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]\displaystyle{ k }[/math] of an AFSp is:

[math]\displaystyle{ 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]\displaystyle{ 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, 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