  # 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  (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:

• exactly 0.01 (= 1/100) of a 12-edo semitone
• exactly 0.0158,3... (= 19/1200) of a 19-edo third-tone
• exactly 0.02 (= 1/50) of a quarter-tone
• exactly 0.0358,3... (= 43/1200) of a méride
• approximately 0.06023552099133 (= ~1/17) of a cleonides-morion
• exactly 0.06 (= 3/50) of a 72-edo morion
• exactly 0.12 (= 3/25) of a farab
• exactly 0.1708,3... (= 41/240) of a mem
• exactly 0.225 (= 9/40) of a tredek
• exactly 0.25 (= 1/4) of a savart
• exactly 0.2508,3... (= 301/1200) of an heptaméride (old savart)
• exactly 0.2591,6... (= 311/1200) of a gene
• exactly 0.4841,6... (= 581/1200) of a spook
• exactly 0.5 (= 1/2) of a centitone (or Iring)
• exactly 0.51 (= 51/100) of a sk (612-edo schisma)
• approximately 0.511508725 (= ~43/86) of a grad
• exactly 0.608,3... (= 73/120) of a woolhouse-unit
• exactly 0.64 (= 16/25 ~= 2/3) of a 6mu
• exactly 0.8,3... (= 5/6) of a millioctave
• exactly 0.85,3... (= 64/75) of a yamaha-unit
• exactly 2.05 (= 2 1/20) minas
• exactly 7.1158,3... (= 7 139/1200 = ~ 7 1/8 ) tinas
• exactly 8.8,3... (= 8 5/6) türk-sents
• exactly 25.0858,3... (= 25 103/1200) jots
• approximately 30.6905235 (~= 30 2/3) temperament-units
• exactly 38.36 (= 38 9/25) flus
• exactly 40.96 (= 40 24/25) 12mus
• exactly 163.84 (= 163 21/25) 14mus

. . . . . . . . .

### 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 100Â¢ 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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