User:Cmloegcmluin/EPD

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An EPD, or equal pitch division, is a kind of arithmetic and harmonotonic tuning.

Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to ED, or equal division.

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-EPDp is equivalent to a rank-1 temperament of p/n, or an equal multiplication of p/n.

Vs. APS

One period of 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).

example: 4-EPDO = 4-EDO

quantity	(0)	1	2	3	4
frequency (f, ratio)	(1)	1.19	1.41	1.68	2
pitch (log₂f, octaves)	(2⁰⸍⁴)	2¹⸍⁴	2²⸍⁴	2³⸍⁴	2⁴⸍⁴
length (1/f, ratio)	(1)	0.84	0.71	0.59	0.5