28812/28561 - Revision history

Lucius Chiaraviglio: four 14/13 semitones exceed the 4/3 perfect fourth, not fall short of it

2025-07-19T08:07:51Z

four 14/13 semitones exceed the 4/3 perfect fourth, not fall short of it

← Older revision		Revision as of 08:07, 19 July 2025
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	\| Comma = yes		\| Comma = yes
	}}		}}
	'''28812/28561''', the '''tesseract comma''', is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones ~~fall short of~~ the [[4/3]] perfect fourth.		'''28812/28561''', the '''tesseract comma''', is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones exceed the [[4/3]] perfect fourth.

	It can be factored into the [[28672/28561\|voltage comma]] and the [[1029/1024\|gamelisma]], which provides the 77 & 87 temperament '''cubical''' (see below); it can also be factored into the [[octaphore]] plus four [[729/728\|squbemas]], which makes the tesseract comma a useful extension to the rank-3 octaphore and to rank-2 unicorn temperaments.		It can be factored into the [[28672/28561\|voltage comma]] and the [[1029/1024\|gamelisma]], which provides the 77 & 87 temperament '''cubical''' (see below); it can also be factored into the [[octaphore]] plus four [[729/728\|squbemas]], which makes the tesseract comma a useful extension to the rank-3 octaphore and to rank-2 unicorn temperaments.

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

2025-06-06T16:39:06Z

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

← Older revision		Revision as of 16:39, 6 June 2025
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	{{Optimal ET sequence\|legend=1\| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }}		{{Optimal ET sequence\|legend=1\| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }}

	[[Badness]] (~~Dirichlet~~): 2.528		[[Badness]] (Sintel): 2.528

	==== 2.3.5.7.13 subgroup ====		==== 2.3.5.7.13 subgroup ====

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

2025-06-06T16:38:39Z

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

← Older revision		Revision as of 16:38, 6 June 2025
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	{{Optimal ET sequence\|legend=0\| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }}		{{Optimal ET sequence\|legend=0\| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }}

	Badness (~~Dirichlet~~): 1.818		Badness (Sintel): 1.818

	=== Cubical ===		=== Cubical ===
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	{{Optimal ET sequence\|legend=0\| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }}		{{Optimal ET sequence\|legend=0\| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }}

	Badness (~~Dirichlet~~): 1.261		Badness (Sintel): 1.261

	=== Other temperaments ===		=== Other temperaments ===

Jerdle: /* 2.3.5.7.13 subgroup */ Mentioned negri

2025-01-19T00:18:16Z

2.3.5.7.13 subgroup: Mentioned negri

← Older revision		Revision as of 00:18, 19 January 2025
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	==== 2.3.5.7.13 subgroup ====		==== 2.3.5.7.13 subgroup ====
	By noticing that three generators is almost exactly 5/4, we can add prime 5 to the subgroup by tempering out the [[cantonisma]]. We can equivalently temper out the [[105/104\|animist comma]] by noticing that the difference between 4/3 and 5/4 (that is, 16/15) is equivalent in mapping to 14/13.		By noticing that three generators is almost exactly 5/4, we can add prime 5 to the subgroup by tempering out the [[cantonisma]]. We can equivalently temper out the [[105/104\|animist comma]] by noticing that the difference between 4/3 and 5/4 (that is, 16/15) is equivalent in mapping to 14/13. As such, this is also a form of [[negri]].

	Subgroup: 2.3.5.7.13		Subgroup: 2.3.5.7.13

FloraC: Style. Cleanup. Note the badness metric being used. -redundant categories

2025-01-17T10:19:56Z

Style. Cleanup. Note the badness metric being used. -redundant categories

← Older revision		Revision as of 10:19, 17 January 2025
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	{{Infobox Interval\|Name=Tesseract ~~Comma~~\|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma\|Ratio=28812/28561\|Comma=yes}}		{{Infobox Interval
			\| Name= Tesseract comma
			\| Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma
			\| Ratio = 28812/28561
			\| Comma = yes
			}}
			'''28812/28561''', the '''tesseract comma''', is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.

