Gregorian leap day: Difference between revisions
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400 is | 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. | ||
== Theory == | |||
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]]. | |||
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. | |||
[[Category: Rank 2]] | |||
== Temperament data == | |||
Subgroup: 2.3.5.7 | |||
Comma list: 67108864/66976875, 6281023877654589/6250000000000000 | |||
{{mapping|legend=1| 1 10 -7 8 | 0 -102 113 131 }} | |||
: mapping generators: ~2 = 1\1, 160000/151263 = 98.998 | |||
[[Support]]ing [[ET]]s: {{EDOs|97, 400}}, ... | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
[[ | Comma list: 166698/166375, 422576/421875, 67108864/66976875 | ||
[[ | |||
{{mapping|legend=1| 1 10 -7 8 16 | 0 -102 113 131 -152 }} | |||
: mapping generators: ~2 = 1\1, 69120/65219 = 98.998 | |||
[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ... | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 676/675, 4096/4095, 166698/166375, 105644/105625 | |||
{{mapping|legend=1| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }} | |||
: mapping generators: ~2 = 1\1, 55/52 = 98.998 | |||
[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ... | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500 | |||
{{mapping|legend=1| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }} | |||
: mapping generators: ~2 = 1\1, 55/52 = 98.998 | |||
[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ... | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399 | |||
{{mapping|legend=1| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3}} | |||
: mapping generators: ~2 = 1\1, 55/52 = 98.998 | |||
[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ... | |||
Revision as of 00:52, 5 June 2023
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.
Theory
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.
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.
Temperament data
Subgroup: 2.3.5.7
Comma list: 67108864/66976875, 6281023877654589/6250000000000000
Mapping: [⟨1 10 -7 8], ⟨0 -102 113 131]]
- mapping generators: ~2 = 1\1, 160000/151263 = 98.998
Supporting ETs: 97, 400, ...
11-limit
Subgroup: 2.3.5.7.11
Comma list: 166698/166375, 422576/421875, 67108864/66976875
Mapping: [⟨1 10 -7 8 16], ⟨0 -102 113 131 -152]]
- mapping generators: ~2 = 1\1, 69120/65219 = 98.998
Supporting ETs: 97, 303, 400, ...
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 4096/4095, 166698/166375, 105644/105625
Mapping: [⟨1 10 -7 8 16 7], ⟨0 -102 113 131 -152 -40]]
- mapping generators: ~2 = 1\1, 55/52 = 98.998
Supporting ETs: 97, 303, 400, ...
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500
Mapping: [⟨1 10 -7 8 16 7 -12], ⟨0 -102 113 131 -152 -40 195]]
- mapping generators: ~2 = 1\1, 55/52 = 98.998
Supporting ETs: 97, 303, 400, ...
19-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399
Mapping: [⟨1 10 -7 8 16 7 -12 4], ⟨0 -102 113 131 -152 -40 195 3]]
- mapping generators: ~2 = 1\1, 55/52 = 98.998
Supporting ETs: 97, 303, 400, ...