# 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]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.

### 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 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) (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