	'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.		It can be factored into the [[28672/28561\|voltage comma]] and the [[1029/1024\|gamelisma]], which provides the 77 & 87 temperament '''cubical''' (see below); it can also be factored into the [[octaphore]] plus four [[729/728\|squbemas]], which makes the tesseract comma a useful extension to the rank-3 octaphore and to rank-2 unicorn temperaments.

	It can be factored into the [[28672/28561\|~~Voltage Comma~~]] and the [[1029/1024\|~~Gamelisma~~]], which provides the 77 & 87 temperament '''~~Cubical~~''' (see below); it can also be factored into the [[~~Octaphore~~]] plus four [[729/728\|~~Squbemas~~]] ~~(squbemae?)~~, which makes the ~~Tesseract Comma~~ a useful extension to the rank-3 ~~Octaphore~~ and to rank-2 ~~Unicorn~~ temperaments.

	== Temperaments ==		== Temperaments ==

	=== Tesseract ===		=== Tesseract ===
	Tempering out the ~~Tesseract Comma~~ in its minimal subgroup, 2.3.7.13, yields the rank-3 '''~~Tesseract~~''' temperament.		Tempering out the tesseract comma in its minimal subgroup, 2.3.7.13, yields the rank-3 '''tesseract''' temperament.

	[[~~Just intonation subgroup\|~~Subgroup]]: 2.3.7.13		[[Subgroup]]: 2.3.7.13

	[[Comma list]]: 28812/28561		[[Comma list]]: 28812/28561

	[[Mapping~~]]: [⟨1~~ 2 2 3~~], ⟨0~~ -4 0 -1~~], ⟨0~~ 0 1 1]]		{{Mapping\|legend=1\| 1 2 2 3 \| 0 -4 0 -1 \| 0 0 1 1 }}

	[[~~Optimization\|~~Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~14/13 = 124.539, ~7/4 = 967.452		[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~14/13 = 124.539, ~7/4 = 967.452

	[[Optimal ET sequence~~]]: [[9edo~~\|9]], ~~[[10edo\|~~10]], ~~[[19edo\|~~19]], ~~[[29edo\|~~29]], ~~[[37edo\|~~37b]], ~~[[48edo\|~~48]], ~~[[49edo\|~~49f]], ~~[[58edo\|~~58]], ~~[[67edo\|~~67]], ~~[[68edo\|~~68]], ~~[[77edo\|~~77]], ~~[[87edo\|~~87]]		{{Optimal ET sequence\|legend=1\| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }}

	[[Badness]]: 2.528		[[Badness]] (Dirichlet): 2.528

	==== 2.3.5.7.13 subgroup ====		==== 2.3.5.7.13 subgroup ====
	By noticing that three generators is almost exactly 5/4, we can add prime 5 to the subgroup by tempering out the [[~~Cantonisma~~]]. We can equivalently temper out the [[105/104\|~~Animist~~ comma]] by noticing that the difference between 4/3 and 5/4 (that is, 16/15) is equivalent in mapping to 14/13.		By noticing that three generators is almost exactly 5/4, we can add prime 5 to the subgroup by tempering out the [[cantonisma]]. We can equivalently temper out the [[105/104\|animist comma]] by noticing that the difference between 4/3 and 5/4 (that is, 16/15) is equivalent in mapping to 14/13.

	Subgroup: 2.3.5.7.13		Subgroup: 2.3.5.7.13

	Comma list: 28812/28561~~, 10985/10976~~		Comma list: 10985/10976, 28812/28561

	Mapping: ~~[⟨1~~ 2 2 2 3~~], ⟨0~~ -4 3 0 -1~~], ⟨0~~ 0 0 1 1]]		Mapping: {{mapping\| 1 2 2 2 3 \| 0 -4 3 0 -1 \| 0 0 0 1 1 }}

	Optimal tuning (CTE): ~2 = ~~1\1~~, ~14/13 = 126.679, ~7/4 = 962.564		Optimal tuning (CTE): ~2 = 1200.000, ~14/13 = 126.679, ~7/4 = 962.564

