Septimal ennealimma - Revision history

FloraC: Update data

2026-02-23T11:05:37Z

Update data

← Older revision		Revision as of 11:05, 23 February 2026
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	== Temperaments ==		== Temperaments ==
	In the 2.3.7 subgroup, tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal]] temperament, which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.		In the [[2.3.7 subgroup]], tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal]] temperament, which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.

	In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.		In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.
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	{{Mapping\|legend=1\| 9 0 0 11 \| 0 1 0 1 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 9 0 0 11 \| 0 1 0 1 \| 0 0 1 0 }}

	: mapping generators: ~2592/2401, ~3, ~5		: mapping generators: ~2592/2401, ~3, ~5

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2592/2401 = 133.~~3333~~, ~3/2 = ~~702~~.~~0044~~, ~5/4 = 386.~~3137~~		* [[WE]]: ~2592/2401 = 133.3357{{c}}, ~3/2 = 701.9772{{c}}, ~5/4 = 386.2717{{c}}
	: [[error map]]: {{val\| 0.~~0000~~ +0.~~0494~~ 0.~~0000~~ -0.~~1549~~ }}		: [[error map]]: {{val\| +0.021 +0.043 -0.000 -0.135 }}
	* [[CWE]]: ~2592/2401 = 133.3333, ~3/2 = 701.9833, ~5/4 = 386.2877		* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9833{{c}}, ~5/4 = 386.2877{{c}}
	: error map: {{val\| 0.~~0000~~ +0.~~0283~~ -0.~~0260~~ -0.~~1759~~ }}		: error map: {{val\| 0.000 +0.028 -0.026 -0.176 }}

	{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}		{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}

	[[Badness]] (~~Smith~~): 0.~~502 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 2.21

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

FloraC: +tunings

2025-03-06T10:14:17Z

+tunings

← Older revision		Revision as of 10:14, 6 March 2025
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	{{Mapping\|legend=1\| 9 0 0 11 \| 0 1 0 1 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 9 0 0 11 \| 0 1 0 1 \| 0 0 1 0 }}

	: mapping generators: ~2592/2401, ~3, ~5		: mapping generators: ~2592/2401, ~3, ~5

			[[Optimal tuning]]s:
			* [[CTE]]: ~2592/2401 = 133.3333, ~3/2 = 702.0044, ~5/4 = 386.3137
			: [[error map]]: {{val\| 0.0000 +0.0494 0.0000 -0.1549 }}
			* [[CWE]]: ~2592/2401 = 133.3333, ~3/2 = 701.9833, ~5/4 = 386.2877
			: error map: {{val\| 0.0000 +0.0283 -0.0260 -0.1759 }}

	{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}		{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}

FloraC: Cleanup

2025-03-06T10:09:49Z

Cleanup

← Older revision		Revision as of 10:09, 6 March 2025
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	== Temperaments ==		== Temperaments ==
	In the 2.3.7 subgroup, tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal ~~temperament~~]], which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.		In the 2.3.7 subgroup, tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal]] temperament, which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.

	In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.		In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.
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	{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}		{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}

	Badness (Smith): 0.502 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.502 × 10<sup>-3</sup>

	== See also ==		== See also ==
	* [[Unnoticeable comma]]
	* [[Ennealimma]]		* [[Ennealimma]]

	[[Category:Septiennealimmal]]		[[Category:Septiennealimmal]]
	[[Category:Commas named for their periods per equave]]		[[Category:Commas named for their periods per equave]]

FloraC: Undo the redefinition since ennealimmal is now discussed under the septiennealimmal clan. Rework and -no-5 septiennealimmal

2025-03-06T10:09:01Z

Undo the redefinition since ennealimmal is now discussed under the septiennealimmal clan. Rework and -no-5 septiennealimmal

← Older revision		Revision as of 10:09, 6 March 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Ratio = 40353607/40310784		\| Ratio = 40353607/40310784
	\| Name = septimal ennealimma, no-five ennealimma		\| Name = septimal ennealimma, septiennealimma, no-five ennealimma
	\| Color name = tritrizo comma		\| Color name = tritrizo comma
	\| Comma = yes		\| Comma = yes
	}}		}}
	The '''septimal ennealimma''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma ~~defined as~~ ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup> ~~and measuring 1.838 cents~~.		The '''septimal ennealimma''', '''septiennealimma''' or '''no-five ennealimma''' is an [[unnoticeable comma\|unnoticeable]] [[7-limit]] [[comma]] measuring 1.838 [[cent]]s. It is the difference between a stack of nine [[7/6\|septimal minor thirds (7/6)]] and two [[2/1\|octaves]], or ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup>.

