Chalmersia - Revision history

FloraC: Update data

2026-02-19T15:14:04Z

Update data

← Older revision		Revision as of 15:14, 19 February 2026
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 2 \|\| 1 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 2 \|\| 1 \|\| ]]
	\|}		\|}

	: mapping generators: ~2, ~3, ~5, ~7, ~351/280		: mapping generators: ~2, ~3, ~5, ~7, ~351/280

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2 = 1200.~~0000~~{{c}}, ~3/2 = 701.~~9539~~{{c}}, ~5/4 = 386.~~3145~~{{c}}, ~7/4 = ~~3368~~.~~8265~~{{c}}, ~351/280 = 391.~~2462~~{{c}}		* [[WE]]: ~2 = 1200.000379{{c}}, ~3/2 = 701.953671{{c}}, ~5/4 = 386.313637{{c}}, ~7/4 = 968.825646{{c}}, ~351/280 = 391.246147{{c}}
	* [[CWE]]: ~2 = 1200.~~0000~~{{c}}, ~3/2 = 701.~~9536~~{{c}}, ~5/4 = 386.~~3140~~{{c}}, ~7/4 = ~~3368~~.~~8259~~{{c}}, ~351/280 = 391.~~2461~~{{c}}		* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.953639{{c}}, ~5/4 = 386.313976{{c}}, ~7/4 = 968.825869{{c}}, ~351/280 = 391.246091{{c}}

	{{Optimal ET sequence\|legend=1\| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}		{{Optimal ET sequence\|legend=1\| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}

			[[Badness]] (Sintel): 0.0267

	== Etymology ==		== Etymology ==
	The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html Yahoo! Tuning Group \| ''Nameable 13-limit'']</ref>.		The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html Yahoo! Tuning Group \| ''Nameable 13-limit'']</ref>.

	~~:''~~The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.''		<blockquote>The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.</blockquote>

	—Gene Ward Smith		—Gene Ward Smith

FloraC: Cleanup

2026-02-19T15:00:54Z

Cleanup

← Older revision		Revision as of 15:00, 19 February 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}
			The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] [[comma]] with a [[ratio]] of '''123201/123200''' and a size of approximately 0.014 [[cent]]s. It is the smallest 13-limit [[superparticular]] comma.

	The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two: these are properties characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]s; it splits [[11/7]] into two [[351/280]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].		It factors into the two smallest [[17-limit]] superparticular ratios: 123201/123200 = ([[194481/194480]])⋅([[336141/336140]]).

	It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = ([[194481/194480]])([[336141/336140]]).

	== Temperaments ==		== Temperaments ==
	Tempering out ~~the~~ comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''.		[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''. It equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]'s; it splits [[11/7]] into two [[351/280]]'s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].

	[[Subgroup]]: 2.3.5.7.11.13		[[Subgroup]]: 2.3.5.7.11.13
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~3/2 = 701.9539, ~5/4 = 386.3145, ~7/4 = 3368.8265, ~351/280 = 391.2462		* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9539{{c}}, ~5/4 = 386.3145{{c}}, ~7/4 = 3368.8265{{c}}, ~351/280 = 391.2462{{c}}
	* [[CWE]]: ~2 = ~~1\1~~, ~3/2 = 701.9536, ~5/4 = 386.3140, ~7/4 = 3368.8259, ~351/280 = 391.2461		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9536{{c}}, ~5/4 = 386.3140{{c}}, ~7/4 = 3368.8259{{c}}, ~351/280 = 391.2461{{c}}

	{{Optimal ET sequence\|legend=1\| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}		{{Optimal ET sequence\|legend=1\| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}
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	== See also ==		== See also ==
	* [[Unnoticeable comma]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]

	== ~~Notes~~ ==		== References ==

	[[Category:Chalmersic]]		[[Category:Chalmersic]]
	[[Category:Commas named after music theorists]]		[[Category:Commas named after music theorists]]

Squib at 21:28, 18 February 2026

2026-02-18T21:28:55Z

← Older revision		Revision as of 21:28, 18 February 2026
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	}}		}}

	The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two: these are properties characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].		The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two: these are properties characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]s; it splits [[11/7]] into two [[351/280]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].