	Optimal ET sequence~~: [[9edo~~\|9]], ~~[[10edo\|~~10]], ~~[[19edo\|~~19]], ~~[[29edo\|~~29]], ~~[[37edo\|~~37b]], ~~[[38edo\|~~38]], ~~[[47edo\|~~47]], ~~[[57edo\|~~57]], ~~[[58edo\|~~58]], ~~[[67edo\|~~67c]], 76, ~~[[87edo\|~~86c]]		{{Optimal ET sequence\|legend=0\| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }}

	Badness: 1.818		Badness (Dirichlet): 1.818

	=== Cubical ===		=== Cubical ===
	By factoring the ~~Tesseract~~ comma into the ~~Voltage Comma~~ and ~~Gamelisma~~, we get the rank-2 temperament '''~~Cubical~~'''. This temperament is so named because its lattice is the same as ~~Tesseract~~, but with one dimension collapsed; similarly, a cube can be thought of as a ~~Tesseract~~ with one of its dimensions collapsed.		By factoring the tesseract comma into the voltage comma and gamelisma, we get the rank-2 temperament '''cubical'''. This temperament is so named because its lattice is the same as tesseract, but with one dimension collapsed; similarly, a cube can be thought of as a tesseract with one of its dimensions collapsed.

	Subgroup: 2.3.7.13		Subgroup: 2.3.7.13

	Comma list: 28672/28561~~, 1029/1024~~		Comma list: 1029/1024, 28672/28561

	Mapping: [⟨1 10 0 3], ⟨0 0 4], ⟨0 3 1]]		Mapping: [⟨1 10 0 3], ⟨0 0 4], ⟨0 3 1]]

	Optimal tuning (CTE): ~2 = ~~1\1~~, ~13/8 = 841.527		Optimal tuning (CTE): ~2 = 1200.000, ~13/8 = 841.527

	Optimal ET sequence~~: [[10edo~~\|10]], ~~[[37edo\|~~37b]], ~~[[47edo\|~~47]], ~~[[57edo\|~~57]], ~~[[67edo\|~~67]], ~~[[77edo\|~~77]], ~~[[87edo\|~~87]], ~~[[97edo\|~~97]], ~~[[107edo\|~~107]], ~~[[124edo\|~~124b]], ~~[[144edo\|~~144]]		{{Optimal ET sequence\|legend=0\| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }}

	Badness: 1.261		Badness (Dirichlet): 1.261

	=== Other temperaments ===		=== Other temperaments ===
	Temperaments discussed elsewhere that temper out the ~~Tesseract~~ comma include:		Temperaments discussed elsewhere that temper out the tesseract comma include:

	Tridecimal ~~Octaphore~~ → [[Octaphore#Tridecimal%20Octaphore\|Octaphore]]		Tridecimal octaphore → [[Octaphore #Tridecimal%20Octaphore\|Octaphore]]

	2.3.5.7.13 subgroup ~~Unicorn~~ (+351/350 ~~+126/125~~) → [[Unicorn family#2.3.5.7.13%20subgroup\|Unicorn ~~Family~~]]		2.3.5.7.13 subgroup unicorn (+126/125 and 351/350) → [[Unicorn family #2.3.5.7.13%20subgroup\|Unicorn family]]

	== Etymology ==		== Etymology ==
	The name ~~Tesseract Comma~~ was chosen by [[User:Unque\|Unque]] in 2025. This name was chosen because tempering the comma cleaves the ~~Perfect Fourth~~ into four parts, and a tesseract is the 4D regular polytope made from four-sided regular polygons.		The name tesseract comma was chosen by [[User:Unque\|Unque]] in 2025. This name was chosen because tempering the comma cleaves the perfect fourth into four parts, and a tesseract is the 4D regular polytope made from four-sided regular polygons.

