ALS: Difference between revisions

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-An '''ALS''', or '''arithmetic length sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+An '''ALS''', or '''arithmetic length sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-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.
+== 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.
-A [[US|US, or utonal sequence]], is a specific (rational) type of ALS. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an ALS. 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.
+== 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)]].
+=== 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"
 |+example: (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2)
+''(arranged so that the pitches are in ascending order and still begin on 1/1)''
 |-
 ! quantity
@@ Line 21: / Line 52: @@
 ! 8
 |-
-! frequency (f)
+! frequency (''f'', ratio)
 |(1)
 |1.12
@@ Line 32: / Line 63: @@
 |7.73
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
 |(0)
 |0.17
@@ Line 43: / Line 74: @@
 |2.95
 |-
-! length (1/f)
+! length (1/''f'', ratio)
 |(1)
 |0.89
@@ Line 55: / Line 86: @@
 |}
-[[Category:Undertone]]
+{| class="wikitable"
-[[Category:Undertone series]]
+|+example: (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2)
-[[Category:Utonality]]
+''(descending pitches)''
+|-
+! quantity
+! (0)
+! 1
+! 2
+! 3
+! 4
+! 5
+! 6
+! 7
+! 8
+|-
+! frequency (''f'', ratio)
+|(1)
+|0.54
+|0.37
+|0.28
+|0.23
+|0.19
+|0.17
+|0.15
+|0.13
+|-
+! pitch (log₂''f'', octaves)
+|(0)
+| -0.88
+| -1.42
+| -1.82
+| -2.13
+| -2.38
+| -2.60
+| -2.78
+| -2.95
+|-
+! length (1/''f'', ratio)
+|(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
+|}
 [[Category:Subharmonic]]
 [[Category:Subharmonic series‏‎]]
+[[Category:Utonality]]
+[[Category:Xenharmonic series]]