Maximal evenness: Difference between revisions

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In particular, within every [[EDO|edo]] one can specify such a scale for every smaller number of notes.In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo. Mathematically, ME scales of ''n'' notes in ''m'' edo are any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function rounds down to the nearest integer.

~~A special case~~ of the ~~maximally even~~ scale is ~~the [[Irvian mode]], which originates from~~ a ~~calendar reform to smoothly spread inaccuracies arising from the uneven~~ number of ~~days or weeks per year~~. ~~For example, the major mode of the basic [[5L 2s|diatonic]]~~ scale ~~from~~ [[~~12edo~~]]~~, <span style="font~~-~~family: monospace;">2 2 1 2 2 2 1</span>,~~ is ~~not only~~ a ~~maximally even scale~~, ~~but also~~ the ~~Irvian mode of such~~ scale~~. Every mode of any diatonic scale is maximally even, but not necessarily Irvian~~.

== Mathematics ==

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.

== Sound perception ==

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Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel_Mandelbaum|Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.

[[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic [[5L 2s|diatonic]] scale from [[12edo]], <span style="font-family: monospace;">2 2 1 2 2 2 1</span>, is not only a maximally even scale, but also the Irvian mode of such scale. Such a mode is best shown in odd EDOs, which truly have a "middle" note owing to being odd, and therefore allowing for true symmetric arrangements of notes.

== Real life counterparts ==

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 In particular, within every [[EDO|edo]] one can specify such a scale for every smaller number of notes.In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo. Mathematically, ME scales of ''n'' notes in ''m'' edo are any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function rounds down to the nearest integer.
-A special case of the maximally even scale is the [[Irvian mode]], which originates from a calendar reform to smoothly spread inaccuracies arising from the uneven number of days or weeks per year. For example, the major mode of the basic [[5L 2s|diatonic]] scale from [[12edo]], <span style="font-family: monospace;">2 2 1 2 2 2 1</span>, is not only a maximally even scale, but also the Irvian mode of such scale. Every mode of any diatonic scale is maximally even, but not necessarily Irvian.
+== Mathematics ==
+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.
 == Sound perception ==
@@ Line 17: / Line 18: @@
 Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel_Mandelbaum|Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.
+[[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic [[5L 2s|diatonic]] scale from [[12edo]], <span style="font-family: monospace;">2 2 1 2 2 2 1</span>, is not only a maximally even scale, but also the Irvian mode of such scale. Such a mode is best shown in odd EDOs, which truly have a "middle" note owing to being odd, and therefore allowing for true symmetric arrangements of notes.
 == Real life counterparts ==