	It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = ([[194481/194480]])([[336141/336140]]).		It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = ([[194481/194480]])([[336141/336140]]).

	== Temperaments ==		== Temperaments ==

Squib: added links to 17-limit factorization

2026-01-25T15:59:54Z

added links to 17-limit factorization

← Older revision		Revision as of 15:59, 25 January 2026
Line 8:		Line 8:
	The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two: these are properties characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].		The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two: these are properties characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].

	It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).		It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = ([[194481/194480]])([[336141/336140]]).

	== Temperaments ==		== Temperaments ==

Recentlymaterialized at 17:34, 25 December 2024

2024-12-25T17:34:25Z

← Older revision		Revision as of 17:34, 25 December 2024
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	}}		}}

	The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two – these are ~~features highly~~ characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third.		The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two: these are properties characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].

	It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).		It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).

BudjarnLambeth at 02:23, 4 November 2024

2024-11-04T02:23:45Z

← Older revision		Revision as of 02:23, 4 November 2024
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	[[Category:Chalmersic]]		[[Category:Chalmersic]]
	[[Category:Commas named after ~~individuals~~]]		[[Category:Commas named after music theorists]]

BudjarnLambeth at 09:09, 3 November 2024

2024-11-03T09:09:57Z

← Older revision		Revision as of 09:09, 3 November 2024
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	[[Category:Chalmersic]]		[[Category:Chalmersic]]
			[[Category:Commas named after individuals]]

FloraC: /* Temperaments */ +optimal tunings; note that there are more ETs in the sequence

2024-05-09T07:41:04Z

Temperaments: +optimal tunings; note that there are more ETs in the sequence

← Older revision		Revision as of 07:41, 9 May 2024
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	: mapping generators: ~2, ~3, ~5, ~7, ~351/280		: mapping generators: ~2, ~3, ~5, ~7, ~351/280

	{{Optimal ET sequence\|legend=1\|~~5e, 7, 12de,~~ 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539}}		[[Optimal tuning]]s:
			* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9539, ~5/4 = 386.3145, ~7/4 = 3368.8265, ~351/280 = 391.2462
			* [[CWE]]: ~2 = 1\1, ~3/2 = 701.9536, ~5/4 = 386.3140, ~7/4 = 3368.8259, ~351/280 = 391.2461

			{{Optimal ET sequence\|legend=1\| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}

	== Etymology ==		== Etymology ==

Francium: +temperaments

2024-05-09T06:33:08Z

+temperaments

← Older revision		Revision as of 06:33, 9 May 2024
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	It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).		It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).

			== Temperaments ==
			Tempering out the comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''.

			[[Subgroup]]: 2.3.5.7.11.13

			[[Mapping]]: <br>
			{\| class="right-all"
			\| [⟨ \|\| 1 \|\| 1 \|\| 2 \|\| 2 \|\| 2 \|\| 4 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| -3 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| 1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 2 \|\| 1 \|\| ]]
			\|}

			: mapping generators: ~2, ~3, ~5, ~7, ~351/280

			{{Optimal ET sequence\|legend=1\|5e, 7, 12de, 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539}}

	== Etymology ==		== Etymology ==

FloraC: +17-limit factorization

2023-02-10T11:58:51Z

+17-limit factorization

← Older revision		Revision as of 11:58, 10 February 2023
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	The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two – these are features highly characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third.		The '''chalmersia''' is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two – these are features highly characteristic of '''chalmersic temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third.

			It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).

	== Etymology ==		== Etymology ==
	The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]].		The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html Yahoo! Tuning Group \| ''Nameable 13-limit'']</ref>.

	:''The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.''		:''The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.''
	—Gene Ward Smith~~, [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html ''Nameable 13-limit''].~~		—Gene Ward Smith

	== See also ==		== See also ==
	* [[Unnoticeable comma]]		* [[Unnoticeable comma]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]

			== Notes ==

	[[Category:Chalmersic]]		[[Category:Chalmersic]]