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" |
Every step of EPD is a period so it doesn't make sense to compare n-APS to EPD. It's the unspecified APS that's equivalent to EPD. Try to clarify the dimensionality of the "p" |
||
| Line 7: | Line 7: | ||
== Formula == | == Formula == | ||
The pitch of the ''k''-th step of an APS-''p'' is quite simply ''k''⋅''p''. | The pitch of the ''k''-th step of an 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 17: | ||
=== 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 log-frequency quantity ''p''. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢. | ||
=== Vs. AS === | === Vs. AS === | ||
| Line 58: | Line 58: | ||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||
[[Category:Xenharmonic series]] | [[Category:Xenharmonic series]] | ||
Revision as of 10:31, 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 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 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¢.
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 |