# 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

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 |

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 |