Diapente

In harmony, diapente is the ratio 3:2 between a pair of frequencies or, equivalently, the ratio 2:3 between a pair of wavelengths. It is the arithmetic mean of diapason and unison (considered as frequencies):

It is 1.1 in binary, and it is the sum of the first three reciprocals of triangular numbers:

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It is the basis (together with diapason) of the Pythagorean tuning system. It is derived from the number 3. 1:3 is smaller than an octave (1:2): to bring it back to within one octave of unison, it is multiplied by 2:1 (diapason) yielding an equivalent note but an octave lower: 1:3 x 2:1 = 2:3 .

In Pythagorean tuning, 12 diapentes are approximately equal to 7 diapasons:

The proportional error in the approximation is called Pythagorean comma. The base 2 logarithm of diapente is approximately 7/12;

and the error in the approximation is +1.95 cents, which is the Pythagorean comma in cents.

The ratio 12/7 is the sum of the first 6 reciprocals of triangular numbers:

Diapason is equal to 12 semitones, and diapente is equal to 7 semitones. A piano has 7 (and one third) diapasons and 12 (and four sevenths) diapentes. Therefore it is possible to play the circle of fifths on a piano without wrapping it into a single octave, but rather playing it as a spiral of fifths.

The diapente is also called perfect fifth.

See also: diapason, diatessaron.