Maximal evenness: Difference between revisions

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A '''maximally even''' ('''ME''') scale is a [[scale]] inscribed in an [[equal-step tuning]] which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of '''maximal evenness'''. ~~These conditions entail that an ME scale is necessarily an [[MOS scale]].~~ Mathematically, 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.

A '''maximally even''' ('''ME''') scale is a [[scale]] inscribed in an [[equal-step tuning]] which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of '''maximal evenness'''.

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 is easy to prove 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.

In particular, within every [[EDO|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.

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 {{Distinguish|Distributional evenness}}
-A '''maximally even''' ('''ME''') scale is a [[scale]] inscribed in an [[equal-step tuning]] which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of '''maximal evenness'''. These conditions entail that an ME scale is necessarily an [[MOS scale]]. Mathematically, 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.
+A '''maximally even''' ('''ME''') scale is a [[scale]] inscribed in an [[equal-step tuning]] which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of '''maximal evenness'''.
+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 is easy to prove 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.
 In particular, within every [[EDO|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.