# AFS

An **AFS**, or **arithmetic frequency sequence**, is a kind of arithmetic and harmonotonic tuning.

## Specification

Its full specification is (n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by (irrtional) interval p. The n is optional. If not provided, the sequence is open-ended.

## Formula

The formula for step [math]k[/math] of an AFSp is:

[math] f(k) = 1 + k⋅p [/math]

## Relationship to other tunings

### vs. OS

The only difference between an OS (overtone sequence) and AFS is that for OS the p is rational.

### As shifted overtone series

An AFS could also be described as a shifted overtone series (± frequency). Both AFS and OS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see derivation of OS.

### vs. EFD

By specifying n, your sequence will be equivalent to some EFD (equal frequency division). Specifically, n-EFDp = n-AFS((p-1)/n).

### vs. ALS

The analogous utonal equivalent of an AFS is an ALS (arithmetic length sequence).

## Examples

If we wanted to move by steps of φ, like this: [math]1, 1+φ, 1+2φ, 1+3φ...[/math] etc. we could have the AFSφ.

Here's another example:

quantity | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

frequency (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 |

pitch (log₂f) | (0) | 0.88 | 1.42 | 1.82 | 2.13 | 2.38 | 2.60 | 2.78 | 2.95 |

length (1/f) | (1) | 0.54 | 0.37 | 0.28 | 0.23 | 0.19 | 0.17 | 0.15 | 0.13 |