Pentacircle clan - Revision history

FloraC: Preliminary cleanup on the intros. - redundant category

2026-03-16T13:44:42Z

Preliminary cleanup on the intros. - redundant category

← Older revision		Revision as of 13:44, 16 March 2026
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	== Pentafrost ==		== Pentafrost ==
	Pentafrost tempers out the [[245/242\|frostma]] in addition to 896/891 which also means that the [[schisma]] is tempered out, mapping prime 5 to 8 [[4/3\|perfect fourths]] and ~~-1 octaves~~.		Pentafrost tempers out the [[245/242\|frostma]] in addition to 896/891 which also means that the [[schisma]] is tempered out, mapping prime 5 to eight [[4/3\|perfect fourths]] minus an octave.

			It was named by [[Tristan Bay]] in 2024 as a portmanteau of ''pentacircle'' and ''frost''.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	== Trienparapyth ==		== Trienparapyth ==
	~~Trienparapyth~~ can be described as the ~~no-17's 23-limit~~ {{nowrap\| 80 & 87 ~~& 109~~ }} temperament. It splits the ~4/3 generator of [[~~#Parapythic\|parapythic~~]] ~~into three ~~~[[11/10]]'s by tempering out [[4000/3993\|~~4000/3993 =~~ S10/S11]] in the 11-limit ~~and it~~ interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299\|~~2300/2299 =~~ S20/S22]], or optionally less accurately as ~[[17/14]], though because this mapping only really makes much sense in [[80edo]] it is not included here; however, its connection to parapyth structure comes from later in the generator chain; specifically, from (11/10)<sup>7</sup> onwards. We may simplify (11/10)<sup>7</sup> as [[16/9\|(4/3)<sup>2</sup>]]([[11/10]]) = [[88/45]], the octave-complement of [[45/44]]. Notice that parapythic wants a slightly flat ~4/3 corresponding to an 11/10 being tuned anywhere from around just (in an extremely sharp-for-parapyth tuning) to a little less than 1-cent sharp, a very narrow tuning range; therefore 88/45 is flattened so that 2/(11/10)<sup>7</sup>~45/44 is sharpened so that we can equate it with [[40/39]], tempering out (40/39)/(45/44) = [[352/351]], and because we know we want prime 19 later on, we equate this with [[39/38]] by tempering out the pinkanberry, [[1521/1520~~\|1521/1520 = S39~~]]. Next, for eight generator steps, observe that (11/10)<sup>9</sup>/(11/10)/2 = (4/3)<sup>3</sup>/(11/10)/2 = ([[32/27]])/(11/10) = 320/297 is sharp of [[15/14]] by [[896/891]], which is reasonable to equate it with because in an optimal tuning 11/10 will be very slightly sharp so that the interval of eight generator steps is eight times as sharp. Thus, tempering out [[896/891]] and [[4000/3993]] defines trienparapyth in the 11-limit, which also tempers out [[3388/3375]], the 13-limit adds [[352/351]], the no-17's 19-limit equates 40/39 with 39/38 and the no-17's 23-limit equates 23/19 with (11/10)<sup>2</sup> as already mentioned.		Named by [[Godtone]] in 2024, trienparapyth can be described as the {{nowrap\| 58 & 80 & 87 }} temperament, with an extension to the no-17's 23-limit. It splits the ~4/3 generator of parapythic into three [[~]][[11/10]]'s by tempering out [[4000/3993]] ([[S-expression\|S10/S11]]) in the 11-limit. It further interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]] ([[S-expression\|S20/S22]]), or optionally less accurately as ~[[17/14]], though because this mapping only really makes much sense in [[80edo]] it is not included here; however, its connection to parapyth structure comes from later in the generator chain; specifically, from (11/10)<sup>7</sup> onwards. We may simplify (11/10)<sup>7</sup> as [[16/9\|(4/3)<sup>2</sup>]]([[11/10]]) = [[88/45]], the octave-complement of [[45/44]]. Notice that parapythic wants a slightly flat ~4/3 corresponding to an 11/10 being tuned anywhere from around just (in an extremely sharp-for-parapyth tuning) to a little less than 1-cent sharp, a very narrow tuning range; therefore 88/45 is flattened so that 2/(11/10)<sup>7</sup>~45/44 is sharpened so that we can equate it with [[40/39]], tempering out (40/39)/(45/44) = [[352/351]], and because we know we want prime 19 later on, we equate this with [[39/38]] by tempering out the pinkanberry, [[1521/1520]] ({{S\|39}}). Next, for eight generator steps, observe that (11/10)<sup>9</sup>/(11/10)/2 = (4/3)<sup>3</sup>/(11/10)/2 = ([[32/27]])/(11/10) = 320/297 is sharp of [[15/14]] by [[896/891]], which is reasonable to equate it with because in an optimal tuning 11/10 will be very slightly sharp so that the interval of eight generator steps is eight times as sharp. Thus, tempering out [[896/891]] and [[4000/3993]] defines trienparapyth in the 11-limit, which also tempers out [[3388/3375]], the 13-limit adds [[352/351]], the no-17's 19-limit equates 40/39 with 39/38 and the no-17's 23-limit equates 23/19 with (11/10)<sup>2</sup> as already mentioned.

