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An '''AS''', or '''ambitonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[ | {{Editable user page}} | ||
An '''AS''', or '''ambitonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | |||
== Specification == | |||
(n-) | Its full specification is (''n''-)AS-''p'': (''n'' pitches of an) [[ambitonal]] sequence adding by rational interval ''p''. | ||
These are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. | '''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 == | |||
=== 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 === | |||
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 == | |||
{| class="wikitable" | {| class="wikitable" | ||
|+example: | |+example: AS5/4 = 1D JI lattice of 5/4 = 5/4 chain | ||
|- | |- | ||
! quantity | ! quantity | ||
! (0) | |||
! 1 | ! 1 | ||
! 2 | ! 2 | ||
| Line 19: | Line 35: | ||
! 7 | ! 7 | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency | ! frequency (''f'', ratio) | ||
| | |(5⁰/4⁰) | ||
| | |5¹/4¹ | ||
| | |5²/4² | ||
| | |5³/4³ | ||
| | |5⁴/4⁴ | ||
| | |5⁵/4⁵ | ||
| | |5⁶/4⁶ | ||
| | |5⁷/4⁷ | ||
| | |5⁸/4⁸ | ||
|- | |- | ||
! pitch | ! pitch (log₂''f'', octaves) | ||
| | |(0) | ||
| | |0.32 | ||
| | |0.64 | ||
| | |0.97 | ||
| | |1.29 | ||
| | |1.61 | ||
| | |1.93 | ||
| | |2.25 | ||
| | |2.58 | ||
|- | |- | ||
! length | ! 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 == | |||
{{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 divisions of the octave]] | |||
[[Category:Xenharmonic series]] | |||