Lesser tendoneutralisma - Revision history

FloraC: Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

2025-06-06T16:40:18Z

Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

← Older revision		Revision as of 16:40, 6 June 2025
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	{{Optimal ET sequence\|legend=1\| 10, 47, 57, 67, 77, 164, 241, 405, 646f }}		{{Optimal ET sequence\|legend=1\| 10, 47, 57, 67, 77, 164, 241, 405, 646f }}

	[[Badness]] (~~Dirichlet~~): 3.930		[[Badness]] (Sintel): 3.930

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

BudjarnLambeth at 23:15, 15 November 2024

2024-11-15T23:15:28Z

← Older revision		Revision as of 23:15, 15 November 2024
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	* [[512/507]]		* [[512/507]]
	* [[Tridecapyth comma]]		* [[Tridecapyth comma]]

			[[Category:Commas with unknown etymology]]

Godtone: /* Lesser Tendoneutralic */

2024-10-04T17:57:51Z

Lesser Tendoneutralic

← Older revision		Revision as of 17:57, 4 October 2024
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	== Temperaments ==		== Temperaments ==
	=== Lesser Tendoneutralic ===		=== Lesser Tendoneutralic ===
			{{ See also \| Restles }}
	Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence).		Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence).

BudjarnLambeth: BudjarnLambeth moved page 70368744177664/69894255367443 to Lesser tendoneutralisma over redirect: Long comma number

2024-09-26T05:41:33Z

BudjarnLambeth moved page 70368744177664/69894255367443 to Lesser tendoneutralisma over redirect: Long comma number

← Older revision	Revision as of 05:41, 26 September 2024
(No difference)

TallKite: add color name

2024-09-26T01:46:40Z

add color name

← Older revision		Revision as of 01:46, 26 September 2024
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	\| Ratio = 70368744177664/69894255367443		\| Ratio = 70368744177664/69894255367443
	\| Name = lesser tendoneutralisma		\| Name = lesser tendoneutralisma
			\| Color name = Sasa-quadtrithu comma
	\| Comma = yes		\| Comma = yes
	}}		}}

Godtone: note a 3rd mapping of 3

2024-05-29T19:03:45Z

note a 3rd mapping of 3

← Older revision		Revision as of 19:03, 29 May 2024
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	\| Comma = yes		\| Comma = yes
	}}		}}
	The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to ([[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings.		The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to ([[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. (A much simpler but (relatively) much higher error way of mapping 3 for those that prefer sharp fifths is by tempering ([[16/13]])<sup>2</sup>/([[3/2]]) = [[512/507]].)

	== Temperaments ==		== Temperaments ==
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	== See also ==		== See also ==
	* [[Greater tendoneutralisma]]		* [[Greater tendoneutralisma]]
			* [[512/507]]
	* [[Tridecapyth comma]]		* [[Tridecapyth comma]]

Godtone: missing bracket

2024-05-29T19:00:09Z

missing bracket

← Older revision		Revision as of 19:00, 29 May 2024
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to [[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings.		The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to ([[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings.

	== Temperaments ==		== Temperaments ==

Godtone: add important note

2024-05-29T18:59:40Z

add important note

← Older revision		Revision as of 18:59, 29 May 2024
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to [[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]].		The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to [[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings.

	== Temperaments ==		== Temperaments ==

Godtone: /* Lesser Tendoneutralic */ increase clarity that the CTE gen is not the same as the mapping gen

2024-05-29T18:56:09Z

Lesser Tendoneutralic: increase clarity that the CTE gen is not the same as the mapping gen

← Older revision		Revision as of 18:56, 29 May 2024
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	[[Comma list]]: {{monzo\| 46 -1 (0 0 0) -12 }}		[[Comma list]]: {{monzo\| 46 -1 (0 0 0) -12 }}

	{{Mapping\|legend=1\| 1 10 3 \| 0 -12 1 }}		{{Mapping\|legend=2\| 1 10 3 \| 0 -12 1 }}
			: sval mapping generators: 2, ~13

	[[Optimal tuning]] ([[CTE]]): ~13/8 = 841.503{{cent}}		[[Optimal tuning]] ([[CTE]]): ~13/8 = 841.503{{cent}}

Godtone: Created page with "{{Infobox Interval | Ratio = 70368744177664/69894255367443 | Name = lesser tendoneutralisma | Comma = yes }} The '''lesser tendoneutralisma''' is a small comma of the 2.3...."

2024-05-29T18:52:41Z

Created page with "{{Infobox Interval | Ratio = 70368744177664/69894255367443 | Name = lesser tendoneutralisma | Comma = yes }} The '''lesser tendoneutralisma''' is a small comma of the 2.3...."

New page

{{Infobox Interval
| Ratio = 70368744177664/69894255367443
| Name = lesser tendoneutralisma
| Comma = yes
}}
The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])12 / ([[3/2]]) / ([[2/1]])3 and so equivalently also to [[16/13]])11 / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]].

== Temperaments ==
=== Lesser Tendoneutralic ===
Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence).

[[Subgroup]]: 2.3.13

[[Comma list]]: {{monzo| 46 -1 (0 0 0) -12 }}

{{Mapping|legend=1| 1 10 3 | 0 -12 1 }}

[[Optimal tuning]] ([[CTE]]): ~13/8 = 841.503{{cent}}

{{Optimal ET sequence|legend=1| 10, 47, 57, 67, 77, 164, 241, 405, 646f }}

[[Badness]] (Dirichlet): 3.930

== See also ==
* [[Greater tendoneutralisma]]
* [[Tridecapyth comma]]