User:Cmloegcmluin/APS: Difference between revisions
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m Added to category "xenharmonic series" |
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. | 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-) | 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. | ||
== Formula == | == Formula == | ||
The pitch of the | The pitch of the ''k''-th step of an APS-''p'' is quite simply ''k''⋅''p''. | ||
== Relationship to other tunings == | == Relationship to other tunings == | ||
=== Vs. rank-1 temperaments | === Vs. rank-1 temperaments === | ||
By applying a [[mapping]], APS-''p'' becomes an [[equal temperament]] with generator ''p''. | |||
=== Vs. EPD === | === Vs. EPD === | ||
If specified, an APS will be equivalent to one period of some [[EPD|EPD, or equal pitch division]]. Specifically, n- | If specified, an APS will be equivalent to one period of some [[EPD|EPD, or equal pitch division]]. Specifically, ''n''-EPD-''x'' = ''n''-APS(''x''/''n''), for example 12-EPD1200¢ = 12-APS(1200¢/12) = 12-APS100¢. | ||
=== Vs. AS === | === Vs. AS === | ||
The only difference between an APS and an [[AS|AS (ambitonal sequence)]] is that the p for an AS must be rational. | The only difference between an APS and an [[AS|AS (ambitonal sequence)]] is that the ''p'' for an AS must be rational. | ||
== Examples == | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+Example: APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢ | ||
|- | |- | ||
! | ! Quantity | ||
! (0) | ! (0) | ||
! 1 | ! 1 | ||
| Line 35: | Line 35: | ||
! 4 | ! 4 | ||
|- | |- | ||
! | ! Frequency (''f'') | ||
|(1) | | (1) | ||
|1.19 | | 1.19 | ||
|1.41 | | 1.41 | ||
|1.68 | | 1.68 | ||
|2 | | 2 | ||
|- | |- | ||
! | ! Pitch (log₂''f'') | ||
|(2⁰⸍⁴) | | (2⁰⸍⁴) | ||
|2¹⸍⁴ | | 2¹⸍⁴ | ||
|2²⸍⁴ | | 2²⸍⁴ | ||
|2³⸍⁴ | | 2³⸍⁴ | ||
|2⁴⸍⁴ | | 2⁴⸍⁴ | ||
|- | |- | ||
! | ! Length (1/''f'') | ||
|(1) | | (1) | ||
|0.84 | | 0.84 | ||
|0.71 | | 0.71 | ||
|0.59 | | 0.59 | ||
|0.5 | | 0.5 | ||
|} | |} | ||
Revision as of 09:25, 14 October 2023
An APS, or arithmetic pitch sequence, is a kind of arithmetic and 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.
Formula
The pitch of the k-th step of an APS-p is quite simply k⋅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 specified, an APS will be equivalent to one period of some EPD, or equal pitch division. Specifically, n-EPD-x = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12) = 12-APS100¢.
Vs. AS
The only difference between an APS and an AS (ambitonal sequence) is that the p for an AS must be rational.
Examples
| Quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency (f) | (1) | 1.19 | 1.41 | 1.68 | 2 |
| Pitch (log₂f) | (2⁰⸍⁴) | 2¹⸍⁴ | 2²⸍⁴ | 2³⸍⁴ | 2⁴⸍⁴ |
| Length (1/f) | (1) | 0.84 | 0.71 | 0.59 | 0.5 |