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]]. It can also be called an '''equal multiplication'''.

== Specification ==

Its full specification is (''n''-)APS-''p'': (''n'' pitches of an) arithmetic pitch sequence adding by 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~~ ''k''~~-th step~~ of an APS-''p'' is quite simply ''k''⋅''p'' for a pitch (log-frequency) quantity ''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 ==

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=== 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 ~~log-frequency~~ quantity ''p''. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.

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 ===

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! 4

|-

! ~~Frequency~~ (''f'')

! frequency (''f'', ratio)

| (1)

| 1.19

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| 2

|-

! ~~Pitch~~ (~~log₂~~''f'')

! length (1/''f'', ratio)

| (0/4)

| 1/4

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| 0.5

|}

== List of notable APSs ==

* 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:Xenharmonic series]]

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+{{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''-)APS-''p'': (''n'' pitches of an) arithmetic pitch sequence adding by 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 ''k''-th step of an APS-''p'' is quite simply ''k''⋅''p'' for a pitch (log-frequency) quantity ''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 ==
@@ Line 17: / Line 22: @@
 === 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 log-frequency quantity ''p''. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.
+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 ===
@@ Line 35: / Line 40: @@
 ! 4
 |-
-! Frequency (''f'')
+! frequency (''f'', ratio)
 | (1)
 | 1.19
@@ Line 42: / Line 47: @@
 | 2
 |-
-! Pitch (log₂''f'')
+! length (1/''f'', ratio)
 | (0/4)
 | 1/4
@@ Line 56: / Line 61: @@
 | 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:Xenharmonic series]]