Cent

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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.

Cents are often used to express the size of intervals in different tuning systems, sometimes to express the accuracy of the representation of a just intonation interval in a given system.

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.

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 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

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): 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.

In the Xenharmonic wiki there is broad agreement to stick to cents as a general interval measure. Under certain circumstances, alternative interval size measures are provided in addition.

See also

External links