User:Cmloegcmluin/EPD
An EPD, or equal pitch division, is a kind of arithmetic and harmonotonic tuning.
Specification
Its full specification is n-EPDp: n equal (pitch) divisions of interval p.
Formula
To find the step size for an n-EPDp, take the nth root of p. For example, the step of 12-EDO is [math]\displaystyle{ 2^{\frac{1}{12}} }[/math]. So the formula for the kth step of an n-EPDp is:
[math]\displaystyle{ c(k) = p^{\frac{k}{n}} }[/math]
This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ c(k) }[/math] is simply [math]\displaystyle{ 1 }[/math], because any number to the 0th power is 1. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ c(k) }[/math] is simply [math]\displaystyle{ p }[/math], because any number to the 1st power is itself.
Relationship to other tunings
Vs. rank-1 temperaments & equal multiplications
An n-EPDn is equivalent to a rank-1 temperament of p/n, or an equal multiplication of p/n.
Vs. APS
An EPD will be equivalent to some APS, or arithmetic pitch sequence, which has had its count of pitches specified by prefixing "n-". Specifically, n-EPDx = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12=100¢).
Examples
The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal pitch divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (1) | 1.19 | 1.41 | 1.68 | 2 |
| pitch (log₂f) | (2⁰⸍⁴) | 2¹⸍⁴ | 2²⸍⁴ | 2³⸍⁴ | 2⁴⸍⁴ |
| length (1/f) | (1) | 0.84 | 0.71 | 0.59 | 0.5 |