323edo - Revision history

FloraC: /* Regular temperament properties */ + heptacot

2026-05-20T12:11:52Z

Regular temperament properties: + heptacot

← Older revision		Revision as of 12:11, 20 May 2026
Line 108:		Line 108:
	\| 200/189		\| 200/189
	\| [[Hemiluna]] (323)		\| [[Hemiluna]] (323)
			\|-
			\| 1
			\| 27\323
			\| 100.31
			\| 675/637
			\| [[Heptacot]] (323)
	\|-		\|-
	\| 1		\| 1

FloraC: Improve linking

2026-04-23T06:46:44Z

Improve linking

← Older revision		Revision as of 06:46, 23 April 2026
Line 5:		Line 5:
	323edo is a strong [[5-limit]] system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].		323edo is a strong [[5-limit]] system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

	As an equal temperament, it [[tempering out\|tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], ~~supporting~~ 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].		As an equal temperament, it [[tempering out\|tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the 5-limit; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], [[support]]ing 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

	In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. [[1375/1372]], [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], [[12005/11979]], and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.		In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. [[1375/1372]], [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], [[12005/11979]], and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] temperament tempering out 14641/14580.

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: + subsets and supersets. + secondary harmonics table

2026-04-23T06:45:12Z

+ subsets and supersets. + secondary harmonics table

← Older revision		Revision as of 06:45, 23 April 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].		323edo is a strong [[5-limit]] system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

	As an equal temperament, it [[tempering out\|tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].		As an equal temperament, it [[tempering out\|tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

	In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.		In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. [[1375/1372]], [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], [[12005/11979]], and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

	~~Since 323 factors into~~ {{~~factorisation~~\|323}}~~, 323edo shares the excellent approximations of [[25/24]]~~ in ~~[[17edo]] and~~ of ~~[[28/27]] and the [[6/5]]~~ in ~~[[19edo]].~~		=== Prime harmonics ===
			{{Harmonics in equal\|323\|columns=11}}
			{{Harmonics in equal\|323\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 323edo (continued)}}

	=== ~~Prime harmonics~~ ===		=== Subsets and supersets ===
	{{~~Harmonics in equal~~\|~~323~~}}		Since 323 factors into primes as {{nowrap\| 17 × 19 }}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[6/5]] and [[28/27]] in [[19edo]].

