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.

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Case 1: ME(''n'', ''m'') where {{nowrap|''n'' < {{frac|''m''|2}}}}. This is a maximally even subset of Z/mZ with step sizes {{nowrap|L > s > 1}}, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are {{nowrap|L − 2}} and {{nowrap|s − 2}} (0 is a valid chunk size), and the sizes form a maximally even MOS.

Case 2: ME(''n'', ''m'') where {{nowrap|''n'' > {{frac|''m''|2}}}}. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since {{nowrap|''n'' > {{frac|''m''|2}}}} occupy a maximally even subset of the slots of ME(''n'', ''m'') (*). Now replace each 1 with "|" and each 2 with "$|".

Case 2: ME(''n'', ''m'') where {{nowrap|''n'' > {{frac|''m''|2}}}}. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since {{nowrap|''n'' > {{frac|''m''|2}}}}) occupy a maximally even subset of the slots of ME(''n'', ''m'') (*). Now replace each 1 with "|" and each 2 with "$|".

(e.g.) {{nowrap|2112111 → <nowiki>$|||$||||</nowiki>}}

Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*). The desired complement, occupied by the "$"'s, thus forms a maximally even subset.

~~== Concoctic scales ==~~

The generator of a maximally even scale is a modular multiplicative inverse of it's number of notes and the EDO size. A maximal even 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.

== Sound perception ==

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Let's say we want to see what would repeatedly stacking 11th harmonic do well in all of 11-limit, in an EDO that presents it well.

11/8 amounts to 17 steps of 37edo, and the solution to the problem {{nowrap|17*''x'' (mod 1) {{=}} 37}} is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is 24 & 37, which can be interpreted as [[freivald]] or [[emka]].

11/8 amounts to 17 steps of 37edo, and the solution to the problem {{nowrap|17*''x'' (mod 1) {{=}} 37}} is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is {{nowrap|24 & 37}}, which can be interpreted as [[freivald]] or [[emka]].

=== Example 3: On-request maximum evenness scales ===

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We simply merge {{nowrap|52 & 293}} in a selected limit to get our answer. Let's say 17 limit, we get a {{nowrap|52 & 243c}} temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025.

~~Let's see what temperament does the Tabular Persian or Dee calendar offer {{nowrap|(29 & 33)}}. In the 5-limit, we get a contorted Lala-Quinyo (553584375:536870912).~~

~~== Sonifications ==~~

* Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the {{w|Hebrew calendar}}.

* Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the {{w|Iranian calendars|Persian calendar}}.

[[Category:Scale]]

[[Category:Todo:cleanup]]

@@ 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.
@@ Line 21: / Line 21: @@
 Case 1: ME(''n'',&nbsp;''m'') where {{nowrap|''n'' &lt; {{frac|''m''|2}}}}. This is a maximally even subset of Z/mZ with step sizes {{nowrap|L &gt; s &gt; 1}}, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are {{nowrap|L − 2}} and {{nowrap|s − 2}} (0 is a valid chunk size), and the sizes form a maximally even MOS.
-Case 2: ME(''n'',&nbsp;''m'') where {{nowrap|''n'' &gt; {{frac|''m''|2}}}}. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since {{nowrap|''n'' &gt; {{frac|''m''|2}}}} occupy a maximally even subset of the slots of ME(''n'',&nbsp;''m'') (*). Now replace each 1 with "|" and each 2 with "$|".
+Case 2: ME(''n'',&nbsp;''m'') where {{nowrap|''n'' &gt; {{frac|''m''|2}}}}. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since {{nowrap|''n'' &gt; {{frac|''m''|2}}}}) occupy a maximally even subset of the slots of ME(''n'',&nbsp;''m'') (*). Now replace each 1 with "|" and each 2 with "$|".
 (e.g.) {{nowrap|2112111 → <nowiki>$|||$||||</nowiki>}}
 Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*). The desired complement, occupied by the "$"'s, thus forms a maximally even subset.
-== Concoctic scales ==
-The generator of a maximally even scale is a modular multiplicative inverse of it's number of notes and the EDO size. A maximal even 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.
 == Sound perception ==
@@ Line 61: / Line 58: @@
 Let's say we want to see what would repeatedly stacking 11th harmonic do well in all of 11-limit, in an EDO that presents it well.
-/8 amounts to 17 steps of 37edo, and the solution to the problem {{nowrap|17*''x'' (mod 1) {{=}} 37}} is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is 24 & 37, which can be interpreted as [[freivald]] or [[emka]].
+/8 amounts to 17 steps of 37edo, and the solution to the problem {{nowrap|17*''x'' (mod 1) {{=}} 37}} is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is {{nowrap|24 &amp; 37}}, which can be interpreted as [[freivald]] or [[emka]].
 === Example 3: On-request maximum evenness scales ===
@@ Line 67: / Line 64: @@
 We simply merge {{nowrap|52 &amp; 293}} in a selected limit to get our answer. Let's say 17 limit, we get a {{nowrap|52 &amp; 243c}} temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025.
-Let's see what temperament does the Tabular Persian or Dee calendar offer {{nowrap|(29 &amp; 33)}}. In the 5-limit, we get a contorted Lala-Quinyo (553584375:536870912).
-== Sonifications ==
-* Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the {{w|Hebrew calendar}}.
-* Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the {{w|Iranian calendars|Persian calendar}}.
 [[Category:Scale]]
 [[Category:Todo:cleanup]]