AFS: Difference between revisions

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

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.

== Specification ==

The n is optional. If not provided, the sequence is open-ended~~. By specifying n, your sequence will be equivalent to some [[EFD|EFD (equal frequency division)]]. Specifically, n-EFDp = n-AFS((p-1)/n)~~.

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.

~~The analogous utonal equivalent of an AFS is an [[ALS|ALS (arithmetic length sequence)]].~~

== Formula ==

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

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]].

The formula for step <span><math>k</math></span> of an AFSp is:

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</math>

=== Examples ===

== Relationship to other tunings ==

=== vs. OS ===

The only difference between an [[OS|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 [[OS#Derivation|derivation of OS]].

=== vs. EFD ===

By specifying n, your sequence will be equivalent to some [[EFD|EFD (equal frequency division)]]. Specifically, n-EFDp = n-AFS((p-1)/n).

=== vs. ALS ===

The analogous utonal equivalent of an AFS is an [[ALS|ALS (arithmetic length sequence)]].

== Examples ==

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.

@@ Line 1: / Line 1: @@
 An '''AFS''', or '''arithmetic frequency sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-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.
+== Specification ==
-The n is optional. If not provided, the sequence is open-ended. By specifying n, your sequence will be equivalent to some [[EFD|EFD (equal frequency division)]]. Specifically, n-EFDp = n-AFS((p-1)/n).
+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.
-The analogous utonal equivalent of an AFS is an [[ALS|ALS (arithmetic length sequence)]].
+== Formula ==
-An AFS could also be described as a shifted [[overtone series]] (± frequency).
-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]].
 The formula for step <span><math>k</math></span> of an AFSp is:
@@ Line 17: / Line 13: @@
 </math>
-=== Examples ===
+== Relationship to other tunings ==
+=== vs. OS ===
+The only difference between an [[OS|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 [[OS#Derivation|derivation of OS]].
+=== vs. EFD ===
+By specifying n, your sequence will be equivalent to some [[EFD|EFD (equal frequency division)]]. Specifically, n-EFDp = n-AFS((p-1)/n).
+=== vs. ALS ===
+The analogous utonal equivalent of an AFS is an [[ALS|ALS (arithmetic length sequence)]].
+== Examples ==
 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.