6656/6655 - Revision history

FloraC: Complete data

2026-02-25T06:52:33Z

Complete data

← Older revision		Revision as of 06:52, 25 February 2026
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 3 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 3 \|\| ]]
	\|}		\|}
			: mapping generators: ~2, ~3, ~5, ~7, ~11

	: ~~mapping generators~~: ~2, ~3, ~5, ~7, ~11		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9887{{c}}, ~3/2 = 701.9663{{c}}, ~5/4 = 386.3430{{c}}, ~7/4 = 968.8484{{c}}, ~11/8 = 551.3967{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9661{{c}}, ~5/4 = 386.3383{{c}}, ~7/4 = 968.8455{{c}}, ~11/8 = 551.3980{{c}}

	{{Optimal ET sequence\|legend=1\| 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}		{{Optimal ET sequence\|legend=1\| 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}

			[[Badness]] (Sintel): 0.416

	==== Septendecimal jacobin ====		==== Septendecimal jacobin ====
Line 55:		Line 60:
	\|}		\|}

	Optimal ~~ET sequence~~: {{Optimal ET sequence\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}		Optimal tunings:
			* WE: ~2 = 1199.9890{{c}}, ~3/2 = 701.9675{{c}}, ~5/4 = 386.3466{{c}}, ~7/4 = 968.8454{{c}}, ~11/8 = 551.3945{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9670{{c}}, ~5/4 = 386.3410{{c}}, ~7/4 = 968.8434{{c}}, ~11/8 = 551.3964{{c}}

			{{Optimal ET sequence\|legend=0\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}

			Badness (Sintel): 0.239

	== Etymology ==		== Etymology ==

FloraC: Cleanup

2026-02-25T06:47:16Z

Cleanup

← Older revision		Revision as of 06:47, 25 February 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}
	'''6656/6655''', the '''jacobin comma''', [~~https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by~~ [~~[Gene Ward Smith~~]] ~~in 2014, is a~~ [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] ~~interval~~ of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]] ([[13/5]]). In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into ([[12376/12375]])([[14400/14399]]).		'''6656/6655''', the '''jacobin comma''', is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] [[comma]] of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]] ([[13/5]]).

			In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the [[17-limit]], it factors neatly into ([[12376/12375]])⋅([[14400/14399]]).

	== Temperaments ==		== Temperaments ==
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	Optimal ET sequence: {{Optimal ET sequence\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}		Optimal ET sequence: {{Optimal ET sequence\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}

			== Etymology ==
			This comma was apparently named by [[Gene Ward Smith]] in 2014 for reasons unknown<ref>[https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 Xenharmonic Wiki \| ''List of superparticular intervals (Revision as of 23:46, 13 June 2014 by Wikispaces>genewardsmith)'']</ref>.

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

			== References ==

	[[Category:Jacobin]]		[[Category:Jacobin]]
	[[Category:Commas with unknown etymology]]		[[Category:Commas with unknown etymology]]
			{{Todo\|add etymology\|comment=I know it's something to do with the French Revolution, but what, exactly? Spell it all out please, why exactly does the name "jacobin" uniquely fit this particular comma?}}
	{{~~todo~~\|add etymology\|comment=I know ~~it’s~~ something to do with the French Revolution, but what, exactly? Spell it all out please, why exactly does the name ~~“jacobin”~~ uniquely fit this particular comma?}}

Jdood: added 13/5 to "13/10 naiadic + an octave)

2025-12-22T06:09:59Z

added 13/5 to "13/10 naiadic + an octave)

← Older revision		Revision as of 06:09, 22 December 2025
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into ([[12376/12375]])([[14400/14399]]).		'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]] ([[13/5]]). In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into ([[12376/12375]])([[14400/14399]]).

	== Temperaments ==		== Temperaments ==

BudjarnLambeth at 03:34, 4 November 2024

2024-11-04T03:34:00Z

← Older revision		Revision as of 03:34, 4 November 2024
Line 60:		Line 60:

	[[Category:Jacobin]]		[[Category:Jacobin]]
			[[Category:Commas with unknown etymology]]

			{{todo\|add etymology\|comment=I know it’s something to do with the French Revolution, but what, exactly? Spell it all out please, why exactly does the name “jacobin” uniquely fit this particular comma?}}

Eliora: /* Temperaments */ link to The Jacobins

2023-07-19T16:24:56Z

Temperaments: link to The Jacobins

← Older revision		Revision as of 16:24, 19 July 2023
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	== Temperaments ==		== Temperaments ==
	By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament.		By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament. Miscellaneous temperaments tempering out this comma are collected in [[The Jacobins]].

	The 17-limit factorization shows us a natural path of extension, also given below.		The 17-limit factorization shows us a natural path of extension, also given below.

