Octaphore - Revision history

FloraC: Cleanup

2025-04-03T10:41:35Z

Cleanup

← Older revision		Revision as of 10:41, 3 April 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Name = ~~the~~ octaphore, enneagari comma		\| Name = octaphore, enneagari comma
	\| Ratio = 94450499584/94143178827		\| Ratio = 94450499584/94143178827
	\| Monzo = 14 -23 0 8		\| Monzo = 14 -23 0 8
	}}		}}
			The '''octaphore''', also known as the '''enneagari comma''', is a [[small comma\|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 5.64 [[cent]]s. It is so named because it is the amount by which eight [[28/27]] third-tones exceed the [[4/3]] perfect fourth. It can also be found as the amount by which seven 28/27 third-tones exceed the [[9/7]] supermajor third, or as the sum of the [[garischisma\|garischisma (33554432/33480783)]] and the [[septimal ennealimma\|septimal ennealimma (40353607/40310784)]].
	The '''octaphore''', also known as the '''enneagari comma''', is a [[small comma\|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 5.64 [[cent]]s. It is so named because it is the amount by which eight [[28/27]] third-tones exceed the [[4/3]] perfect fourth. It can also be found as the amount by which seven 28/27 third-tones exceed the [[9/7]] supermajor third, or as the sum of the [[garischisma\|garischisma (33554432/33480783)]] and the [[septimal ennealimma\|septimal ennealimma (40353607/40310784)]].

	== Temperaments ==		== Temperaments ==
	Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and excluding prime 5 from the subgroup leads to the 2.3.7 subgroup rank-2 ~~Unicorn~~ temperament.		[[Tempering out]] the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and excluding prime 5 from the subgroup leads to the 2.3.7 subgroup rank-2 unicorn temperament.

	=== Octaphore ===		=== Octaphore ===
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	: mapping generators: ~2, ~28/27, ~5		: mapping generators: ~2, ~28/27, ~5

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~28/27 = 62.233, ~5/4 = 386.314		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233, ~5/4 = 386.314

	{{Optimal ET sequence\|legend=1\| 19, 39d, 58, 77, 96d, 135 }}		{{Optimal ET sequence\|legend=1\| 19, 39d, 58, 77, 96d, 135 }}

	==== Undecimal ~~Octaphore~~====		==== Undecimal octaphore ====
	By noticing that the interval at ⟨4 2 -2] is quite close to 11/8, we can add prime 11 to the mapping by tempering out the [[~~Reef~~ comma]].		By noticing that the interval at {{monzo\| 4 2 -2 }} is quite close to 11/8, we can add prime 11 to the mapping by tempering out the [[reef comma]].

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

	Comma list: 94450499584/94143178827~~, 200704/200475~~		Comma list: 200704/200475, 94450499584/94143178827

	Mapping: ~~[⟨1~~ 2 2 4 4~~], ⟨0~~ -8 0 -23 2~~], ⟨0~~ 0 1 0 -2]]		Mapping: {{mapping\| 1 2 2 4 4 \| 0 -8 0 -23 2 \| 0 0 1 0 -2 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~28/27 = 62.233, ~5/4 = 386.481		Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 62.233, ~5/4 = 386.481

	==== Tridecimal ~~Octaphore~~ ====		==== Tridecimal octaphore ====
	By noticing that two generators is extremely close to 14/13, we can add prime 13 to the mapping by tempering out the [[729/728\|~~Squbema~~]], or equivalently by tempering out the [[28812/28561\|~~Tesseract Comma~~]].		By noticing that two generators is extremely close to 14/13, we can add prime 13 to the mapping by tempering out the [[729/728\|squbema]], or equivalently by tempering out the [[28812/28561\|tesseract comma]].

	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13
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	Comma list: 729/728, 3584/3575, 660275/657072		Comma list: 729/728, 3584/3575, 660275/657072

	Mapping: ~~[⟨1~~ 2 2 4 4 5~~], ⟨0~~ -8 0 -23 2 -25~~], ⟨0~~ 0 1 0 -2 0]]		Mapping: {{mapping\| 1 2 2 4 4 5 \| 0 -8 0 -23 2 -25 \| 0 0 1 0 -2 0 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~27/26 = 62.281, ~5/4 = 386.512		Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 62.281, ~5/4 = 386.512

	===2.3.7 ~~Unicorn~~===		=== Unicorn (2.3.7 subgroup) ===
	{{See also \| Unicorn }}		{{See also \| Unicorn }}
	If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering [[126/125]].		If we temper out the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering out [[126/125]].

