Cent
The cent (symbol: ¢) is a unit of interval size equal to exactly 1/100th (or 1%) of a 12edo semitone. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by Alexander J. Ellis, the cent may also be defined as the 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.
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.
Conversion
Ratio to cents
To find the size s of an interval in cents from its ratio c, you have to calculate the binary logarithm (log2) of its frequency ratio, and multiply this by 1200.
[math]\displaystyle s = 1200\log_2 (c)[/math]
Example (just perfect fifth): log22(3/2) × 1200 ≈ 0.584 × 1200 ≈ 701.955 cents.
If your pocket calculator has no log2 key, but does have a log (log10) or ln (loge) key, you can key it this way:
(frequency ratio) log ÷ 2 log =
This makes use of the property of logarithms that log2(x) = logn(x) / logn(2).
For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.
Edosteps to cents
For edosteps, 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 = [m1 m2 m3 …⟩, left-multiply m by the just tuning map in cents TJ = ⟨1200.000 1901.955 2786.314 …]
[math]\displaystyle s = T_J \cdot \vec m[/math]
Other interval size units
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.
In the Xenharmonic Wiki there is broad agreement to stick to cents as a general interval measure. Under certain circumstances, alternative interval size measures are provided in addition.
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