FloraC: /* Temperaments */ formatting

2023-07-19T12:25:53Z

Temperaments: formatting

← Older revision		Revision as of 12:25, 19 July 2023
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	[[Mapping]]: <br>		[[Mapping]]: <br>
	[~~{{val~~\| 1 0 0 0 0 -9 }}, ~~<br>~~		{\| class="right-all"
	~~{{val~~\| 0 1 0 0 0 0 }}, ~~<br>~~		\|-
	~~{{val~~\| 0 0 1 0 0 1 }}, ~~<br>~~		\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -9 \|\| ],
	~~{{val~~\| 0 0 0 1 0 0 }}, ~~<br>~~		\|-
	~~{{val~~\| 0 0 0 0 1 3 }}]		\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 3 \|\| ]]
			\|}

	: mapping generators: ~2, ~3, ~5, ~7, ~11		: mapping generators: ~2, ~3, ~5, ~7, ~11
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	Mapping: <br>		Mapping: <br>
	[~~{{val~~\| 1 0 0 0 0 -9 6 }}, ~~<br>~~		{\| class="right-all"
	~~{{val~~\| 0 1 0 0 0 0 2 }}, ~~<br>~~		\|-
	~~{{val~~\| 0 0 1 0 0 1 2 }}, ~~<br>~~		\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -9 \|\| 6 \|\| ],
	~~{{val~~\| 0 0 0 1 0 0 -1 }}, ~~<br>~~		\|-
	~~{{val~~\| 0 0 0 0 1 3 -2 }}]		\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 2 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| 2 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| -1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 3 \|\| -2 \|\| ]]
			\|}

	Optimal ET sequence: {{Optimal ET sequence\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}		Optimal ET sequence: {{Optimal ET sequence\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}

FloraC: Invert the clause of 1789edo vs good edos; +natural 17-limit extension

2023-07-19T10:22:00Z

Invert the clause of 1789edo vs good edos; +natural 17-limit extension

← Older revision		Revision as of 10:22, 19 July 2023
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into [[12376/12375]] × [[~~sparkisma\|~~14400/14399]].		'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into ([[12376/12375]])([[14400/14399]]).

	== Temperaments ==		== Temperaments ==
	By tempering it out, the '''jacobin temperament''' is defined. ~~Perhaps most remarkably~~, ~~'''~~[[1789edo]]~~'''~~ is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although ~~it is more rational to use such edos for this temperament,~~ 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution.		By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament.

			The 17-limit factorization shows us a natural path of extension, also given below.

			=== Jacobin ===
	[[Subgroup]]: 2.3.5.7.11.13		[[Subgroup]]: 2.3.5.7.11.13

			[[Comma list]]: 6656/6655

	[[Mapping]]: <br>		[[Mapping]]: <br>
Line 20:		Line 25:
	: mapping generators: ~2, ~3, ~5, ~7, ~11		: mapping generators: ~2, ~3, ~5, ~7, ~11

	{{Optimal ET sequence\|legend=1\| 9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}		{{Optimal ET sequence\|legend=1\| 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}

			==== Septendecimal jacobin ====
			Subgroup: 2.3.5.7.11.13.17

			Comma list: 6656/6655, 12376/12375

			Mapping: <br>
			[{{val\| 1 0 0 0 0 -9 6 }}, <br>
			{{val\| 0 1 0 0 0 0 2 }}, <br>
			{{val\| 0 0 1 0 0 1 2 }}, <br>
			{{val\| 0 0 0 1 0 0 -1 }}, <br>
			{{val\| 0 0 0 0 1 3 -2 }}]

			Optimal ET sequence: {{Optimal ET sequence\| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}

	== See also ==		== See also ==

FloraC: Link and formatting

2023-07-19T09:59:46Z

Link and formatting

← Older revision		Revision as of 09:59, 19 July 2023
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into 12376/12375 × [[sparkisma\|14400/14399]].		'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into [[12376/12375]] × [[sparkisma\|14400/14399]].

	== Temperaments ==		== Temperaments ==
	By tempering it out, the '''jacobin temperament''' is defined. Perhaps most remarkably, '''[[1789edo]]''' is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although it is more rational to use such edos for this temperament, 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution.		By tempering it out, the '''jacobin temperament''' is defined. Perhaps most remarkably, '''[[1789edo]]''' is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although it is more rational to use such edos for this temperament, 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution.

	Subgroup: 2.3.5.7.11.13		[[Subgroup]]: 2.3.5.7.11.13

	[[Mapping]]: <br>		[[Mapping]]: <br>
Line 18:		Line 18:
	{{val\| 0 0 0 0 1 3 }}]		{{val\| 0 0 0 0 1 3 }}]

	~~Mapping~~ generators: ~2, ~3, ~5, ~7, ~11		: mapping generators: ~2, ~3, ~5, ~7, ~11

	{{Optimal ET sequence\|legend=1\| 9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}		{{Optimal ET sequence\|legend=1\| 9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}

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

	[[Category:Jacobin]]		[[Category:Jacobin]]

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T18:31:14Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 18:31, 7 May 2023
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	Mapping generators: ~2, ~3, ~5, ~7, ~11		Mapping generators: ~2, ~3, ~5, ~7, ~11

	{{~~Val list~~\|legend=1\| 9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}		{{Optimal ET sequence\|legend=1\| 9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}

	== See also ==		== See also ==

TallKite at 07:22, 14 January 2023

2023-01-14T07:22:54Z

← Older revision		Revision as of 07:22, 14 January 2023
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	{{Infobox Interval		{{Infobox Interval
	\| Name = jacobin comma		\| Name = jacobin comma
	\| Color name = ~~thotrilu~~-agu comma		\| Color name = Thotrilu-agu comma
	\| Comma = yes		\| Comma = yes
	}}		}}