# Gregorian leap day

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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.

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, [-13 3 -17 17⟩

Mapping: [⟨1 10 -7 8], ⟨0 -102 113 131]]

- mapping generators: ~2, 160000/151263

Optimal tuning (CTE): ~2 = 1\1, 160000/151263 = 98.9982

Optimal ET sequence: 97, 303, 400, 1297, 1697c

Badness: 1.10

### 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]]

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

Optimal ET sequence: 97, 303, 400

Badness: 0.251

### 13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 10 -7 8 16 7], ⟨0 -102 113 131 -152 -40]]

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

Optimal ET sequence: 97, 303, 400

Badness: 0.126

### 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]]

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

Optimal ET sequence: 97, 303, 400

Badness: 0.0776

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

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]]

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

Optimal ET sequence: 97, 303, 400

Badness: 0.0541