User:Cmloegcmluin/APS: Difference between revisions

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-An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+{{Editable user page}}
+An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] [[tuning]]. It can also be called an '''equal multiplication'''.
-(n-)APSp: (n pitches of an) arithmetic pitch sequence adding by p (equivalent to rank-1 temperament with generator p)
+== Specification ==
-An AS is a specific (rational) type of APS.
+Its full specification is (''n''-)APS-''p'': (''n'' pitches of an) arithmetic pitch sequence adding by interval ''p''.
-equal multiplications https://en.xen.wiki/w/Equal-step_tuning#Equal_multiplications
+'''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.
+== Formula ==
+The pitch of ''k'' steps of APS-''p'' is quite simply ''k''⋅''p'' for a pitch (log-frequency) quantity ''p''.
+== Relationship to other tunings ==
+=== Vs. rank-1 temperaments ===
+By applying a [[mapping]], APS-''p'' becomes an [[equal temperament]] with generator ''p''.
+=== Vs. EPD ===
+If the ''n'' is not specified, an APS will be equivalent to an [[EPD|equal pitch division (EPD)]]. Specifically, ''n''-EPD-''p'' = APS(''p''/''n'') for a pitch quantity ''p''. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.
+=== Vs. AS ===
+The only difference between an APS and an [[AS|AS (ambitonal sequence)]] is that the ''p'' for an AS must be rational.
+== Examples ==
 {| class="wikitable"
-|+example: 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189
+|+Example:  APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢
 |-
-! quantity
+! Quantity
 ! (0)
 ! 1
@@ Line 17: / Line 40: @@
 ! 4
 |-
-! frequency (f)
+! frequency (''f'', ratio)
-|(1)
+| (1)
-|1.19
+| 1.19
-|1.41
+| 1.41
-|1.68
+| 1.68
-|2
+| 2
 |-
-! pitch (log₂f)
+! length (1/''f'', ratio)
-|(2⁰⸍⁴)
+| (0/4)
-|2¹⸍⁴
+| 1/4
-|2²⸍⁴
+| 2/4
-|2³⸍⁴
+| 3/4
-|2⁴⸍⁴
+| 4/4
 |-
-! length (1/f)
+! Length (1/''f'')
-|(1)
+| (1)
-|0.84
+| 0.84
-|0.71
+| 0.71
-|0.59
+| 0.59
-|0.5
+| 0.5
 |}
+== List of notable APSs ==
+{{See also| AS #List of ASs }}
+* APS35.099¢, tuning of [[Carlos Gamma]]
+* APS63.833¢, tuning of [[Carlos Beta]]
+* [[1ed69c|APS69¢]]
+* APS77.965¢, tuning of [[Carlos Alpha]]
+* [[1ed86.4c|APS86.4¢]]
+* [[88cET|APS88¢]]
+* [[1ed97.5c|APS97.5¢]]
+* [[1ed125c|APS125¢]]
+For a more complete list, see [[Gallery of arithmetic pitch sequences]]. But do note that the gallery includes many obscure tunings that are of less importance to most xenharmonicists compared to the more curated selection listed above.
 [[Category:Equal-step tuning‏‎]]
-[[Category:Equal divisions of the octave‏‎ ]]
+[[Category:Xenharmonic series]]