AFS

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

An OS is a specific (rational) type of AFS.

(n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by p

shifted overtone series (± frequency) (equivalent to AFS)

AFS(1/7π,7) is saying start at 1/7π and then just move by 1's, so the next step is 1+1/7π which equals (7π+1)/7π, and the next step would be 2+1/7π = (14π+1)/7π, and you'd keep going until you had 7 pitches, so the last one would be (42π+1)/7π. Though that's just the first step, because as I mentioned in my previous comment here, you want the first pitch to be 1/1, so you multiply everything by 7π, so in the end the scale is 1, 7π+1, 14π+1, 21π+1 ... 42π+1. A good way to read AFS(1/p,n) is "start on 1, then add p each step, and go until you have n pitches."

If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, 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.

OS and AFS 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 the later section on the derivation of OS.

example: (1/⁴√2)-shifted overtone series segment = 9-AFS(1/⁴√2)

quantity	1	2	3	4	5	6	7	8	9
frequency	1.00	1.84	2.68	3.52	4.36	5.20	6.05	6.89	7.73
pitch	0.00	0.88	1.42	1.82	2.13	2.38	2.60	2.78	2.95
length	1.00	0.54	0.37	0.28	0.23	0.19	0.17	0.15	0.13