Maximal evenness: Difference between revisions
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Mathematically, if 0 < ''n'' < ''m'', a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set | Mathematically, if 0 < ''n'' < ''m'', a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set | ||
<math>\operatorname{ME}(n, m) = \{ \mathbb{Z} + \lceil \frac{im}{n} \rceil : i \in \{0, ..., n-1\} \} \subseteq \mathbb{Z}/m\mathbb{Z},</math> | <math>\operatorname{ME}(n, m) = \{ m\mathbb{Z} + \lceil \frac{im}{n} \rceil : i \in \{0, ..., n-1\} \} \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|MOS subset]] of '''Z'''/''m'''''Z''' where the two step sizes differ by exactly 1, and that the set of degrees where each step size occurs is itself maximally even in '''Z'''/''n'''''Z''', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically first 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. | 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 set of degrees where each step size occurs is itself maximally even in '''Z'''/''n'''''Z''', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically first 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. | ||