User:Cmloegcmluin/AS: Difference between revisions
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An '''AS''', or '''ambitonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | An '''AS''', or '''ambitonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | == Specification == | ||
Its full specification is (n-) | Its full specification is (''n''-)AS-''p'': (''n'' pitches of an) [[ambitonal]] sequence adding by rational interval ''p''. | ||
'''Note''': | |||
* The ''n'' is optional. If not provided, the sequence is open-ended. | |||
* The ''p'' can be dimensionless, in which case it refers to an interval by its [[frequency ratio]]. It can also take a unit proportional to [[octave]]s, in which case it refers to an interval by its pitch relation. | |||
== Relationship to other tunings == | == Relationship to other tunings == | ||
| Line 9: | Line 14: | ||
=== Vs. 1D JI Lattice & equal multiplications === | === Vs. 1D JI Lattice & equal multiplications === | ||
AS-''p'' is equivalent to a 1-dimensional [[Harmonic lattice diagram|JI lattice]] of ''p''. These are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. It is also equivalent to an [[equal multiplication]] of a rational interval ''p''. | |||
=== Vs. APS === | === Vs. APS === | ||
The only difference between an (n-) | The only difference between an (''n''-)AS-''p'' and an [[APS|(''n''-)APS-''p'' (arithmetic pitch sequence)]] is that the ''p'' for an AS must be rational. | ||
== Examples == | == Examples == | ||
| Line 31: | Line 36: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(5⁰/4⁰) | |(5⁰/4⁰) | ||
|5¹/4¹ | |5¹/4¹ | ||
| Line 42: | Line 47: | ||
|5⁸/4⁸ | |5⁸/4⁸ | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.32 | |0.32 | ||
| Line 53: | Line 58: | ||
|2.58 | |2.58 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(1/1) | |(1/1) | ||
|4/5 | |4/5 | ||
| Line 64: | Line 69: | ||
|65536/390625 | |65536/390625 | ||
|} | |} | ||
== List of ASs == | |||
{{See also| APS #List of APSs }} | |||
; [[Superparticular]] | |||
* [[1ed8/7|AS8/7]] | |||
* [[1ed9/8|AS9/8]] | |||
* [[1ed10/9|AS10/9]] | |||
* [[1ed15/14|AS15/14]] | |||
* [[1ed16/15|AS16/15]] | |||
* [[1ed18/17|AS18/17]] | |||
* [[1ed21/20|AS21/20]] | |||
* [[1ed25/24|AS25/24]] | |||
* [[1ed26/25|AS26/25]] | |||
* [[1ed33/32|AS33/32]] | |||
* [[1ed81/80|AS81/80]] | |||
; Others | |||
* [[1ed13/10|AS13/10]] | |||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||
[[Category:Equal divisions of the | [[Category:Equal divisions of the octave]] | ||
[[Category:Xenharmonic series]] | |||
Latest revision as of 01:40, 19 May 2025
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An AS, or ambitonal sequence, is a kind of arithmetic and harmonotonic tuning.
Specification
Its full specification is (n-)AS-p: (n pitches of an) ambitonal sequence adding by rational interval p.
Note:
- The n is optional. If not provided, the sequence is open-ended.
- The p can be dimensionless, in which case it refers to an interval by its frequency ratio. It can also take a unit proportional to octaves, in which case it refers to an interval by its pitch relation.
Relationship to other tunings
Vs. 1D JI Lattice & equal multiplications
AS-p is equivalent to a 1-dimensional JI lattice of p. These are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. It is also equivalent to an equal multiplication of a rational interval p.
Vs. APS
The only difference between an (n-)AS-p and an (n-)APS-p (arithmetic pitch sequence) is that the p for an AS must be rational.
Examples
| quantity | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| frequency (f, ratio) | (5⁰/4⁰) | 5¹/4¹ | 5²/4² | 5³/4³ | 5⁴/4⁴ | 5⁵/4⁵ | 5⁶/4⁶ | 5⁷/4⁷ | 5⁸/4⁸ |
| pitch (log₂f, octaves) | (0) | 0.32 | 0.64 | 0.97 | 1.29 | 1.61 | 1.93 | 2.25 | 2.58 |
| length (1/f, ratio) | (1/1) | 4/5 | 16/25 | 64/125 | 256/625 | 1024/3125 | 4096/15625 | 16384/78125 | 65536/390625 |
List of ASs
- Others