User:Cmloegcmluin/APS

An APS, or arithmetic pitch sequence, is a kind of arithmetic and harmonotonic tuning. It can also be called an equal multiplication.

Specification

Its full specification is (n-)APS-p: (n pitches of an) arithmetic pitch sequence adding by interval p.

Note:

• The n is optional. If not provided, the sequence is open-ended.
• The p can be dimensionless, in which case it refers to an interval by its frequency ratio. It can also take a unit proportional to octaves, in which case it refers to an interval by its pitch relation.

Formula

The pitch of k steps of APS-p is quite simply k⋅p for a pitch (log-frequency) quantity p.

Relationship to other tunings

Vs. rank-1 temperaments

By applying a mapping, APS-p becomes an equal temperament with generator p.

Vs. EPD

If the n is not specified, an APS will be equivalent to an equal pitch division (EPD). Specifically, n-EPD-p = APS(p/n) for a pitch quantity p. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.

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, ratio)	(1)	1.19	1.41	1.68	2
length (1/f, ratio)	(0/4)	1/4	2/4	3/4	4/4
Length (1/f)	(1)	0.84	0.71	0.59	0.5