# 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 $2^{\frac{1}{12}}$. So the formula for the kth step of an n-EPDp is:

$c(k) = p^{\frac{k}{n}}$

This way, when $k$ is $0$, $c(k)$ is simply $1$, because any number to the 0th power is 1. And when $k$ is $n$, $c(k)$ is simply $p$, 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).

example: 4-EPDO = 4-EDO
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