AFS

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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:

example: (1/⁴√2)-shifted overtone series segment = 8-AFS(1/⁴√2) ≈ 8-AFS0.841
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