	[[Subgroup]]: 2.3.7		[[Subgroup]]: 2.3.7
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	: mapping generators: ~2, ~28/27		: mapping generators: ~2, ~28/27

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~28/27 = 62.233		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233

	{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}		{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}

	==See also==		== See also ==
	*[[Unicorn family]]		* [[Unicorn family]]
	*[[Unicorn comma]]		* [[Unicorn comma]]

	[[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]]

FloraC: Undo revision 188752 by Unque (talk). No duplicate data. Restore mapping generators

2025-04-03T10:37:40Z

Undo revision 188752 by Unque (talk). No duplicate data. Restore mapping generators

← Older revision		Revision as of 10:37, 3 April 2025
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	{{Mapping\|legend=1\| 1 2 2 4 \| 0 -8 0 -23 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 1 2 2 4 \| 0 -8 0 -23 \| 0 0 1 0 }}

			: mapping generators: ~2, ~28/27, ~5

	[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314
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	===2.3.7 Unicorn===		===2.3.7 Unicorn===
	{{See also \| Unicorn }}		{{See also \| Unicorn }}
	If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]]:		If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering [[126/125]].

	[[Subgroup]]: 2.3.7		[[Subgroup]]: 2.3.7
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	{{Mapping\|legend=1\| 1 2 4 \| 0 -8 -23 }}		{{Mapping\|legend=1\| 1 2 4 \| 0 -8 -23 }}

			: mapping generators: ~2, ~28/27

	[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233

	{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}		{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}

	~~==== Septimal Unicorn ====~~
	~~By noticing that the interval at five generators is nearly [[6/5]], we can introduce prime 5 to the mapping by tempering out [[126/125]].~~

	~~[[Subgroup]]: 2.3.5.7~~

	~~[[Comma list]]: 10976/10935, 126/125~~

	~~{{Mapping\|legend=1\| 1 2 3 4\| 0 -8 -13 -23 }}~~

	~~[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.324~~

	~~{{Optimal ET sequence\|legend=1\| 19, 39d, 58, 77, 135c, 212c}}~~

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

Unque: Minor formatting changes for the temperaments section

2025-03-29T14:17:34Z

Minor formatting changes for the temperaments section

← Older revision		Revision as of 14:17, 29 March 2025
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	{{Mapping\|legend=1\| 1 2 2 4 \| 0 -8 0 -23 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 1 2 2 4 \| 0 -8 0 -23 \| 0 0 1 0 }}

	~~: mapping generators: ~2, ~28/27, ~5~~

	[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314
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	===2.3.7 Unicorn===		===2.3.7 Unicorn===
	{{See also \| Unicorn }}		{{See also \| Unicorn }}
	If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]]~~, where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering [[126/125]].~~		If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]]:

	[[Subgroup]]: 2.3.7		[[Subgroup]]: 2.3.7
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	{{Mapping\|legend=1\| 1 2 4 \| 0 -8 -23 }}		{{Mapping\|legend=1\| 1 2 4 \| 0 -8 -23 }}

	~~: mapping generators: ~2, ~28/27~~

	[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233

	{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}		{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}

			==== Septimal Unicorn ====
			By noticing that the interval at five generators is nearly [[6/5]], we can introduce prime 5 to the mapping by tempering out [[126/125]].

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 10976/10935, 126/125

			{{Mapping\|legend=1\| 1 2 3 4\| 0 -8 -13 -23 }}

			[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.324

			{{Optimal ET sequence\|legend=1\| 19, 39d, 58, 77, 135c, 212c}}

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

Unque: Fixed typo in generator list

2025-01-22T20:30:24Z

Fixed typo in generator list

← Older revision		Revision as of 20:30, 22 January 2025
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	Mapping: [⟨1 2 2 4 4], ⟨0 -8 0 -23 2], ⟨0 0 1 0 -2]]		Mapping: [⟨1 2 2 4 4], ⟨0 -8 0 -23 2], ⟨0 0 1 0 -2]]

	Optimal tuning (POTE): ~2 = 1\1, ~27~~/26~~ = 62.233, ~5/4 = 386.481		Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.481

	==== Tridecimal Octaphore ====		==== Tridecimal Octaphore ====

Unque: Added 11-limit and 13-limit extensions of the rank-3 temperament

2025-01-16T22:51:12Z

Added 11-limit and 13-limit extensions of the rank-3 temperament

← Older revision		Revision as of 22:51, 16 January 2025
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	[[Comma list]]: 94450499584/94143178827		[[Comma list]]: 94450499584/94143178827

	{{Mapping\|legend=1\| 1 2 0 4 \| 0 -8 0 -23 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 1 2 2 4 \| 0 -8 0 -23 \| 0 0 1 0 }}

	: mapping generators: ~2, ~28/27, ~5		: mapping generators: ~2, ~28/27, ~5
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	{{Optimal ET sequence\|legend=1\| 19, 39d, 58, 77, 96d, 135 }}		{{Optimal ET sequence\|legend=1\| 19, 39d, 58, 77, 96d, 135 }}

	=== 2.3.7 Unicorn ===		==== Undecimal Octaphore====
	{{ See also \| Unicorn }}		By noticing that the interval at ⟨4 2 -2] is quite close to 11/8, we can add prime 11 to the mapping by tempering out the [[Reef comma]].

