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. | An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
The n is optional. If not provided, the sequence is open-ended. 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¢). | 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. | ||
== Formula == | |||
The pitch of the kth step of an APSp is quite simply k⋅p. | |||
== Relationship to other tunings == | |||
=== vs. rank-1 temperaments & equal multiplications === | |||
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. | |||
=== 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¢). | |||
=== 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 APS is irrational. | ||
== Examples == | |||
{| class="wikitable" | {| class="wikitable" | ||
Revision as of 20:38, 24 March 2021
An APS, or arithmetic pitch sequence, is a kind of arithmetic and harmonotonic tuning.
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.
Formula
The pitch of the kth step of an APSp is quite simply k⋅p.
Relationship to other tunings
vs. rank-1 temperaments & equal multiplications
An APSp is equivalent to a rank-1 temperament with generator p. It is also equivalent to an equal multiplication of p.
vs. EPD
If specified, an APS will be equivalent to some EPD, or equal pitch division. Specifically, n-EPDx = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12=100¢).
vs. AS
The only difference between an APS and an AS (ambitonal sequence) is that the p for an APS is irrational.
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 |