Maximal evenness: Difference between revisions

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In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.

== ~~Mathematics~~ ==

== Formal definition ==

Mathematically, if ''n'' < ''m'', a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set

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It is easy to show that using round() (rounding half-integers up) gives an equivalent definition; floor() does too, since ME(''n'', ''m'') is a MOS and thus achiral.

== Concoctic scales ==

From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. ~~Maximal evenness~~ scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.

From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. A maximal even scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.

== Sound perception ==

@@ Line 5: / Line 5: @@
 In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.
-== Mathematics ==
+== Formal definition ==
 Mathematically, if ''n'' < ''m'', a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set
@@ Line 13: / Line 13: @@
 It is easy to show that using round() (rounding half-integers up) gives an equivalent definition; floor() does too, since ME(''n'', ''m'') is a MOS and thus achiral.
+== Concoctic scales ==
-From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. Maximal evenness scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.
+From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. A maximal even scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.
 == Sound perception ==