What a cent is
The cent is a logarithmic unit for musical intervals, introduced by the phonetician Alexander Ellis in the 1880s. An octave, a doubling of frequency, is divided into 1,200 cents, so each equal-tempered semitone is exactly 100 cents. The interval between two frequencies f1 and f2 is:
cents = 1200 × log2(f2 / f1)
Because the scale is logarithmic, intervals add up: a fifth (700 cents) plus a fourth (500 cents) makes an octave (1,200 cents), even though the frequency ratios multiply.
Example
From 440 Hz (A4) to 660 Hz, the ratio is 1.5, or 3:2. The interval is 1200 × log2(1.5) = 701.955 cents. The nearest equal-tempered interval is 7 semitones, a perfect fifth of 700 cents, so this pure fifth is 1.96 cents wider. The equal-tempered fifth above 440 Hz is 659.26 Hz.
Just intervals compared with equal temperament
| Interval | Just ratio | Just, cents | Equal temperament | Difference |
|---|---|---|---|---|
| Minor third | 6:5 | 315.64 | 300 | +15.64 |
| Major third | 5:4 | 386.31 | 400 | −13.69 |
| Perfect fourth | 4:3 | 498.04 | 500 | −1.96 |
| Perfect fifth | 3:2 | 701.96 | 700 | +1.96 |
| Major sixth | 5:3 | 884.36 | 900 | −15.64 |
| Octave | 2:1 | 1200 | 1200 | 0 |
Equal temperament makes every key sound the same by spreading small errors across all intervals. Fifths and fourths are almost pure, but the major thirds are noticeably wide, which is why choirs and string quartets often adjust their thirds by ear.
Interval names by semitones
0 unison, 1 minor second, 2 major second, 3 minor third, 4 major third, 5 perfect fourth, 6 tritone, 7 perfect fifth, 8 minor sixth, 9 major sixth, 10 minor seventh, 11 major seventh, 12 octave. Beyond an octave, the same names repeat as compound intervals: 19 semitones is an octave plus a fifth.
Practical uses
- Tuning: tuners often show the error in cents. Most listeners notice differences of about 5 to 10 cents between two notes played one after the other.
- Pitch correction and synthesizers: detune settings are given in cents.
- Comparing temperaments: historical tunings such as meantone or Werckmeister are described by their deviations in cents.