A cent (¢) is the interval equal to exactly 1/100th (or 1%) of a 12-EDO semitone. In other words, cents divide the half step (semitone) of 12-EDO into 100 equal parts.
The 12-EDO perfect fifth is exactly 700 cents, and the 12-EDO major third is exactly 400 cents. In contrast, the just perfect fifth, which corresponds to two notes in a frequency ratio of 3/2, is approximately 702 cents, and the just major third of 5/4 is about 386 cents. The 24-EDO neutral third is exactly 350 cents. The 22-EDO approximation to 3/2 is ca. 709 cents.
How to calculate the size of an interval in cents
Example (just perfect fifth): log22(3/2) × 1200 = ~0.584 × 1200 = ~701.955 cents.
If your pocket calculator has no log2 key, but does have a log (log10) or ln (loge) key, you can key it this way:
(frequency ratio) log ÷ 2 log =
(This makes use of the property of logarithms that log2(x) = logn(x) / logn(2).)
For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.
For EDO steps, which are already logarithmic, simply divide 1200 by the EDO size, then multiply by the number of steps.
For example, 1 step of 31-EDO is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.
Other interval size units
The cent is commonly used because of its ease in communicating information about intervals to a 12-EDO-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12-EDO inherent importance over any other decent tuning. In contrast, others have suggested that cents are a useful unit of interval measure for purely mathematical reasons, even despite of 12-EDO's current status as the dominant tuning in Western society.
- Relative cent -- a useful generalization for the cent measure to any equal tuning
- Millioctave -- one prominent alternative interval measure
- interval size measure -- overview