	== Regular temperament properties ==		== Regular temperament properties ==

FloraC: Interpretation of vulture generator

2026-04-23T06:40:42Z

Interpretation of vulture generator

← Older revision		Revision as of 06:40, 23 April 2026
Line 20:		Line 20:
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
Line 27:		Line 27:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 512 -323 }}		\| {{Monzo\| 512 -323 }}
	\| {{~~mapping~~\| 323 512 }}		\| {{Mapping\| 323 512 }}
	\| −0.0669		\| −0.0669
	\| 0.0669		\| 0.0669
Line 34:		Line 34:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 24 -21 4 }}, {{monzo\| 38 -2 -15 }}		\| {{Monzo\| 24 -21 4 }}, {{monzo\| 38 -2 -15 }}
	\| {{~~mapping~~\| 323 512 750 }}		\| {{Mapping\| 323 512 750 }}
	\| −0.0538		\| −0.0538
	\| 0.0577		\| 0.0577
Line 42:		Line 42:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 589824/588245, 703125/702464		\| 4375/4374, 589824/588245, 703125/702464
	\| {{~~mapping~~\| 323 512 750 907 }}		\| {{Mapping\| 323 512 750 907 }}
	\| −0.1146		\| −0.1146
	\| 0.1165		\| 0.1165
Line 49:		Line 49:
	\| 2.3.5.7.13		\| 2.3.5.7.13
	\| 676/675, 4096/4095, 4375/4374, 16848/16807		\| 676/675, 4096/4095, 4375/4374, 16848/16807
	\| {{~~mapping~~\| 323 512 750 907 1195 }}		\| {{Mapping\| 323 512 750 907 1195 }}
	\| −0.0431		\| −0.0431
	\| 0.1770		\| 0.1770
Line 56:		Line 56:
	\| 2.3.5.7.13.17		\| 2.3.5.7.13.17
	\| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374		\| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374
	\| {{~~mapping~~\| 323 512 750 907 1195 1320 }}		\| {{Mapping\| 323 512 750 907 1195 1320 }}
	\| +0.0020		\| +0.0020
	\| 0.1905		\| 0.1905
Line 63:		Line 63:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 1375/1372, 4375/4374, 5632/5625, 14641/14580		\| 1375/1372, 4375/4374, 5632/5625, 14641/14580
	\| {{~~mapping~~\| 323 512 750 907 1117 }} (323)		\| {{Mapping\| 323 512 750 907 1117 }} (323)
	\| −0.0066		\| −0.0066
	\| 0.2399		\| 0.2399
Line 70:		Line 70:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374		\| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
	\| {{~~mapping~~\| 323 512 750 907 1117 1195 }} (323)		\| {{Mapping\| 323 512 750 907 1117 1195 }} (323)
	\| +0.0350		\| +0.0350
	\| 0.2380		\| 0.2380
Line 77:		Line 77:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 540/539, 4375/4374, 12005/11979, 16384/16335		\| 540/539, 4375/4374, 12005/11979, 16384/16335
	\| {{~~mapping~~\| 323 512 750 907 1118 }} (323e)		\| {{Mapping\| 323 512 750 907 1118 }} (323e)
	\| −0.2213		\| −0.2213
	\| 0.2375		\| 0.2375
Line 84:		Line 84:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 364/363, 540/539, 676/675, 4096/4095, 4375/4374		\| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
	\| {{~~mapping~~\| 323 512 750 907 1118 1195 }} (323e)		\| {{Mapping\| 323 512 750 907 1118 1195 }} (323e)
	\| −0.1440		\| −0.1440
	\| 0.2773		\| 0.2773
Line 95:		Line 95:
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 134:		Line 134:
	\| 128\323		\| 128\323
	\| 475.54		\| 475.54
	\| ~~320~~/~~243~~		\| 25/19
	\| [[Vulture]]		\| [[Vulture]]
	\|-		\|-
Line 149:		Line 149:
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|}		\|}
	<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	[[Category:Deuteromere]]		[[Category:Deuteromere]]
	[[Category:Lambeth]]		[[Category:Lambeth]]
	[[Category:Stockhausenic]]		[[Category:Stockhausenic]]

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:31Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
Line 149:		Line 149:
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	[[Category:Deuteromere]]		[[Category:Deuteromere]]
	[[Category:Lambeth]]		[[Category:Lambeth]]
	[[Category:Stockhausenic]]		[[Category:Stockhausenic]]

ArrowHead294 at 19:13, 20 February 2025

2025-02-20T19:13:17Z

← Older revision		Revision as of 19:13, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|323~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T15:54:41Z

Theory: cleanup

← Older revision		Revision as of 15:54, 17 January 2025
Line 5:		Line 5:
	323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].		323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

	It [[tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].		As an equal temperament, it [[tempering out\|tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

	In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.		In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

ArrowHead294 at 15:21, 16 January 2025

2025-01-16T15:21:21Z

← Older revision		Revision as of 15:21, 16 January 2025
Line 5:		Line 5:
	323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].		323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

	~~The equal temperament~~ [[~~tempering out\|~~tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]] and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].		It [[tempers out]] the [[vulture comma]], {{monzo\| 24 -21 4 }} and the [[luna comma]], {{monzo\| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

	In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.		In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

