AFS: Difference between revisions

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An '''AFS''', or '''arithmetic frequency sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.

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

Its full specification is (n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by (irrtional) interval p. The only difference between an [[OS|OS (overtone sequence)]] and AFS is that for OS the p is rational.

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

An AFS could also be described as a shifted [[overtone series]] (± frequency).

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

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 [[OS#Derivation|derivation of OS]].

~~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."

=== Examples ===

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: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> etc. we could have the AFSφ.

If we wanted to move by steps of φ, like this: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> 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 [[Monotonic tunings#Derivation of OS|derivation of OS]].

Here's another example:

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 An '''AFS''', or '''arithmetic frequency sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
-An OS is a specific (rational) type of AFS.
+Its full specification is (n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by (irrtional) interval p. The only difference between an [[OS|OS (overtone sequence)]] and AFS is that for OS the p is rational.
-(n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by p
+An AFS could also be described as a shifted [[overtone series]] (± frequency).
-shifted overtone series (± frequency) (equivalent to AFS)
+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 [[OS#Derivation|derivation of OS]].
-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."
+=== Examples ===
-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: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> etc. we could have the AFSφ.
+If we wanted to move by steps of φ, like this: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> 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 [[Monotonic tunings#Derivation of OS|derivation of OS]].
+Here's another example:
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