Maximal evenness: Difference between revisions

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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.

In this paradigm, years correspond to individual steps of the scale, and leap years correspond to steps that are part of the mode. The length of the cycle is the size of an Edo.

The pattern is defined by the following:

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''K'' = (''C''-1)/2 if ''C'' is odd, can choose between (''C''-1)/2 and ''C''/2 if ''C'' is even</blockquote>

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.

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 of [[293edo]]. 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.

Even-length Irvian modes with odd number of years per cycle (that is notes) 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. Thus it is called ''almost symmetrical'' in the calendar lore. That being said, they fulfill their function just like odd cycles do, and therefore belong in Irvian modes.

=== Example on a standard 12edo piano ===

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 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.
+In this paradigm, years correspond to individual steps of the scale, and leap years correspond to steps that are part of the mode. The length of the cycle is the size of an Edo.
 The pattern is defined by the following:
@@ Line 34: / Line 36: @@
 ''K'' = (''C''-1)/2 if ''C'' is odd, can choose between (''C''-1)/2 and ''C''/2 if ''C'' is even</blockquote>
-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.
+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 of [[293edo]]. 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.
+Even-length Irvian modes with odd number of years per cycle (that is notes) 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. Thus it is called ''almost symmetrical'' in the calendar lore. That being said, they fulfill their function just like odd cycles do, and therefore belong in Irvian modes.
 === Example on a standard 12edo piano ===