	Since {{~~nowrap~~\|323 ~~{{=}} 17 × 19~~}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].		Since 323 factors into {{factorisation\|323}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 29:		Line 29:
	\| {{monzo\| 512 -323 }}		\| {{monzo\| 512 -323 }}
	\| {{mapping\| 323 512 }}		\| {{mapping\| 323 512 }}
	\| ~~−0~~.0669		\| −0.0669
	\| 0.0669		\| 0.0669
	\| 1.80		\| 1.80
Line 36:		Line 36:
	\| {{monzo\| 24 -21 4 }}, {{monzo\| 38 -2 -15 }}		\| {{monzo\| 24 -21 4 }}, {{monzo\| 38 -2 -15 }}
	\| {{mapping\| 323 512 750 }}		\| {{mapping\| 323 512 750 }}
	\| ~~−0~~.0538		\| −0.0538
	\| 0.0577		\| 0.0577
	\| 1.55		\| 1.55
Line 43:		Line 43:
	\| 4375/4374, 589824/588245, 703125/702464		\| 4375/4374, 589824/588245, 703125/702464
	\| {{mapping\| 323 512 750 907 }}		\| {{mapping\| 323 512 750 907 }}
	\| ~~−0~~.1146		\| −0.1146
	\| 0.1165		\| 0.1165
	\| 3.14		\| 3.14
Line 50:		Line 50:
	\| 676/675, 4096/4095, 4375/4374, 16848/16807		\| 676/675, 4096/4095, 4375/4374, 16848/16807
	\| {{mapping\| 323 512 750 907 1195 }}		\| {{mapping\| 323 512 750 907 1195 }}
	\| ~~−0~~.0431		\| −0.0431
	\| 0.1770		\| 0.1770
	\| 4.76		\| 4.76
Line 64:		Line 64:
	\| 1375/1372, 4375/4374, 5632/5625, 14641/14580		\| 1375/1372, 4375/4374, 5632/5625, 14641/14580
	\| {{mapping\| 323 512 750 907 1117 }} (323)		\| {{mapping\| 323 512 750 907 1117 }} (323)
	\| ~~−0~~.0066		\| −0.0066
	\| 0.2399		\| 0.2399
	\| 6.46		\| 6.46
Line 78:		Line 78:
	\| 540/539, 4375/4374, 12005/11979, 16384/16335		\| 540/539, 4375/4374, 12005/11979, 16384/16335
	\| {{mapping\| 323 512 750 907 1118 }} (323e)		\| {{mapping\| 323 512 750 907 1118 }} (323e)
	\| ~~−0~~.2213		\| −0.2213
	\| 0.2375		\| 0.2375
	\| 6.39		\| 6.39
Line 85:		Line 85:
	\| 364/363, 540/539, 676/675, 4096/4095, 4375/4374		\| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
	\| {{mapping\| 323 512 750 907 1118 1195 }} (323e)		\| {{mapping\| 323 512 750 907 1118 1195 }} (323e)
	\| ~~−0~~.1440		\| −0.1440
	\| 0.2773		\| 0.2773
	\| 7.47		\| 7.47
Line 149:		Line 149:
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:Deuteromere]]		[[Category:Deuteromere]]
	[[Category:Lambeth]]		[[Category:Lambeth]]
	[[Category:Stockhausenic]]		[[Category:Stockhausenic]]

ArrowHead294: Partial undo

2024-11-16T12:14:43Z

Partial undo

← Older revision		Revision as of 12:14, 16 November 2024
Line 15:		Line 15:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
Line 79:		Line 88:
	\| 0.2773		\| 0.2773
	\| 7.47		\| 7.47
	~~{{comma basis end}~~}		\|}
	* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo\|289]] and followed by [[388edo\|388]].		* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo\|289]] and followed by [[388edo\|388]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
Line 132:		Line 148:
	\| 4/3<br>(225/224)		\| 4/3<br>(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	[[Category:Deuteromere]]		[[Category:Deuteromere]]
	[[Category:Lambeth]]		[[Category:Lambeth]]
	[[Category:Stockhausenic]]		[[Category:Stockhausenic]]

ArrowHead294 at 05:15, 16 November 2024

2024-11-16T05:15:03Z

← Older revision		Revision as of 05:15, 16 November 2024
Line 9:		Line 9:
	In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.		In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

