Musical Interval Calculator
Find the frequency ratio, cents, and name of any interval — from a semitone count or from two frequencies.
A musical interval’s size is a frequency ratio: ratio = 2^(semitones ÷ 12). A perfect fifth spans 7 semitones, so its equal-tempered ratio is 2^(7 ÷ 12) ≈ 1.4983 — very close to the pure 3:2. In cents, that’s 700 cents, since every equal-tempered semitone equals exactly 100 cents.
What a musical interval is
An interval is the distance in pitch between two notes. Because we hear pitch on a ratio scale, an interval is best described not by a difference in hertz but by the ratio of the two frequencies. In twelve-tone equal temperament (12-TET) — the tuning of a standard piano — the octave is split into twelve equal semitones, and every semitone multiplies the frequency by the same factor, the twelfth root of two (about 1.0595).
Musicians measure fine differences in pitch using cents: 100 cents to a semitone, 1200 cents to an octave. Cents are logarithmic, so equal steps in cents sound like equal musical steps, which is why they add up neatly where ratios multiply.
From a semitone count use the left form; from two frequencies use the right. Semitones = cents ÷ 100.
Worked example
Two notes at 440 Hz and 660 Hz — what interval is that?
- 1 Form the frequency ratio. Divide the higher frequency by the lower: 660 ÷ 440 = 1.5.
- 2 Convert the ratio to cents. 1200 × log₂(1.5) = 1200 × 0.58496 ≈ 701.96 cents.
- 3 Convert cents to semitones. 701.96 ÷ 100 ≈ 7.02 semitones — or directly, 12 × log₂(1.5) ≈ 7.02.
- 4 Round to the nearest named interval. 7 semitones is a perfect fifth, so 440 Hz to 660 Hz is a perfect fifth.
- 5 Compare with the pure ratio. 3:2 = 1.5 exactly (701.96 cents); the equal-tempered fifth is 2^(7 ÷ 12) ≈ 1.4983 (700 cents) — about 2 cents narrower.
Intervals within one octave (12-TET)
Equal-tempered frequency ratios rounded to four places; cents = 100 × semitones. The ratio column is the exact equal-tempered value, which for many intervals sits close to a simple whole-number ratio.
| Semitones | Interval | Ratio | Cents |
|---|---|---|---|
| 0 | Unison | 1.0000 | 0 |
| 1 | Minor second | 1.0595 | 100 |
| 2 | Major second | 1.1225 | 200 |
| 3 | Minor third | 1.1892 | 300 |
| 4 | Major third | 1.2599 | 400 |
| 5 | Perfect fourth | 1.3348 | 500 |
| 6 | Tritone | 1.4142 | 600 |
| 7 | Perfect fifth | 1.4983 | 700 |
| 8 | Minor sixth | 1.5874 | 800 |
| 9 | Major sixth | 1.6818 | 900 |
| 10 | Minor seventh | 1.7818 | 1000 |
| 11 | Major seventh | 1.8877 | 1100 |
| 12 | Octave | 2.0000 | 1200 |
Equal-tempered vs. just ratios
The simple whole-number ratios — 3:2 for a fifth, 5:4 for a major third, 2:1 for an octave — come from just intonation, where intervals are tuned to pure harmonics and sound perfectly consonant. Equal temperament trades that purity for flexibility: every key is equally in tune, at the cost of small deviations. A tempered fifth is 2^(7 ÷ 12) ≈ 1.4983 rather than the pure 3:2 = 1.5 — about 2 cents flat, too small to notice. The tempered major third, at 1.2599 versus the pure 5:4 = 1.25, is about 14 cents sharp, which trained ears can hear.
Only the octave is pure in equal temperament: 2^(12 ÷ 12) = 2, exactly 2:1. Everything in between is a compromise. Cents make the size of that compromise easy to read — 100 cents per semitone, 1200 per octave — so a 14-cent error is simply 14 hundredths of a semitone.