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

[math] c(k) = p^{\frac{k}{n}} [/math]

This way, when [math]k[/math] is [math]0[/math], [math]c(k)[/math] is simply [math]1[/math], because any number to the 0th power is 1. And when [math]k[/math] is [math]n[/math], [math]c(k)[/math] is simply [math]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 |