6691edo - Revision history

ArrowHead294 at 15:46, 20 February 2025

2025-02-20T15:46:43Z

← Older revision		Revision as of 15:46, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.		6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.

FloraC: Readability

2024-08-21T11:54:56Z

Readability

← Older revision		Revision as of 11:54, 21 August 2024
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	6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.		6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.

	We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, ~~281484423828125/281474976710656~~, ~~8936733825332544112/8936247052719140625~~}.		We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, {{monzo\| -48 0 11 8 }}, {{monzo\| 4 -28 -8 21 }}}.

	The approximation to [[harmonic]] [[13/1\|13]] is weaker, though it is still [[consistent]] to the [[15-odd-limit]]. In fact, it is consistent to the no-13 or no-17 no-23 [[29-odd-limit]]. In the 13-limit we may note it [[tempering out\|tempers out]] [[10648/10647]], and is a good tuning for the corresponding rank-5 temperament. It also tempers out [[140625/140608]]. In the 17-limit, [[194481/194480]]; in the 19-limit, 14080/14079, 23409/23408, 43681/43680, 89376/89375, 165376/165375; and in the 23-limit, 21505/21504, 23276/23275, 25921/25920, 52326/52325, 76545/76544 among others.		The approximation to [[harmonic]] [[13/1\|13]] is weaker, though it is still [[consistent]] to the [[15-odd-limit]]. In fact, it is consistent to the no-13 or no-17 no-23 [[29-odd-limit]]. In the 13-limit we may note it [[tempering out\|tempers out]] [[10648/10647]], and is a good tuning for the corresponding rank-5 temperament. It also tempers out [[140625/140608]]. In the 17-limit, [[194481/194480]]; in the 19-limit, 14080/14079, 23409/23408, 43681/43680, 89376/89375, 165376/165375; and in the 23-limit, 21505/21504, 23276/23275, 25921/25920, 52326/52325, 76545/76544 among others.

FloraC: Expand and unstub

2024-08-21T11:52:48Z

Expand and unstub

← Older revision		Revision as of 11:52, 21 August 2024
Line 4:		Line 4:
	6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.		6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.

	A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.

			The approximation to [[harmonic]] [[13/1\|13]] is weaker, though it is still [[consistent]] to the [[15-odd-limit]]. In fact, it is consistent to the no-13 or no-17 no-23 [[29-odd-limit]]. In the 13-limit we may note it [[tempering out\|tempers out]] [[10648/10647]], and is a good tuning for the corresponding rank-5 temperament. It also tempers out [[140625/140608]]. In the 17-limit, [[194481/194480]]; in the 19-limit, 14080/14079, 23409/23408, 43681/43680, 89376/89375, 165376/165375; and in the 23-limit, 21505/21504, 23276/23275, 25921/25920, 52326/52325, 76545/76544 among others.

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	6691edo is the 863rd [[prime edo]].		6691edo is the 863rd [[prime edo]].


	~~{{Stub}}~~

FloraC: Cleanup; +subsets and supersets

2024-08-20T16:50:02Z

Cleanup; +subsets and supersets

← Older revision		Revision as of 16:50, 20 August 2024
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	{{Infobox ET}}		{{Infobox ET}}
	~~The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It~~ is a very strong [[11-limit]] ~~division~~, with a lower 11-limit [[Tenney-~~Euclidean_temperament_measures~~#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.		{{EDO intro}}

			6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness\|relative error]] than any smaller division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.

	A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.

			=== Prime harmonics ===
	{{Harmonics in equal\|6691}}		{{Harmonics in equal\|6691}}
	~~{{stub}}~~

	[[~~Category:Equal divisions of the octave\|####~~]] ~~<!-- 4-digit number -->~~		=== Subsets and supersets ===
			6691edo is the 863rd [[prime edo]].


			{{Stub}}

BudjarnLambeth at 07:13, 14 January 2024

2024-01-14T07:13:26Z

← Older revision		Revision as of 07:13, 14 January 2024
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	A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.
	{{Harmonics in equal}}		{{Harmonics in equal\|6691}}
	{{stub}}		{{stub}}

	[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->		[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->

BudjarnLambeth at 07:12, 14 January 2024

2024-01-14T07:12:40Z

← Older revision		Revision as of 07:12, 14 January 2024
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	{{Infobox ET}}		{{Infobox ET}}
	The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong [[11-limit]] division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]].		The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong [[11-limit]] division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the [[7-limit]], where only [[3125edo\|3125]] is both smaller and with a lesser relative error.

	~~It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error.~~ A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.
	{{Harmonics in equal}}		{{Harmonics in equal}}
	{{stub}}		{{stub}}

	[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->		[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->

BudjarnLambeth: Added paragraph break, added table of harmonics

2024-01-14T07:12:22Z

Added paragraph break, added table of harmonics

← Older revision		Revision as of 07:12, 14 January 2024
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	{{Infobox ET}}		{{Infobox ET}}
	The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong [[11-limit]] division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]].

			It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.
			{{Harmonics in equal}}
			{{stub}}

	[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->		[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->

BudjarnLambeth at 07:09, 9 July 2023

2023-07-09T07:09:36Z

← Older revision		Revision as of 07:09, 9 July 2023
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	~~{{novelty}}{{stub}}~~{{Infobox ET}}		{{Infobox ET}}
	The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.

	[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->		[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->

BudjarnLambeth at 04:20, 9 July 2023

2023-07-09T04:20:33Z

← Older revision		Revision as of 04:20, 9 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{novelty}}{{stub}}{{Infobox ET}}
	The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.

	[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->		[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->

Plumtree: Infobox ET added

2022-10-04T22:20:03Z

Infobox ET added

← Older revision		Revision as of 22:20, 4 October 2022
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			{{Infobox ET}}
	The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.		The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong 11-limit division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]] than any division until [[40006edo\|40006]]. It is also strong in the 7-limit, where only [[3125edo\|3125]] is both smaller and with a lesser relative error. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.

	[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->		[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->