Cent: Difference between revisions
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m "1/100" or "100th", not "1/100th" |
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{{interwiki | {{interwiki | ||
| en = Cent | |||
| de = Cent | | de = Cent | ||
| es = Centésimas | | es = Centésimas | ||
| ja = | | ja = セント | ||
| ko = 센트 | |||
| ro = Centisunet | |||
}} | }} | ||
{{Wikipedia|Cent (music)}} | {{Wikipedia|Cent (music)}} | ||
The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/ | The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/100 (or 1%) of a [[12edo]] [[semitone (interval size measure)|semitone]]. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by {{w|Alexander John Ellis|Alexander J. Ellis}}, the cent may also be defined as the {{w|logarithm}} base 1200th root of 2 of a ratio. | ||
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. | 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. | ||
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== Conversion == | == Conversion == | ||
{{See also| Ratio #Conversion }} | |||
=== Ratio to cents === | === Ratio to cents === | ||
To find the size ''s'' of an interval in cents from its ratio '' | To find the size ''s'' of an interval in cents from its ratio ''r'', calculate the [[log2|binary logarithm]] (log<sub>2</sub>) of its [[frequency ratio]], and multiply it by 1200. | ||
<math>\displaystyle s = 1200\log_2 ( | <math>\displaystyle s = 1200 \cdot \log_2 (r)</math> | ||
For example, the size in cents of a just perfect fifth is 1200⋅log<sub>2</sub>(3/2) ≈ 1200 × 0.584 ≈ 701.955 cents. | |||
If your pocket calculator has no | If your pocket calculator has no <code>log2</code> key, but does have a <code>log</code> (log<sub>10</sub>) or <code>ln</code> (log<sub>''e''</sub>) key, you can key it this way: | ||
(frequency ratio) log ÷ 2 log = | <code>(frequency ratio)</code> <code>log</code> <code>÷</code> <code>2</code> <code>log</code> <code>=</code> | ||
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. | ||
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=== Monzo to 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 … }} | 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> | <math>\displaystyle s = T_J \cdot \vec m</math> | ||
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[[Category:Elementary math]] | [[Category:Elementary math]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||