User:Cmloegcmluin/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 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 one period of 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 AS must be rational.

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