Maximal evenness: Difference between revisions

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Mathematically, if ''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} + \~~lfloor~~ \frac{im}{n} \~~rfloor~~ : 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|~~floor~~]] function fixes integers and rounds ~~down~~ non-integers to the next ~~lower~~ 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 ~~darkest~~ mode among its rotations.

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.

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.

@@ Line 8: / Line 8: @@
 Mathematically, if ''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} + \lfloor \frac{im}{n} \rfloor : 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|floor]] function fixes integers and rounds down non-integers to the next lower 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 darkest mode among its rotations.
+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.
 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.