Maximal evenness: Difference between revisions

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where the [[Wikipedia:Floor and ceiling functions|ceiling]] function fixes integers and rounds up non-integers to the next higher integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''⟨''E''⟩ when interpreted as a subset of ''m''-ed''E'', and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed''E'', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically brightest mode among its rotations.

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.

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.

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 where the [[Wikipedia:Floor and ceiling functions|ceiling]] function fixes integers and rounds up non-integers to the next higher integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''⟨''E''⟩ when interpreted as a subset of ''m''-ed''E'', and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed''E'', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically brightest mode among its rotations.
+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.
 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.