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Encyclopedia of Microtonal Music Theory

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cent, ¢, 1200-ed2

[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

A unit of interval measurement invented by Alexander Ellis and appearing in his appendix to his translation of Helmhotz's On the Sensations of Tone [1875] (p. 41, and Appendix XX Section C, p 446-451, in the 1954 Dover edition).

A cent is the logarithmic division of the 12-tone equal temperament (12-edo) semitone into 100 equal parts. It is calculated as the 1200th root of 2, expressed mathematically as 1200√2, or 2(1/1200), with a ratio of approximately 1:1.0005777895. It is an irrational number. The formula for calculating the cents-value of any ratio is: cents = log10r * [1200 / log102] or cents = log2r * 1200 , where r is the ratio.

Some tuning-theorists prefer to use millioctaves or other small units for interval-measurement, but cents are almost universally employed as the small logarithmic measurement used to compare interval sizes.

A cent is:

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cents calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = cents

. . . . . . . . .
[John Chalmers, Divisions of the Tetrachord]

A measure of musical interval width invented by A. J. Ellis. There are 100¢ to the 12-tone equal temperament semitone and 1200 cents to the octave. Cents are equivalent to 1200 times the base 2 logarithm of the frequency ratio.

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