	[[Category:Commas named for how they divide the fourth]]		[[Category:Commas named for how they divide the fourth]]
	[[Category:Commas named for the intervals they stack]]		[[Category:Commas named for the intervals they stack]]
	~~[[Category:Rational intervals]]~~
	~~[[Category:13-limit intervals]]~~

BudjarnLambeth at 06:36, 17 January 2025

2025-01-17T06:36:19Z

← Older revision		Revision as of 06:36, 17 January 2025
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	{{Infobox Interval\|Name=Tesseract Comma\|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma\|Ratio=28812/28561\|~~comma~~=yes}}		{{Infobox Interval\|Name=Tesseract Comma\|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma\|Ratio=28812/28561\|Comma=yes}}

	'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.		'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.

BudjarnLambeth: Comma=yes in infobox

2025-01-17T06:35:56Z

Comma=yes in infobox

← Older revision		Revision as of 06:35, 17 January 2025
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	{{Infobox Interval\|Name=Tesseract Comma\|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma\|Ratio=28812/28561}}		{{Infobox Interval\|Name=Tesseract Comma\|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma\|Ratio=28812/28561\|comma=yes}}

	'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.		'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.

TallKite: add color name

2025-01-17T06:00:31Z

add color name

← Older revision		Revision as of 06:00, 17 January 2025
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	{{Infobox Interval\|Name=Tesseract Comma\|Ratio=28812/28561}}		{{Infobox Interval\|Name=Tesseract Comma\|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma\|Ratio=28812/28561}}

	'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.		'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.

Unque at 23:58, 16 January 2025

2025-01-16T23:58:58Z

← Older revision		Revision as of 23:58, 16 January 2025
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	{{Infobox Interval\|Name=Tesseract Comma\|Ratio=28812/28561}}		{{Infobox Interval\|Name=Tesseract Comma\|Ratio=28812/28561}}

	'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths~~, when octave-reduced,~~ fall short of the [[3/2]] ~~perfect fifth~~, and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.		'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[12/1\|twelfth harmonic]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.

	It can be factored into the [[28672/28561\|Voltage Comma]] and the [[1029/1024\|Gamelisma]], which provides the 77 & 87 temperament '''Cubical''' (see below); it can also be factored into the [[Octaphore]] plus four [[729/728\|Squbemas]] (squbemae?), which makes the Tesseract Comma a useful extension to the rank-3 Octaphore and to rank-2 Unicorn temperaments.		It can be factored into the [[28672/28561\|Voltage Comma]] and the [[1029/1024\|Gamelisma]], which provides the 77 & 87 temperament '''Cubical''' (see below); it can also be factored into the [[Octaphore]] plus four [[729/728\|Squbemas]] (squbemae?), which makes the Tesseract Comma a useful extension to the rank-3 Octaphore and to rank-2 Unicorn temperaments.

Unque at 23:57, 16 January 2025

2025-01-16T23:57:12Z

← Older revision		Revision as of 23:57, 16 January 2025
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	{{Infobox Interval\|Name=Tesseract Comma\|Ratio=28812/28561}}		{{Infobox Interval\|Name=Tesseract Comma\|Ratio=28812/28561}}

	'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths fall short of the [[6/~~1\|sixth harmonic~~]], and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.		'''28812/28561''' (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four [[13/7]] sevenths, when octave-reduced, fall short of the [[3/2]] perfect fifth, and the amount by which four [[14/13]] semitones fall short of the [[4/3]] perfect fourth.

	It can be factored into the [[28672/28561\|Voltage Comma]] and the [[1029/1024\|Gamelisma]], which provides the 77 & 87 temperament '''Cubical''' (see below); it can also be factored into the [[Octaphore]] plus four [[729/728\|Squbemas]] (squbemae?), which makes the Tesseract Comma a useful extension to the rank-3 Octaphore and to rank-2 Unicorn temperaments.		It can be factored into the [[28672/28561\|Voltage Comma]] and the [[1029/1024\|Gamelisma]], which provides the 77 & 87 temperament '''Cubical''' (see below); it can also be factored into the [[Octaphore]] plus four [[729/728\|Squbemas]] (squbemae?), which makes the Tesseract Comma a useful extension to the rank-3 Octaphore and to rank-2 Unicorn temperaments.