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

List of APSs

See also: AS § List of ASs

• APS28¢
• APS28.5¢
• APS28.7¢
• APS39¢
• APS44¢
• APS35.099¢, tuning of Carlos Gamma
• APS63.833¢, tuning of Carlos Beta
• APS65¢
• APS69¢
• APS77.965¢, tuning of Carlos Alpha
• APS86.4¢
• APS88¢
• APS97.5¢
• APS125¢
• APS148.5¢
• APS237.8¢

For a longer list, see Gallery of arithmetic pitch sequences. But do note that the gallery includes many obscure tunings that are of less importance to most xenharmonicists compared to the more curated selection listed above.