Maximal evenness: Difference between revisions
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<math>\operatorname{ME}(n, m) = \{ \mathbb{Z} + \lceil \frac{im}{n} \rceil : i \in \{1, ..., n\} \} \subseteq \mathbb{Z}/m\mathbb{Z},</math> | <math>\operatorname{ME}(n, m) = \{ \mathbb{Z} + \lceil \frac{im}{n} \rceil : i \in \{1, ..., n\} \} \subseteq \mathbb{Z}/m\mathbb{Z},</math> | ||
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 [[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|MOS subset]] of '''Z'''/''m'''''Z''' where the two step sizes differ by exactly 1, and that the indices for the two step sizes are themselves ME as subsets of '''Z'''/''n'''''Z''', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically brightest mode among its rotations, and combined with the fact that it is a MOS, this implies that ME(''n'', ''m'') is the brightest mode in the MOS sense. | ||
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 [[chirality|achiral]]. | 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 [[chirality|achiral]]. | ||