User:Cmloegcmluin/AS: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
-An '''AS''', or '''ambitonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+{{Editable user page}}
+An '''AS''', or '''ambitonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-An AS is a specific (rational) type of APS.
+== Specification ==
-(n-)ASp: (n pitches of an) ambitonal sequence adding by p (equivalent to 1D JI lattice of p)
+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"
+|+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]]