ALS

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

Specification

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.

Formula

The formula for length [math]\displaystyle{ k }[/math] of an ALSp is:

[math]\displaystyle{ L(k) = 1 + k⋅p }[/math]

Tip about tunings based on length

Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

Relationship to other tunings

As shifted undertone series

By varying the undertone series step size to some number other than 1, if that number is irrational you can produce an ALS, and if rational you can produce a US (which you could also call an ALS if you really wanted to). 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.

Vs. AFS

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

Vs. US

A US, or utonal sequence, is the rational version of an ALS.

Vs. ELD

By specifying n, your sequence will be equivalent to one period of some ELD (equal length division); specifically n-ALSp = n-ELD((p-1)/n).

Examples

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, ratio)	(1)	1.12	1.28	1.48	1.77	2.19	2.88	4.20	7.73
pitch (log₂f, octaves)	(0)	0.17	0.35	0.57	0.82	1.13	1.53	2.07	2.95
length (1/f, ratio)	(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, ratio)	(1)	0.54	0.37	0.28	0.23	0.19	0.17	0.15	0.13
pitch (log₂f, octaves)	(0)	-0.88	-1.42	-1.82	-2.13	-2.38	-2.60	-2.78	-2.95
length (1/f, ratio)	(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