			Subgroup: 2.3.5.7.11

			Comma list: 94450499584/94143178827, 200704/200475

			Mapping: [⟨1 2 2 4 4], ⟨0 -8 0 -23 2], ⟨0 0 1 0 -2]]

			Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 62.233, ~5/4 = 386.481

			==== Tridecimal Octaphore ====
			By noticing that two generators is extremely close to 14/13, we can add prime 13 to the mapping by tempering out the [[729/728\|Squbema]], or equivalently by tempering out the [[28812/28561\|Tesseract Comma]].

			Subgroup: 2.3.5.7.11.13

			Comma list: 729/728, 3584/3575, 660275/657072

			Mapping: [⟨1 2 2 4 4 5], ⟨0 -8 0 -23 2 -25], ⟨0 0 1 0 -2 0]]

			Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 62.281, ~5/4 = 386.512

			===2.3.7 Unicorn===
			{{See also \| Unicorn }}
	If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering [[126/125]].		If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering [[126/125]].

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	{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}		{{Optimal ET sequence\|legend=1\| 19, 20d, 39d, 58, 77, 96d, 135 }}

	== See also ==		==See also==
	* [[Unicorn family]]		*[[Unicorn family]]
	* [[Unicorn comma]]		*[[Unicorn comma]]

	[[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]]

BudjarnLambeth at 00:25, 4 November 2024

2024-11-04T00:25:36Z

← Older revision		Revision as of 00:25, 4 November 2024
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	* [[Unicorn comma]]		* [[Unicorn comma]]

	[[Category:Commas named for ~~their regular temperament properties~~]]		[[Category:Commas named for how they divide the fourth]]
			[[Category:Commas named for the intervals they stack]]

BudjarnLambeth at 09:37, 3 November 2024

2024-11-03T09:37:10Z

← Older revision		Revision as of 09:37, 3 November 2024
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	* [[Unicorn family]]		* [[Unicorn family]]
	* [[Unicorn comma]]		* [[Unicorn comma]]

			[[Category:Commas named for their regular temperament properties]]

Unque: /* Temperaments */

2024-10-03T20:09:37Z

Temperaments

← Older revision		Revision as of 20:09, 3 October 2024
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	== Temperaments ==		== Temperaments ==
	Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and ~~tempering it out in~~ the 2.3.7 subgroup ~~leads to the~~ rank-2 ~~actinide~~ temperament.		Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and excluding prime 5 from the subgroup leads to the 2.3.7 subgroup rank-2 Unicorn temperament.

	=== Octaphore ===		=== Octaphore ===

Godtone: name change

2024-10-03T19:01:43Z

name change

← Older revision		Revision as of 19:01, 3 October 2024
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	}}		}}

	'''~~The~~ octaphore''', also known as the '''enneagari comma''', is a [[small comma\|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 5.64 [[cent]]s. It is so named because it is the amount by which eight [[28/27]] third-tones exceed the [[4/3]] perfect fourth. It can also be found as the amount by which seven 28/27 third-tones exceed the [[9/7]] supermajor third, or as the sum of the [[garischisma\|garischisma (33554432/33480783)]] and the [[septimal ennealimma\|septimal ennealimma (40353607/40310784)]].		The '''octaphore''', also known as the '''enneagari comma''', is a [[small comma\|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 5.64 [[cent]]s. It is so named because it is the amount by which eight [[28/27]] third-tones exceed the [[4/3]] perfect fourth. It can also be found as the amount by which seven 28/27 third-tones exceed the [[9/7]] supermajor third, or as the sum of the [[garischisma\|garischisma (33554432/33480783)]] and the [[septimal ennealimma\|septimal ennealimma (40353607/40310784)]].

	== Temperaments ==		== Temperaments ==

FloraC: FloraC moved page The octaphore to Octaphore: WP:THE

2024-10-03T18:58:41Z

FloraC moved page The octaphore to Octaphore: WP:THE

← Older revision	Revision as of 18:58, 3 October 2024
(No difference)