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.
The formula for length [math]k[/math] of an ALSp is:
[math] 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 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.
The analogous otonal equivalent of an ALS is an AFS (arithmetic frequency sequence).
A US, or utonal sequence, is the (rational) version of an ALS.
By specifying n, your sequence will be equivalent to some ELD (equal length division); specifically n-ALSp = n-ELD((p-1)/n).
|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|