	323 = 17 × 19, ~~and~~ shares the excellent approximations of [[25/24]] in [[17edo]] and of ~~the~~ [[28/27]] and the [[6/5]] in [[19edo]].		Since {{nowrap\|323 {{=}} 17 × 19}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 15:		Line 15:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal 8ve <br>Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{monzo\| 512 -323 }}		\| {{monzo\| 512 -323 }}
	\| {{mapping\| 323 512 }}		\| {{mapping\| 323 512 }}
	\| -0.0669		\| −0.0669
	\| 0.0669		\| 0.0669
	\| 1.80		\| 1.80
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	\| {{monzo\| 24 -21 4 }}, {{monzo\| 38 -2 -15 }}		\| {{monzo\| 24 -21 4 }}, {{monzo\| 38 -2 -15 }}
	\| {{mapping\| 323 512 750 }}		\| {{mapping\| 323 512 750 }}
	\| -0.0538		\| −0.0538
	\| 0.0577		\| 0.0577
	\| 1.55		\| 1.55
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	\| 4375/4374, 589824/588245, 703125/702464		\| 4375/4374, 589824/588245, 703125/702464
	\| {{mapping\| 323 512 750 907 }}		\| {{mapping\| 323 512 750 907 }}
	\| -0.1146		\| −0.1146
	\| 0.1165		\| 0.1165
	\| 3.14		\| 3.14
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	\| 676/675, 4096/4095, 4375/4374, 16848/16807		\| 676/675, 4096/4095, 4375/4374, 16848/16807
	\| {{mapping\| 323 512 750 907 1195 }}		\| {{mapping\| 323 512 750 907 1195 }}
	\| -0.0431		\| −0.0431
	\| 0.1770		\| 0.1770
	\| 4.76		\| 4.76
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	\| 0.1905		\| 0.1905
	\| 5.13		\| 5.13
			\|- style="border-top: double;"
			\| 2.3.5.7.11
			\| 1375/1372, 4375/4374, 5632/5625, 14641/14580
			\| {{mapping\| 323 512 750 907 1117 }} (323)
			\| −0.0066
			\| 0.2399
			\| 6.46
	\|-		\|-
	\| style="border-top: double;" \| 2.3.5.7.11		\| 2.3.5.7.11.13
	~~\| style="border-top: double;"~~ \| 540/539, 4375/4374, 12005/11979, 16384/16335		\| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
	~~\| style="border-top: double;"~~ \| {{mapping\| 323 512 750 907 1118 }} (323e)		\| {{mapping\| 323 512 750 907 1117 1195 }} (323)
	\| ~~style="border-top: double~~;~~" \| -~~0.2213		\| +0.0350
	~~\| style="border-top: double;"~~ \| 0.2375		\| 0.2380
	~~\| style="border-top: double;"~~ \| 6.39		\| 6.40
			\|- style="border-top: double;"
			\| 2.3.5.7.11
			\| 540/539, 4375/4374, 12005/11979, 16384/16335
			\| {{mapping\| 323 512 750 907 1118 }} (323e)
			\| −0.2213
			\| 0.2375
			\| 6.39
	\|-		\|-
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 364/363, 540/539, 676/675, 4096/4095, 4375/4374		\| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
	\| {{mapping\| 323 512 750 907 1118 1195 }} (323e)		\| {{mapping\| 323 512 750 907 1118 1195 }} (323e)
	\| -0.1440		\| −0.1440
	\| 0.2773		\| 0.2773
	\| 7.47		\| 7.47
	\|-		{{comma basis end}}
	~~\| style="border-top: double;" \| 2.3.5.7.11~~
	~~\| style="border-top: double;" \| 1375/1372, 4375/4374, 5632/5625, 14641/14580~~
	~~\| style="border-top: double;" \|~~ {{~~mapping\| 323 512 750 907 1117~~ }~~} (323)~~
	~~\| style="border-top: double;" \| -0.0066~~
	~~\| style="border-top: double;" \| 0.2399~~
	~~\| style="border-top: double;" \| 6.46~~
	\|-
	~~\| 2.3.5.7.11.13~~
	~~\| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374~~
	~~\| {{mapping\| 323 512 750 907 1117 1195 }} (323)~~
	~~\| +0.0350~~
	~~\| 0.2380~~
	~~\| 6.40~~
	\|}
	* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo\|289]] and followed by [[388edo\|388]].		* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo\|289]] and followed by [[388edo\|388]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\| 4/3<br>(225/224)		\| 4/3<br>(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	[[Category:Deuteromere]]		[[Category:Deuteromere]]
			[[Category:Lambeth]]
	[[Category:Stockhausenic]]		[[Category:Stockhausenic]]
	~~[[Category:Lambeth]]~~