Cent: Difference between revisions
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The 12edo perfect fifth is exactly 700 cents, and the 12edo 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 [[24edo]] neutral third is exactly 350 cents. The [[22edo]] approximation to 3/2 is approximately 709 cents. | The 12edo perfect fifth is exactly 700 cents, and the 12edo 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 [[24edo]] neutral third is exactly 350 cents. The [[22edo]] approximation to 3/2 is approximately 709 cents. | ||
== | == Conversion == | ||
=== Ratio to cents === | |||
To find the size of | To find the size ''s'' of an interval in cents from its ratio ''c'', you have to calculate the [[log2|binary logarithm]] (log<sub>2</sub>) of its [[frequency ratio]], and multiply this by 1200. | ||
<math>\displaystyle s = 1200\log_2 (c)</math> | |||
Example (just perfect fifth): log<sub>2</sub>2</sub>(3/2) × 1200 ≈ 0.584 × 1200 ≈ 701.955 cents. | Example (just perfect fifth): log<sub>2</sub>2</sub>(3/2) × 1200 ≈ 0.584 × 1200 ≈ 701.955 cents. | ||
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(frequency ratio) log ÷ 2 log = | (frequency ratio) log ÷ 2 log = | ||
(This makes use of the property of logarithms that log<sub>2</sub>(x) = log<sub>n</sub>(x) / log<sub>n</sub>(2).) | (This makes use of the property of logarithms that log<sub>2</sub>(''x'') = log<sub>''n''</sub>(''x'') / log<sub>''n''</sub>(2).) | ||
For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed. | For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed. | ||
For [[ | === Edosteps to cents === | ||
For [[edo]]steps, which are already logarithmic, simply divide 1200 by the edo number, then multiply by the number of steps. | |||
For example, 1 step of 31edo is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents. | For example, 1 step of 31edo is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents. | ||
=== Monzo to cents === | |||
To find the size ''s'' of a just interval in cents from its [[monzo]] '''m''' = {{monzo| m<sub>1</sub> m<sub>2</sub> m<sub>3</sub> … }}, left-multiply '''m''' by the [[just tuning map]] in cents T<sub>J</sub> = {{val| 1200.000 1901.955 2786.314 … }} | |||
<math>\displaystyle s = T_J \cdot \vec m</math> | |||
== Other interval size units == | == Other interval size units == | ||