User:Cmloegcmluin/APS: Difference between revisions
Equal multiplication isn't an "other tuning". This *is* equal multiplication. Also adopt a stricter def for "equal temperament" |
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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'''. | 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 == | == 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 == | == Formula == | ||
The pitch of | The pitch of ''k'' steps of APS-''p'' is quite simply ''k''⋅''p'' for a pitch (log-frequency) quantity ''p''. | ||
== Relationship to other tunings == | == Relationship to other tunings == | ||
| Line 17: | Line 22: | ||
=== Vs. EPD === | === Vs. EPD === | ||
If specified, an APS will be equivalent to | 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 === | === Vs. AS === | ||
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! 4 | ! 4 | ||
|- | |- | ||
! | ! frequency (''f'', ratio) | ||
| (1) | | (1) | ||
| 1.19 | | 1.19 | ||
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| 2 | | 2 | ||
|- | |- | ||
! | ! length (1/''f'', ratio) | ||
| ( | | (0/4) | ||
| | | 1/4 | ||
| | | 2/4 | ||
| | | 3/4 | ||
| | | 4/4 | ||
|- | |- | ||
! Length (1/''f'') | ! Length (1/''f'') | ||
| Line 56: | Line 61: | ||
| 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-step tuning]] | ||
[[Category:Xenharmonic series]] | [[Category:Xenharmonic series]] | ||