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<span style="display: block; text-align: right;">[[:de:Cent|Deutsch]] - [[Centésimas|Español]]</span>
{{interwiki
| en = Cent
| de = Cent
| es = Centésimas
| ja = セント
| ko = 센트
| ro = Centisunet
}}
{{Wikipedia|Cent (music)}}
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.


=Definitions=
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.
A ''cent'' is an interval equal to exactly 1/100th of a [[12edo|12-EDO]] semitone. In other words, cents equally divide the half step (semitone) of 12-EDO into 100 equal parts. Cents are often used to express the size of intervals in different tuning systems.


The cent, which was first proposed in the late 19th century by [http://en.wikipedia.org/wiki/Alexander_J._Ellis Alexander Ellis], is a logarithmic measure which may also be defined as the [http://en.wikipedia.org/wiki/Logarithm logarithm] to the base 1200th root of 2. It may also be considered as exactly 1 step of 1200-EDO (dividing the octave into 1200 equal parts).
== Examples ==
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.


=Examples=
== Conversion ==
The 12-EDO perfect fifth is exactly 700 cents, and the 12-EDO 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 701.955 cents, and the just major third of 5/4 is ~386.314 cents. The 24-EDO neutral third is exactly 350 cents. The 22-EDO approximation to 3/2 is ~709.091 cents.
{{See also| Ratio #Conversion }}


=How to calculate the size of an interval in cents=
=== Ratio to cents ===
To find the size of a just interval in cents, you have to calculate the [[log2|binary logarithm]] (log<span style="font-size: 80%; vertical-align: sub;">2</span>) of its [[frequency_ratio|frequency ratio]], and multiply this by 1200.
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.


Example (just perfect fifth): 1200 × log<span style="font-size: 80%; vertical-align: sub;">2</span>(3/2) = 1200 × ~0.584 = ~701.955 cents
<math>\displaystyle s = 1200 \cdot \log_2 (r)</math>


If your pocket calculator has no ''log2'' key, but does have a ''log'' (log<span style="font-size: 80%; vertical-align: sub;">10</span>) or ''ln'' (log<span style="font-size: 80%; vertical-align: sub;">e</span>) key, you can key it this way:
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.


(frequency ratio) log ÷ 2 log =
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:


(This makes use of the property of logarithms that log<span style="font-size: 80%; vertical-align: sub;">2</span>(x) = log<span style="font-size: 80%; vertical-align: sub;">n</span>(x) / log<span style="font-size: 80%; vertical-align: sub;">n</span>(2). )
<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.


For EDO steps, which are already logarithmic, simply divide 1200 by the EDO size, then multiply by the number of steps. For example, 1 step of 31-EDO is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.
=== 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.
 
=== 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 Units of Interval Measure=
== Other interval size units ==
The cent is commonly used because of its ease in communicating information about intervals to a 12-EDO-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12-EDO inherent importance over any other decent tuning. In contrast, others have suggested that cents are a useful unit of interval measure for purely mathematical reasons, even despite of 12-EDO's current status as the dominant tuning in Western society.
The cent is commonly used because of its ease in communicating information about intervals to a 12edo-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12edo inherent importance over any other decent tuning. In contrast, others have suggested that cents are a useful unit of interval measure for purely mathematical reasons, even despite of 12edo's current status as the dominant tuning in Western society.  


Whatever your stance, alternative measures of interval size can be found at [[Interval_size_measure|Interval size measure]].
In the Xenharmonic Wiki there is broad agreement to stick to cents as a general interval measure. Under certain circumstances, alternative [[interval size measure]]s are provided in addition.


One prominent alternative interval measure is the [[millioctave|millioctave]] (mO).
== See also ==
* [[Relative cent]] – a useful generalization for the cent measure to ''any'' [[equal]] tuning
* [[Millioctave]] – one prominent alternative interval measure  
* [[Interval size measure]] – overview


Additionally, a useful generalization for the cent measure is the '''[[Relative_cent|relative cent]],''' which is one 100th of two neighboring [[pitch|pitches]] in any [[Equal|equal]] tuning.
== External links ==
* [http://tonalsoft.com/enc/c/cent.aspx cent, ¢, 1200-ed2] on [[Tonalsoft Encyclopedia]]


=References=
[[Category:Interval size measures]]
[http://en.wikipedia.org/wiki/Cent_%28music%29 Wikipedia article on cents]      [[Category:absolute_measure]]
[[Category:Elementary math]]
[[Category:interval]]
[[Category:Terms]]
[[Category:interval_measure]]
[[Category:interval_size_measure]]
[[Category:logarithmic_measure]]
[[Category:measure]]
[[Category:size]]
[[Category:unit]]
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