ALS

An ALS, or arithmetic length sequence, is a kind of arithmetic and harmonotonic tuning.

Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency.

The n is optional. If not provided, the sequence is open-ended. By specifying n, your sequence will be equivalent to some ELD (equal length division); specifically n-ALSp = n-ELD((p-1)/n).

The analogous otonal equivalent of an ALS is an AFS (arithmetic frequency sequence).

A US, or utonal sequence, is a specific (rational) type of ALS. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.

example: (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2) (arranged so that the pitches are in ascending order and still begin on 1/1)

quantity	(0)	1	2	3	4	5	6	7	8
frequency (f)	(1)	1.12	1.28	1.48	1.77	2.19	2.88	4.20	7.73
pitch (log₂f)	(0)	0.17	0.35	0.57	0.82	1.13	1.53	2.07	2.95
length (1/f)	(1)	0.89	0.78	0.67	0.56	0.46	0.35	0.24	0.13

example: (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2) (descending pitches)

quantity	(0)	1	2	3	4	5	6	7	8
frequency (f)	(1)	0.54	0.37	0.28	0.23	0.19	0.17	0.15	0.13
pitch (log₂f)	(0)	-0.88	-1.42	-1.82	-2.13	-2.38	-2.60	-2.78	-2.95
length (1/f)	(1 + 0/⁴√2)	1 + 1/⁴√2	1 + 2/⁴√2	1 + 3/⁴√2	1 + 4/⁴√2	1 + 5/⁴√2	1 + 6/⁴√2	1 + 7/⁴√2	1 + 8/⁴√2