	== Temperaments ==		== Temperaments ==
	In the 2.3.7 subgroup, tempering it out leads to the 2.3.7 ~~version~~ of the [[ennealimmal temperament]], which is ~~a member~~ of the [[~~tritrizo~~ clan]].		In the 2.3.7 subgroup, tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal temperament]], which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.

	~~It can only possibly be tempered~~ out ~~in an EDO if that EDO is~~ a ~~multiple~~ of ~~9. This~~ temperament ~~is documented~~ below:		In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.

	=== Septiennealimmal ===		=== Septiennealimmal ===
	~~The rank 2 temperament septiennealimmal is of interest to anyone who wants a different generator for the "ennealimmal-like structure" by detempering~~ [[~~2401/2400\|S49~~]] ~~and/or because it represents the part of ennealimmal supported by non-ennealimmal EDOs of interest that do well in the~~ 2.3.7 ~~subgroup, such as [[36edo]], which adds [[1029/1024\|S7/S8]], or [[63edo]], which in the 7-limit can be used for [[septimal magic]] and in higher limits for [[parapyth]] (among other things).~~		[[Subgroup]]: 2.3.5.7

	~~[[Subgroup]]: 2.3.7~~		{{Mapping\|legend=1\| 9 0 0 11 \| 0 1 0 1 \| 0 0 1 0 }}

	~~[[Mapping]]~~: ~~[{{val\| 9 0 11 }}~~, ~~{{val\| 0 1 1 }}]~~		: mapping generators: ~2592/2401, ~3, ~5

	~~Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400~~\|~~S49]]~~ = 1\9, ~~~3 = 1902.004{{cent~~}}		{{Optimal ET sequence\|legend=1\| 27, 45, 63, 72, 99, 171, 441, 612 }}

	~~[[Patent val]] non-ennealimmal EDO tunings < 311 with the [[7-limit]]ed [[9-odd-limit]]~~ (~~or here equivalently [[21-odd-limit]]~~) ~~[[consistent]]: 36, 63, 108, 135, 162, 207, 234, 279, 306~~		Badness (Smith): 0.502 × 10<sup>-3</sup>

	~~[[Badness]]~~: 0.~~191~~

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

BudjarnLambeth at 00:42, 4 November 2024

2024-11-04T00:42:04Z

← Older revision		Revision as of 00:42, 4 November 2024
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	[[Category:Septiennealimmal]]		[[Category:Septiennealimmal]]
	[[Category:Commas named for their ~~regular temperament properties~~]]		[[Category:Commas named for their periods per equave]]

BudjarnLambeth at 23:18, 3 November 2024

2024-11-03T23:18:42Z

← Older revision		Revision as of 23:18, 3 November 2024
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	[[Category:Septiennealimmal]]		[[Category:Septiennealimmal]]
			[[Category:Commas named for their regular temperament properties]]

Godtone: correction

2024-04-08T20:49:19Z

correction

← Older revision		Revision as of 20:49, 8 April 2024
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	Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400\|S49]] = 1\9, ~3 = 1902.004{{cent}}		Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400\|S49]] = 1\9, ~3 = 1902.004{{cent}}

	[[Patent val]] non-ennealimmal ~~EDOs~~ with ~~increasingly good approximations of 3~~: ~~9, 18,~~ 36, 63, ~~90, 117~~, 135, ~~333, 360, 378, 747~~, ~~945~~, ~~990~~, ~~1044~~, ~~1089~~		[[Patent val]] non-ennealimmal EDO tunings < 311 with the [[7-limit]]ed [[9-odd-limit]] (or here equivalently [[21-odd-limit]]) [[consistent]]: 36, 63, 108, 135, 162, 207, 234, 279, 306

	[[Badness]]: 0.191		[[Badness]]: 0.191

Godtone at 20:13, 8 April 2024

2024-04-08T20:13:14Z

← Older revision		Revision as of 20:13, 8 April 2024
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	\| Comma = yes		\| Comma = yes
	}}		}}
	The '''septimal ennealimma'''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma defined as ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup> and measuring 1.838 cents.		The '''septimal ennealimma''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma defined as ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup> and measuring 1.838 cents.

