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An '''ALS''', or '''arithmetic length sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | An '''ALS''', or '''arithmetic length 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. | Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency. The n is optional. If not provided, the sequence is open-ended. | ||
== Formula == | |||
The formula for length <span><math>k</math></span> of an ALSp is: | |||
<math> | |||
L(k) = 1 + k⋅p | |||
</math> | |||
== Tip about tunings based on length == | |||
Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available. | |||
== Relationship to other tunings == | |||
=== As shifted undertone series === | |||
By varying the undertone series step size to some number other than 1, if that number is irrational you can produce an ALS, and if rational you can produce a US (which you could also call an ALS if you really wanted to). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes. | |||
=== Vs. AFS === | |||
The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]]. | The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]]. | ||
A [[US|US, or utonal sequence]], is | === Vs. US === | ||
A [[US|US, or utonal sequence]], is the rational version of an ALS. | |||
=== Vs. ELD === | |||
By specifying n, your sequence will be equivalent to one period of some [[ELD|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n). | |||
== Examples == | |||
{| class="wikitable" | {| class="wikitable" | ||
| Line 24: | Line 52: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(1) | |(1) | ||
|1.12 | |1.12 | ||
| Line 35: | Line 63: | ||
|7.73 | |7.73 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.17 | |0.17 | ||
| Line 46: | Line 74: | ||
|2.95 | |2.95 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(1) | |(1) | ||
|0.89 | |0.89 | ||
| Line 73: | Line 101: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(1) | |(1) | ||
|0.54 | |0.54 | ||
| Line 84: | Line 112: | ||
|0.13 | |0.13 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
| -0.88 | | -0.88 | ||
| Line 95: | Line 123: | ||
| -2.95 | | -2.95 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(1 + 0/⁴√2) | |(1 + 0/⁴√2) | ||
|1 + 1/⁴√2 | |1 + 1/⁴√2 | ||
| Line 107: | Line 135: | ||
|} | |} | ||
[[Category:Subharmonic]] | [[Category:Subharmonic]] | ||
[[Category:Subharmonic series]] | [[Category:Subharmonic series]] | ||
[[Category:Utonality]] | |||
[[Category:Xenharmonic series]] | |||