# APS

An APS, or arithmetic pitch sequence, is a kind of arithmetic and harmonotonic tuning.

## Specification

Its full specification is (n-)APSp: (n pitches of an) arithmetic pitch sequence adding by irrational interval p. The n is optional. If not provided, the sequence is open-ended.

## Formula

The pitch of the kth step of an APSp is quite simply k⋅p.

## Relationship to other tunings

### vs. rank-1 temperaments & equal multiplications

An APSp is equivalent to a rank-1 temperament with generator p. It is also equivalent to an equal multiplication of p.

### vs. EPD

If specified, an APS will be equivalent to some EPD, or equal pitch division. Specifically, n-EPDx = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12=100¢).

### vs. AS

The only difference between an APS and an AS (ambitonal sequence) is that the p for an APS is irrational.

## Examples

example: APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢
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