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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.


shifted undertone series (± frequency) (equivalent to ALS)
== Specification ==


(n-)ALSp: (n pitches of an) arithmetic length sequence adding by p
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 is a specific (rational) type of ALS.
== Formula ==


The same principles that were just described for frequency are also possible for length: by varying the undertone series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an arithmetic length sequence (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.
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"
{| class="wikitable"
|+example: (1/⁴√2)-shifted undertone series segment = 9-ALS(1/⁴√2)
|+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
! quantity
! (0)
! 1
! 1
! 2
! 2
Line 21: Line 51:
! 7
! 7
! 8
! 8
! 9
|-
|-
! frequency
! frequency (''f'', ratio)
|1.00
|(1)
|1.12
|1.12
|1.28
|1.28
Line 34: Line 63:
|7.73
|7.73
|-
|-
! pitch
! pitch (log₂''f'', octaves)
|0.00
|(0)
|0.17
|0.17
|0.35
|0.35
Line 45: Line 74:
|2.95
|2.95
|-
|-
! length
! length (1/''f'', ratio)
|1.00
|(1)
|0.89
|0.89
|0.78
|0.78
Line 56: Line 85:
|0.13
|0.13
|}
|}
{| class="wikitable"
|+example: (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2)
''(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]]
Retrieved from "https://en.xen.wiki/w/ALS"