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 (irrational) 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 must be 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 one period of 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, 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 |
| 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) | 0.54 | 0.37 | 0.28 | 0.23 | 0.19 | 0.17 | 0.15 | 0.13 |