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Definitions

A cent is an interval equal to exactly 1/100th of a 12-EDO semitone. In other words, cents equally divide the half step (semitone) of 12-EDO into 100 equal parts. Cents are often used to express the size of intervals in different tuning systems.

The cent, which was first proposed in the late 19th century by Alexander Ellis, is a logarithmic measure which may also be defined as the logarithm to the base 1200th root of 2. It may also be considered as exactly 1 step of 1200-EDO (dividing the octave into 1200 equal parts).

Examples

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 701.955 cents, and the just major third of 5/4 is ~386.314 cents. The 24-EDO neutral third is exactly 350 cents. The 22-EDO approximation to 3/2 is ~709.091 cents.

How to calculate the size of an interval in cents

To find the size of a just interval in cents, you have to calculate the binary logarithm (log2) of its frequency ratio, and multiply this by 1200.

Example (just perfect fifth): 1200 × log2(3/2) = 1200 × ~0.584 = ~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:

(This makes use of the property of logarithms that log2(x) = logn(x) / logn(2). )

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 Units of Interval Measure

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.

Whatever your stance, alternative measures of interval size can be found at Interval size measure.

One prominent alternative interval measure is the millioctave (mO).

Additionally, a useful generalization for the cent measure is the relative cent, which is one 100th of two neighboring pitches in any equal tuning.

References

Wikipedia article on cents