	Structurally, trienparapyth is three copies of parapyth with the independent generator of 7 connected to an equivalent independent generator for 5 through the ~[[15/7]] reached at (11/10)<sup>8</sup> so that ~[[20/7]] is reached at (11/10)<sup>11</sup>, and this is how the last generator can be either 5 or 7.		Structurally, trienparapyth is three copies of parapyth with the independent generator of 7 connected to an equivalent independent generator for 5 through the ~[[15/7]] reached at (11/10)<sup>8</sup> so that ~[[20/7]] is reached at (11/10)<sup>11</sup>, and this is how the last generator can be either 5 or 7.
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	[[Category:Pentacircle clan\| ]] <!-- main article -->		[[Category:Pentacircle clan\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]
	~~[[Category:Tolerant]]~~

FloraC: - CTE, POTE & CEE tunings

2026-03-16T13:31:55Z

- CTE, POTE & CEE tunings

Show changes

FloraC: Switch to WE & CWE tunings

2026-03-16T13:29:49Z

Switch to WE & CWE tunings

Show changes

FloraC: Units & misc. cleanup

2026-03-16T12:49:13Z

Units & misc. cleanup

Show changes

Xenllium at 13:14, 22 January 2026

2026-01-22T13:14:29Z

Show changes

FloraC: Text replacement - "Category:Temperament clans↵Category:Pages with mostly numerical content" to "Category:Temperament clans"

2025-11-27T12:30:54Z

Text replacement - "Category:Temperament clans↵Category:Pages with mostly numerical content" to "Category:Temperament clans"

← Older revision		Revision as of 12:30, 27 November 2025
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	[[Category:Temperament clans]]		[[Category:Temperament clans]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Pentacircle clan\| ]] <!-- main article -->		[[Category:Pentacircle clan\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]
	[[Category:Tolerant]]		[[Category:Tolerant]]

FloraC: Update

2025-08-29T14:12:32Z

Update

← Older revision		Revision as of 14:12, 29 August 2025
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	* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]		* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]
	* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]		* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]
	* ''[[~~Julius]]'' or ''[[varda~~]]'' → [[Diaschismic rank-3 family #~~Julius aka varda~~\|Diaschismic rank-3 family]]		* ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda\|Diaschismic rank-3 family]]
	* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif\|Rastmic rank-3 clan]]		* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif\|Rastmic rank-3 clan]]
	* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]		* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]

FloraC: Fix linking

2025-08-27T13:05:35Z

Fix linking

← Older revision		Revision as of 13:05, 27 August 2025
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	Temperaments discussed elsewhere are:		Temperaments discussed elsewhere are:
	* ''[[Melpomene]]'' → [[Didymus rank ~~three~~ family #Melpomene\|Didymus rank-3 family]]		* ''[[Melpomene]]'' → [[Didymus rank-3 family #Melpomene\|Didymus rank-3 family]]
	* ''[[Apollo]]'' → [[Marvel family #Apollo\|Marvel family]]		* ''[[Apollo]]'' → [[Marvel family #Apollo\|Marvel family]]
	* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic\|Sensamagic family]]		* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic\|Sensamagic family]]
	* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]		* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]
	* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]		* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]
	* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank ~~three~~ family #Julius aka varda\|Diaschismic rank-3 family]]		* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank-3 family #Julius aka varda\|Diaschismic rank-3 family]]
	* ''[[Parahemif]]'' → [[Rastmic rank ~~three~~ clan #Parahemif\|Rastmic rank-3 clan]]		* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif\|Rastmic rank-3 clan]]
	* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]		* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]

Lériendil: Text replacement - "Category:Temperament clans" to "Category:Temperament clans Category:Pages with mostly numerical content"

2025-06-24T00:39:45Z

Text replacement - "Category:Temperament clans" to "Category:Temperament clans Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:39, 24 June 2025
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	[[Category:Temperament clans]]		[[Category:Temperament clans]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Pentacircle clan\| ]] <!-- main article -->		[[Category:Pentacircle clan\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]
	[[Category:Tolerant]]		[[Category:Tolerant]]

BudjarnLambeth: {{Technical data page}}

2025-04-13T14:25:59Z

← Older revision		Revision as of 14:25, 13 April 2025
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			{{Technical data page}}
	The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. This has the effect of identifying [[14/11]] at the [[81/64\|Pythagorean major third]].		The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. This has the effect of identifying [[14/11]] at the [[81/64\|Pythagorean major third]].

← Older revision		Revision as of 14:12, 29 August 2025
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	* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]		* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]
	* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]		* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]
	* ''[[~~Julius]]'' or ''[[varda~~]]'' → [[Diaschismic rank-3 family #~~Julius aka varda~~\|Diaschismic rank-3 family]]		* ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda\|Diaschismic rank-3 family]]
	* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif\|Rastmic rank-3 clan]]		* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif\|Rastmic rank-3 clan]]
	* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]		* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]

← Older revision		Revision as of 13:05, 27 August 2025
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	Temperaments discussed elsewhere are:		Temperaments discussed elsewhere are:
	* ''[[Melpomene]]'' → [[Didymus rank ~~three~~ family #Melpomene\|Didymus rank-3 family]]		* ''[[Melpomene]]'' → [[Didymus rank-3 family #Melpomene\|Didymus rank-3 family]]
	* ''[[Apollo]]'' → [[Marvel family #Apollo\|Marvel family]]		* ''[[Apollo]]'' → [[Marvel family #Apollo\|Marvel family]]
	* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic\|Sensamagic family]]		* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic\|Sensamagic family]]
	* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]		* [[Pele]] → [[Hemifamity family #Pele\|Hemifamity family]]
	* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]		* ''[[Uni]]'' → [[Hemimage family #Uni\|Hemimage family]]
	* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank ~~three~~ family #Julius aka varda\|Diaschismic rank-3 family]]		* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank-3 family #Julius aka varda\|Diaschismic rank-3 family]]
	* ''[[Parahemif]]'' → [[Rastmic rank ~~three~~ clan #Parahemif\|Rastmic rank-3 clan]]		* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif\|Rastmic rank-3 clan]]
	* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]		* ''[[Canta]]'' → [[Canou family #Canta\|Canou family]]