Gregorian leap day - Revision history

Overthink: + Distinguish

2025-12-12T23:43:11Z

+ Distinguish

← Older revision		Revision as of 23:43, 12 December 2025
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	{{Novelty}}		{{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.		'''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.

Eliora: Correct as requested, 97 -> 97g

2025-08-07T20:37:21Z

Correct as requested, 97 -> 97g

← Older revision		Revision as of 20:37, 7 August 2025
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	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.		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.

	~~{{todo\|correct maths\|inline=1\|text=The generator isn't mapped to 18/17 according to the mapping below. Should it be 97g & 400 instead of 97 & 400?}}~~

	== Temperament data ==		== Temperament data ==
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	Mapping: {{mapping\| 1 10 -7 8 16 7 -12 \| 0 -102 113 131 -152 -40 195 }}		Mapping: {{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 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		Badness: 0.0776
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	Mapping: {{mapping\| 1 10 -7 8 16 7 -12 4 \| 0 -102 113 131 -152 -40 195 3 }}		Mapping: {{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 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		Badness: 0.0541

Fredg999: Rephrase relation to 18/17 equal-step tuning

2025-08-07T19:08:34Z

Rephrase relation to 18/17 equal-step tuning

← Older revision		Revision as of 19:08, 7 August 2025
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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.		'''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.		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.

Fredg999: Todo: correct maths

2025-08-07T18:47:06Z

Todo: correct maths

← Older revision		Revision as of 18:47, 7 August 2025
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	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.		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.

			{{todo\|correct maths\|inline=1\|text=The generator isn't mapped to 18/17 according to the mapping below. Should it be 97g & 400 instead of 97 & 400?}}

	== Temperament data ==		== Temperament data ==

FloraC: - parent category

2025-04-28T14:11:52Z

- parent category

← Older revision		Revision as of 14:11, 28 April 2025
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	Badness: 0.0541		Badness: 0.0541

	[[Category: ~~Temperaments~~]]		[[Category:Gregorian leap day\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]

Lériendil at 23:08, 19 April 2025

2025-04-19T23:08:07Z

← Older revision		Revision as of 23:08, 19 April 2025
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	[[Category: Temperaments]]		[[Category: Temperaments]]
			[[Category:Rank-2 temperaments]]

FloraC: Review, cleanup, and mark as novelty

2024-01-19T15:03:29Z

Review, cleanup, and mark as novelty

← Older revision		Revision as of 15:03, 19 January 2024
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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.		{{Novelty}}
	~~== Theory ==~~
			'''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]]. 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.		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 ==		== Temperament data ==
	Subgroup: 2.3.5.7		[[Subgroup]]: 2.3.5.7

	Comma list: 67108864/66976875, ~~6281023877654589/6250000000000000~~		[[Comma list]]: 67108864/66976875, {{monzo\| -13 3 -17 17 }}

	{{~~mapping~~\|legend=1\| 1 10 -7 8 \| 0 -102 113 131 }}		{{Mapping\|legend=1\| 1 10 -7 8 \| 0 -102 113 131 }}

	: mapping generators: ~2 ~~= 1\1~~, 160000/151263 ~~= 98.998~~		: mapping generators: ~2, 160000/151263

	[[~~Support~~]]~~ing~~ [[ET]]s: {{~~EDOs~~\|97, 400}}~~, ..~~.		[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 160000/151263 = 98.9982

			{{Optimal ET sequence\|legend=1\| 97, 303, 400, 1297, 1697c }}

			[[Badness]]: 1.10

	=== 11-limit ===		=== 11-limit ===

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

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

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

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

	: ~~mapping generators: ~2 = 1\1~~, ~~69120/65219 = 98.998~~		Optimal ET sequence: {{Optimal ET sequence\| 97, 303, 400 }}

