Maximal evenness: Difference between revisions

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Mathematically, if {{nowrap|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) = \{ m\mathbb{Z} + \ceil{\frac{im}{n}} : i \in \{0, ..., n-1\} \} \subseteq \mathbb{Z}/m\mathbb{Z},</math>

<math>\operatorname{ME}(n, m) = \left\{ m\mathbb{Z} + \ceil{\frac{im}{n}} : i \in \{0, ..., n-1\} \right\} \subseteq \mathbb{Z}/m\mathbb{Z},</math>

where the {{w|ceiling function}} fixes integers and rounds up non-integers to the next higher integer. It can be proven that when ''n'' does not divide ''m'', 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.

@@ Line 10: / Line 10: @@
 Mathematically, if {{nowrap|0 &lt; ''n'' &lt; ''m''}}, a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set
-<math>\operatorname{ME}(n, m) = \{ m\mathbb{Z} + \ceil{\frac{im}{n}} : i \in \{0, ..., n-1\} \} \subseteq \mathbb{Z}/m\mathbb{Z},</math>
+<math>\operatorname{ME}(n, m) = \left\{ m\mathbb{Z} + \ceil{\frac{im}{n}} : i \in \{0, ..., n-1\} \right\} \subseteq \mathbb{Z}/m\mathbb{Z},</math>
 where the {{w|ceiling function}} fixes integers and rounds up non-integers to the next higher integer. It can be proven that when ''n'' does not divide ''m'', ME(''n'',&nbsp;''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'',&nbsp;''m'') is the lexicographically first mode among its rotations, and combined with the fact that it is a MOS, this implies that ME(''n'',&nbsp;''m'') is the brightest mode in the MOS sense.