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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 == | |||
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 == | |||
=== 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" | |||
|+example: AS5/4 = 1D JI lattice of 5/4 = 5/4 chain | |||
|- | |||
! 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 (log₂''f'', 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 == | |||
{{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]] | |||