	== Temperaments ==		== Temperaments ==

Godtone: redefined septiennealimmal to be more practically useful as there's no real reason to use the rank 3; it's 7/6 = 2\9 that is most useful and this allows future extension to prime 5 in non-ennealimmal ways so that the gen can be something different

2024-04-08T20:11:31Z

redefined septiennealimmal to be more practically useful as there's no real reason to use the rank 3; it's 7/6 = 2\9 that is most useful and this allows future extension to prime 5 in non-ennealimmal ways so that the gen can be something different

← Older revision		Revision as of 20:11, 8 April 2024
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	{{Infobox Interval		{{Infobox Interval
	\| Ratio = 40353607/40310784		\| Ratio = 40353607/40310784
	\| Name = septimal ennealimma~~, septiennealimma~~, no-five ennealimma		\| Name = septimal ennealimma, no-five ennealimma
	\| Color name = tritrizo comma		\| Color name = tritrizo comma
	\| Comma = yes		\| Comma = yes
	}}		}}
	The '''septimal ennealimma'~~'', '''septiennealimma~~''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma defined as ~~40353607/40310784 and measuring 1.838 cents. It arises as a stack between nine~~ [[7/6]]'s and ~~two octaves~~.		The '''septimal ennealimma'''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma defined as ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup> and measuring 1.838 cents.

	== Temperaments ==		== Temperaments ==
	In the 2.3.7 subgroup, tempering it out leads to the 2.3.7 version of the [[ennealimmal temperament]], which is a member of the [[tritrizo clan]]~~. For edos up to circa 3000, it is tempered out if and only if the edo is divisible by 9~~.		In the 2.3.7 subgroup, tempering it out leads to the 2.3.7 version of the [[ennealimmal temperament]], which is a member of the [[tritrizo clan]].

	~~In the full 7-limit, tempering it~~ out ~~leads to the '''septiennealimmal temperament''' (cf. septimal ennealimmal temperament). You may find~~ a ~~list~~ of ~~good equal temperaments that support this~~ temperament below.		It can only possibly be tempered out in an EDO if that EDO is a multiple of 9. This temperament is documented below:

	=== Septiennealimmal ===		=== Septiennealimmal ===
	[[~~Subgroup~~]]: 2.3.5.7		The rank 2 temperament septiennealimmal is of interest to anyone who wants a different generator for the "ennealimmal-like structure" by detempering [[2401/2400\|S49]] and/or because it represents the part of ennealimmal supported by non-ennealimmal EDOs of interest that do well in the 2.3.7 subgroup, such as [[36edo]], which adds [[1029/1024\|S7/S8]], or [[63edo]], which in the 7-limit can be used for [[septimal magic]] and in higher limits for [[parapyth]] (among other things).

	[[~~Mapping~~]]: ~~[{{val\| 9 0 0 11 }}, {{val\| 0 1 0 1 }}, {{val\| 0 0 1 0 }}]~~		[[Subgroup]]: 2.3.7

	Mapping ~~generators~~: ~~~2592/2401~~, ~~~3, ~5~~		[[Mapping]]: [{{val\| 9 0 11 }}, {{val\| 0 1 1 }}]

	~~{{Optimal ET sequence~~\|~~legend~~=1~~\| 27~~, ~~45, 63, 72, 99, 171, 441, 612~~ }}		Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400\|S49]] = 1\9, ~3 = 1902.004{{cent}}

	[[Badness]]: 0.~~502 × 10<sup>-3</sup>~~		[[Patent val]] non-ennealimmal EDOs with increasingly good approximations of 3: 9, 18, 36, 63, 90, 117, 135, 333, 360, 378, 747, 945, 990, 1044, 1089

			[[Badness]]: 0.191

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

Godtone at 16:43, 8 April 2024

2024-04-08T16:43:43Z

← Older revision		Revision as of 16:43, 8 April 2024
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	== See also ==		== See also ==
	* [[Unnoticeable comma]]		* [[Unnoticeable comma]]
			* [[Ennealimma]]

	[[Category:Septiennealimmal]]		[[Category:Septiennealimmal]]