User:Cmloegcmluin/APS: Difference between revisions

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-An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] 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'''.
 == Specification ==
-Its full specification is (n-)APSp: (n pitches of an) arithmetic pitch sequence adding by irrational interval p. The n is optional. If not provided, the sequence is open-ended.
+Its full specification is (''n''-)APS-''p'': (''n'' pitches of an) arithmetic pitch sequence adding by 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.
 == Formula ==
-The pitch of the kth step of an APSp is quite simply k⋅p.
+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 & equal multiplications ===
+=== Vs. rank-1 temperaments ===
-An APSp is equivalent to a [[Tour_of_Regular_Temperaments#Equal_temperaments_.28Rank-1_temperaments.29|rank-1 temperament]] with generator p. It is also equivalent to an [[Equal-step_tuning#Equal_multiplications|equal multiplication]] of p.
+By applying a [[mapping]], APS-''p'' becomes an [[equal temperament]] with generator ''p''.
 === Vs. EPD ===
-If specified, an APS will be equivalent to some [[EPD|EPD, or equal pitch division]]. Specifically, n-EPDx = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12=100¢).
+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 APS is irrational.
+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:  APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢
+|+Example:  APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢
 |-
-! quantity
+! Quantity
 ! (0)
 ! 1
@@ Line 35: / 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]]