	~~[[Support]]ing [[ET]]s~~: ~~{{EDOs\|97, 303, 400}}, ..~~.		Badness: 0.251

	=== 13-limit ===		=== 13-limit ===
			Subgroup: 2.3.5.7.11.13

	~~Subgroup~~: ~~2.3.5.7.11.13~~		Comma list: 676/675, 4096/4095, 105644/105625, 166698/166375

	~~Comma list~~: ~~676/675, 4096/4095, 166698/166375, 105644/105625~~		Mapping: {{mapping\| 1 10 -7 8 16 7 \| 0 -102 113 131 -152 -40 }}

	~~{{mapping\|legend~~=1\| 1 ~~10 -7 8 16 7 \| 0 -102 113 131 -152 -40 }}~~		Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9992

	: ~~mapping generators: ~2 = 1\1~~, ~~55/52 = 98.998~~		Optimal ET sequence: {{Optimal ET sequence\| 97, 303, 400 }}

	~~[[Support]]ing [[ET]]s~~: ~~{{EDOs\|97, 303, 400}}, ..~~.		Badness: 0.126

	=== 17-limit ===		=== 17-limit ===

	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

	Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500		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: {{mapping\| 1 10 -7 8 16 7 -12 \| 0 -102 113 131 -152 -40 195 }}

	~~: mapping generators~~: ~2 = 1\1, 55/52 = 98.~~998~~		Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9993

	~~[[Support]]ing [[~~ET~~]]s~~: {{~~EDOs~~\|97, 303, 400}}~~, ..~~.		Optimal ET sequence: {{Optimal ET sequence\| 97, 303, 400 }}

			Badness: 0.0776

	=== 19-limit ===		=== 19-limit ===

	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17.19

	Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399		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: {{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: {{Optimal ET sequence\| 97, 303, 400 }}

	: ~~mapping generators: ~2 = 1\1, 55/52 = 98~~.~~998~~		Badness: 0.0541

	[[~~Support]]ing [[ET~~]]~~s: {{EDOs\|97, 303, 400}}, ...~~		[[Category: Temperaments]]

Eliora at 00:52, 5 June 2023

2023-06-05T00:52:25Z

@@ Line 1: / Line 1: @@
-is  the number of years in the Gregorian calendar's leap cycle.
+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.
-edo contains GregorianLeapDay[97] scale, which is a [[maximal evenness]] version of the leap rule currently in use in the world today. The scale has a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. The corresponding temperament is 97 & 400, with comma list 2432/2431, 2601/2600, 2926/2925, 6175/6174, 17689/17680, and 22477/22440.
+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.
-[[1L 11s]] mos of this temperament is an octavated variant of [[18/17 equal-step tuning|18/17s equal temperament]], and a barely noticeable circulating temperament for [[12edo]].
+Comma list: 166698/166375, 422576/421875, 67108864/66976875
-[[Category: Rank 2]]

Eliora at 00:39, 5 June 2023

2023-06-05T00:39:21Z

← Older revision		Revision as of 00:39, 5 June 2023
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	[[1L 11s]] mos of this temperament is an octavated variant of [[18/17 equal-step tuning\|18/17s equal temperament]], and a barely noticeable circulating temperament for [[12edo]].		[[1L 11s]] mos of this temperament is an octavated variant of [[18/17 equal-step tuning\|18/17s equal temperament]], and a barely noticeable circulating temperament for [[12edo]].
			[[Category: Rank 2]]

Eliora: once again, i'll elaborate later but for now just want to declutter 400edo page.

2023-06-05T00:36:17Z

once again, i'll elaborate later but for now just want to declutter 400edo page.

New page

400 is the number of years in the Gregorian calendar's leap cycle.

400edo contains GregorianLeapDay[97] scale, which is a [[maximal evenness]] version of the leap rule currently in use in the world today. The scale has a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. The corresponding temperament is 97 & 400, with comma list 2432/2431, 2601/2600, 2926/2925, 6175/6174, 17689/17680, and 22477/22440.

[[1L 11s]] mos of this temperament is an octavated variant of [[18/17 equal-step tuning|18/17s equal temperament]], and a barely noticeable circulating temperament for [[12edo]].