Maximal evenness: Difference between revisions

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~~Within every [[EDO|edo]] one can specify a~~ '''maximally even''' ~~(ME)~~ scale ~~for every smaller edo. The 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~~|~~mode~~]] of ~~the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the [https://en.wikipedia.org/wiki/Floor_and_ceiling_functions "floor"] function rounds down to the nearest integer~~.

The '''maximally even''' scale is a scale which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo), and whose steps are distributed as evenly as possible. Within every [[EDO|edo]] one can specify such a scale for every smaller number of notes.

The maximally even scale ~~will be one~~:

These conditions infer that the ME scale isa [[MOSScales|moment of symmetry scale]]. In terms of sub-edo representation, The 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 variant of the maximal evenness 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.

~~a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).~~

The most popular heptatonic scale in the world, the major scale of [[12edo]]: <span style="font-family: monospace; ">2 2 1 2 2 2 1</span>, is not only a maximal evenness scale but is also an Irvian mode of such scale. Every [http://en.wikipedia.org/wiki/Diatonic_scale diatonic scale] of 12edo is maximally even, but not necessarily Irvian..

~~b. whose steps are distributed as evenly as possible.~~

~~(a) and (b) above imply that the ME scale will be a [[MOSScales|moment of symmetry scale]].~~

The ~~probably~~ most popular heptatonic ME scale is the major scale of [[~~12edo|~~12edo]]: <span style="font-family: monospace; ">2 2 1 2 2 2 1</span>, but also ~~every~~ [http://en.wikipedia.org/wiki/Diatonic_scale diatonic scale] of 12edo is maximally even. ~~Some more detailed examples follow.~~

~~A variant of the maximal evenness scale is the Irvian mode, which originates from leap year arrangements of Sym454 calendar (see below)~~.

== Irvian mode and a relation to a proposed calendar reform ==

In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's ~~uneven~~ distribution of ~~dates~~. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.

In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's unsmooth distribution of days, weeks, months, and leap years. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.

Such a pattern produces a specific maximum evenness mode, which is named '''Irvian mode'''. A stand-alone leap week at the end of year in Sym454 lore is called Irvember, and therefore the constructed name of the mode would be Irvian.

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-Within every [[EDO|edo]] one can specify a '''maximally even''' (ME) scale for every smaller edo. The 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|mode]] of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the [https://en.wikipedia.org/wiki/Floor_and_ceiling_functions "floor"] function rounds down to the nearest integer.
+The '''maximally even''' scale is a scale which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo), and whose steps are distributed as evenly as possible. Within every [[EDO|edo]] one can specify such a scale for every smaller number of notes.
-The maximally even scale will be one:
+These conditions infer that the ME scale isa [[MOSScales|moment of symmetry scale]]. In terms of sub-edo representation, The 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 variant of the maximal evenness 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.
-a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).
+The most popular heptatonic scale in the world, the major scale of [[12edo]]: <span style="font-family: monospace; ">2 2 1 2 2 2 1</span>, is not only a maximal evenness scale but is also an Irvian mode of such scale. Every [http://en.wikipedia.org/wiki/Diatonic_scale diatonic scale] of 12edo is maximally even, but not necessarily Irvian..
-b. whose steps are distributed as evenly as possible.
-(a) and (b) above imply that the ME scale will be a [[MOSScales|moment of symmetry scale]].
-The probably most popular heptatonic ME scale is the major scale of [[12edo|12edo]]: <span style="font-family: monospace; ">2 2 1 2 2 2 1</span>, but also every [http://en.wikipedia.org/wiki/Diatonic_scale diatonic scale] of 12edo is maximally even. Some more detailed examples follow.
-A variant of the maximal evenness scale is the Irvian mode, which originates from leap year arrangements of Sym454 calendar (see below).
 == Irvian mode and a relation to a proposed calendar reform ==
-In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's uneven distribution of dates. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.
+In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's unsmooth distribution of days, weeks, months, and leap years. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.
 Such a pattern produces a specific maximum evenness mode, which is named '''Irvian mode'''. A stand-alone leap week at the end of year in Sym454 lore is called Irvember, and therefore the constructed name of the mode would be Irvian.