Gregorian leap day: Difference between revisions

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'''Gregorian leap day''' is a [[rank-2 temperament]] which is produced by temperament-merging 97edo, which has the cardinality of leap years in Gregorian calendar's cycle, and 400edo, the whole duration of the cycle.

400 is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. ~~Because the generator~~ is ~~mapped~~ to 18/17, ~~the temperament can be seen as an octavated version of~~ [[~~18/17 equal-step~~ tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].

400 is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. The optimal tuning is very close to 18/17, which makes it very similar to [[Galilei's tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].

In the 7-limit, temperament reaches [[15/8]] in 11 generators, entirely contained within the 12-tone well temperament, and also [[7/5]] in 18 generators.

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Comma list: 166698/166375, 422576/421875, 67108864/66976875

~~Mapping:~~ {{~~mapping~~| 1 10 -7 8 16 | 0 -102 113 131 -152 }}

Optimal tuning (CTE): ~2 = 1\1, 69120/65219 = 98.9994

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Comma list: 676/675, 4096/4095, 105644/105625, 166698/166375

~~Mapping:~~ {{~~mapping~~| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }}

Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9992

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Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500

~~Mapping:~~ {{~~mapping~~| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}

{{Mapping|legend=0| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}

Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9993

Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993

Optimal ET sequence: {{Optimal ET sequence| 97, ~~303~~, 400 }}

Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}

Badness: 0.0776

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Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399

~~Mapping:~~ {{~~mapping~~| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3 }}

{{Mapping|legend=0| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3 }}

Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9993

Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993

Optimal ET sequence: {{Optimal ET sequence| 97, ~~303~~, 400 }}

Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}

Badness: 0.0541

[[Category: ~~Temperaments~~]]

[[Category:Gregorian leap day| ]]

[[Category:Rank-2 temperaments]]

@@ Line 1: / Line 1: @@
 {{Novelty}}
+{{Distinguish|Leapday}}
 '''Gregorian leap day''' is a [[rank-2 temperament]] which is produced by temperament-merging 97edo, which has the cardinality of leap years in Gregorian calendar's cycle, and 400edo, the whole duration of the cycle.
-is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. Because the generator is mapped to 18/17, the temperament can be seen as an octavated version of [[18/17 equal-step tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].
+is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. The optimal tuning is very close to 18/17, which makes it very similar to [[Galilei's tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].
 In the 7-limit, temperament reaches [[15/8]] in 11 generators, entirely contained within the 12-tone well temperament, and also [[7/5]] in 18 generators.
@@ Line 27: / Line 27: @@
 Comma list: 166698/166375, 422576/421875, 67108864/66976875
-Mapping: {{mapping| 1 10 -7 8 16 | 0 -102 113 131 -152 }}
+{{Mapping|legend=0| 1 10 -7 8 16 | 0 -102 113 131 -152 }}
 Optimal tuning (CTE): ~2 = 1\1, 69120/65219 = 98.9994
@@ Line 40: / Line 40: @@
 Comma list: 676/675, 4096/4095, 105644/105625, 166698/166375
-Mapping: {{mapping| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }}
+{{Mapping|legend=0| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }}
 Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9992
@@ Line 53: / Line 53: @@
 Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500
-Mapping: {{mapping| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}
+{{Mapping|legend=0| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}
-Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9993
+Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993
-Optimal ET sequence: {{Optimal ET sequence| 97, 303, 400 }}
+Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}
 Badness: 0.0776
@@ Line 67: / Line 67: @@
 Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399
-Mapping: {{mapping| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3 }}
+{{Mapping|legend=0| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3 }}
-Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9993
+Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993
-Optimal ET sequence: {{Optimal ET sequence| 97, 303, 400 }}
+Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}
 Badness: 0.0541
-[[Category: Temperaments]]
+[[Category:Gregorian leap day| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]