AFS: Difference between revisions
Cmloegcmluin (talk | contribs) add formula for mathematician benefit |
Cmloegcmluin (talk | contribs) break up wall of information into helpful sections that are consistent across all arithmetic tuning pages |
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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 (irrtional) 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 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. | 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. | ||
Revision as of 20:38, 24 March 2021
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 (irrtional) 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 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 derivation of OS.
vs. EFD
By specifying n, your sequence will be equivalent to 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:
| quantity | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| frequency (f) | (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) | (0) | 0.88 | 1.42 | 1.82 | 2.13 | 2.38 | 2.60 | 2.78 | 2.95 |
| length (1/f) | (1) | 0.54 | 0.37 | 0.28 | 0.23 | 0.19 | 0.17 | 0.15 | 0.13 |