User:Cmloegcmluin/AS

An AS, or ambitonal sequence, is a kind of arithmetic and harmonotonic tuning.

Specification

Its full specification is (n-)AS-p: (n pitches of an) ambitonal sequence adding by rational 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.

Relationship to other tunings

Vs. 1D JI Lattice & equal multiplications

AS-p is equivalent to a 1-dimensional JI lattice of p. These are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. It is also equivalent to an equal multiplication of a rational interval p.

Vs. APS

The only difference between an (n-)AS-p and an (n-)APS-p (arithmetic pitch sequence) is that the p for an AS must be rational.

Examples

example: AS5/4 = 1D JI lattice of 5/4 = 5/4 chain

quantity	(0)	1	2	3	4	5	6	7	8
frequency (f, ratio)	(5⁰/4⁰)	5¹/4¹	5²/4²	5³/4³	5⁴/4⁴	5⁵/4⁵	5⁶/4⁶	5⁷/4⁷	5⁸/4⁸
pitch (log₂f, octaves)	(0)	0.32	0.64	0.97	1.29	1.61	1.93	2.25	2.58
length (1/f, ratio)	(1/1)	4/5	16/25	64/125	256/625	1024/3125	4096/15625	16384/78125	65536/390625

List of ASs

See also: APS § List of APSs

Superparticular

• AS8/7
• AS9/8
• AS10/9
• AS15/14
• AS16/15
• AS18/17
• AS21/20
• AS25/24
• AS26/25
• AS33/32
• AS81/80

Others

• AS13/10