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.

Its full specification is (n-)APSp: (n pitches of an) arithmetic pitch sequence adding by irrational interval p. It 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.

== 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 pitch of the kth step of an APSp is quite simply k⋅p.~~

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is (n-)APSp: (n pitches of an) arithmetic pitch sequence adding by irrational interval p. It 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.
+== 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 pitch of the kth step of an APSp is quite simply k⋅p.
+== Examples ==
 {| class="wikitable"