User:Cmloegcmluin/APS

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

Its full specification is (n-)APSp: (n pitches of an) arithmetic pitch sequence adding by irrational interval p. It is equivalent to a rank-1 temperament with generator p. It is also equivalent to an equal multiplication of p.

The n is optional. If not provided, the sequence is open-ended. 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¢).

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

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

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