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. | An '''AFS''', or '''arithmetic frequency sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
The n is optional. If not provided, the sequence is open-ended | 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 <span><math>k</math></span> of an AFSp is: | The formula for step <span><math>k</math></span> of an AFSp is: | ||
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</math> | </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 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 [[OS#Derivation|derivation of OS]]. | |||
=== Vs. EFD === | |||
By specifying n, your sequence will be equivalent to one period of 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. | 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. | ||
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!8 | !8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(1 + 0/⁴√2) | |(1 + 0/⁴√2) | ||
|1 + 1/⁴√2 | |1 + 1/⁴√2 | ||
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|1 + 8/⁴√2 | |1 + 8/⁴√2 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
| (0) || 0.88 | | (0) || 0.88 | ||
|1.42 | |1.42 | ||
| Line 56: | Line 70: | ||
|2.95 | |2.95 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
| (1) || 0.54 | | (1) || 0.54 | ||
|0.37 | |0.37 | ||
| Line 67: | Line 81: | ||
|} | |} | ||
[[Category:Otonality]] | [[Category:Otonality]] | ||
[[Category:Harmonic]] | [[Category:Harmonic]] | ||
[[Category:Harmonic series]] | [[Category:Harmonic series]] | ||
[[Category:Xenharmonic series]] | |||