cents |
Limitations
Cents only describe the interval, not the tonal quality (timbre, beating). Two intervals with the same cent value still sound different if their underlying ratios are different — this is the essence of why human ears prefer some equal-cent intervals over others.