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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.  
A '''maximally even''' ('''ME''') scale is a [[scale]] inscribed in an [[equal-step tuning]] which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of '''maximal evenness'''. These conditions infer that an ME scale is necessarily an [[MOS scale]].


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. 
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.


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..  
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.
 
== Sound perception ==
The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th).
 
The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.
 
Maximally even sets tend to be familiar and musically relevant scale collections. Examples:
 
<ul><li>The maximally even heptatonic set of [[19edo|19edo]] is, like the one in 12edo, a diatonic scale.</li><li>The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are Maqamic[7].</li><li>The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</li><li>The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.</li><li>The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[Circulating temperament|circulating temperaments]] with the right timbre.
</li></ul>
 
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 and a relation to a proposed calendar reform ==
== 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 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.
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.
Such a pattern produces a specific mode of a maximally even scale, which is named an 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.


The pattern is defined by the following:<blockquote>'''Year is leap if the remainder of (L x Y + K)/ C is less than L.'''
The pattern is defined by the following:


L = number of leap years per cycle,
<blockquote>'''Year is leap if the remainder of (''L'' x ''Y'' + ''K'')/''C'' is less than ''L''.'''


Y = number of the year
''L'' = number of leap years per cycle,


C = number of years per cycle
''Y'' = number of the year


K = (C-1)/2 if odd, can choose between (C-1)/2 and C/2 if even</blockquote>The current, "canonical" usage of the cycle is that of 52 leap week years in 293 years - year is leap if (52 x Year + 146)/293 is less than 52. Musically, this would correspond to a [[33L 19s|33L 19]]<nowiki/>s MOS scale. In addition, if the remainder of the leap year is less than the count of long intervals in the MOS, the next year will be in a long interval, otherwise in a short interval. For example here, this means if remainder is less than 33, next leap year (or key) will be 6 years later (6 steps above), otherwise 5 years later.
''C'' = number of years per cycle


Even-length symmetrical cycles with an irreducible (that is odd) number of years per cycle have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the K is chosen as (C-1)/2 instead of C/2, the sequence will be leap, nonleap.
''K'' = (''C''-1)/2 if ''C'' is odd, can choose between (''C''-1)/2 and ''C''/2 if ''C'' is even</blockquote>


Every Irvian mode is a maximal evenness scale, but not every maximal evenness scale is Irvian.
The current, "canonical" usage of the cycle is that of 52 leap week years in 293 years - year is leap if the remainder of (52 x Year + 146)/293 is less than 52. Musically, this would correspond to a [[33L 19s]] MOS scale. In addition, if the remainder of the leap year is less than the count of long intervals in the MOS, the next year will be in a long interval, otherwise in a short interval. For example here, this means if remainder is less than 33, next leap year (or key) will be 6 years later (6 steps above), otherwise 5 years later.
 
Even-length symmetrical cycles with an irreducible (that is odd) number of years per cycle have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the ''K'' is chosen as (''C''-1)/2 instead of ''C''/2, the sequence will be leap, nonleap.


=== Example on a standard 12edo piano ===
=== Example on a standard 12edo piano ===
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|}
|}
and so on.
and so on.
== Sound perception ==
The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th).
The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.
Maximally even sets tend to be familiar and musically relevant scale collections. Examples:
<ul><li>The maximally even heptatonic set of [[19edo|19edo]] is, like the one in 12edo, a diatonic scale.</li><li>The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are Maqamic[7].</li><li>The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</li><li>The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.</li><li>The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[Circulating temperament|circulating temperaments]] with the right timbre.
</li></ul>
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.


== Trivia ==
== Trivia ==
* Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the [[Wikipedia:Hebrew calendar|Hebrew calendar]].
* Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the [[Wikipedia:Hebrew calendar|Hebrew calendar]].
* Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the [[Wikipedia:Iranian calendars|Persian calendar]].
* Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the [[Wikipedia:Iranian calendars|Persian calendar]].
== External links ==
== External links ==
 
* [http://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52/293 Symmetry454 Leap Years]
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52/293 Symmetry454 Leap Years]
 
* [http://individual.utoronto.ca/kalendis/leap/index.htm#slc Solar Calendar Leap Rules - subsection Symmetrical Leap Cycles]
* [http://individual.utoronto.ca/kalendis/leap/index.htm#slc Solar Calendar Leap Rules - subsection Symmetrical Leap Cycles]


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