Maximal evenness: Difference between revisions

Line 6:

== Mathematics ==

Mathematically, if ''n'' < ''m'', a ME scale of ''n'' notes in ''m''-ed is any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function rounds down to the ~~nearest~~ integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''ed, and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed, satisfying the informal definition above.

Mathematically, if ''n'' < ''m'', a ME scale of ''n'' notes in ''m''-ed is any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function fixes integers and rounds down non-integers to the next lower integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''ed, and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed, satisfying the informal definition above.

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.

== Sound perception ==

@@ Line 6: / Line 6: @@
 == Mathematics ==
-Mathematically, if ''n'' < ''m'', a ME scale of ''n'' notes in ''m''-ed is any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function rounds down to the nearest integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''ed, and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed, satisfying the informal definition above.
+Mathematically, if ''n'' < ''m'', a ME scale of ''n'' notes in ''m''-ed is any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function fixes integers and rounds down non-integers to the next lower integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''ed, and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed, satisfying the informal definition above.
 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.
 == Sound perception ==