# APS

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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

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 |