193edo - Revision history

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"

2026-03-13T13:32:47Z

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 150:		Line 150:
	\| [[Kwai]]		\| [[Kwai]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==

ArrowHead294 at 14:11, 20 February 2025

2025-02-20T14:11:27Z

← Older revision		Revision as of 14:11, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|193~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T15:29:28Z

Theory: cleanup

← Older revision		Revision as of 15:29, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.		193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong [[23-limit]] system.

	~~193et~~ [[tempers out]] the [[15625/15552\|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], and 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the 19-limit; and [[460/459]], [[507/506]], and [[529/528]] in the 23-limit.		As an equal temperament, it [[tempering out\|tempers out]] the [[15625/15552\|kleisma]] in the [[5-limit]]; [[5120/5103]] and [[16875/16807]] in the [[7-limit]]; [[540/539]], [[1375/1372]], [[3025/3024]], and 4375/4356 in the [[11-limit]]; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the [[13-limit]]; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]; and [[460/459]], [[507/506]], and [[529/528]] in the 23-limit.

	It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.		It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.

ArrowHead294 at 14:55, 16 January 2025

2025-01-16T14:55:46Z

← Older revision		Revision as of 14:55, 16 January 2025
Line 5:		Line 5:
	193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.		193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.

	~~As an equal temperament,~~ 193et [[~~tempering out\|~~tempers out]] the [[15625/15552\|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[460/459]], [[507/506]], [[529/528]] in the 23-limit.		193et [[tempers out]] the [[15625/15552\|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], and 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the 19-limit; and [[460/459]], [[507/506]], and [[529/528]] in the 23-limit.

	It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.		It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.
Line 30:		Line 30:
	\| {{monzo\| 306 -193 }}		\| {{monzo\| 306 -193 }}
	\| {{mapping\| 193 306 }}		\| {{mapping\| 193 306 }}
	\| ~~−0~~.2005		\| −0.2005
	\| 0.2005		\| 0.2005
	\| 3.23		\| 3.23
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	\| 15625/15552, {{monzo\| 50 -33 1 }}		\| 15625/15552, {{monzo\| 50 -33 1 }}
	\| {{mapping\| 193 306 448 }}		\| {{mapping\| 193 306 448 }}
	\| ~~−0~~.0158		\| −0.0158
	\| 0.3084		\| 0.3084
	\| 4.96		\| 4.96
Line 44:		Line 44:
	\| 5120/5103, 15625/15552, 16875/16807		\| 5120/5103, 15625/15552, 16875/16807
	\| {{mapping\| 193 306 448 542 }}		\| {{mapping\| 193 306 448 542 }}
	\| ~~−0~~.1118		\| −0.1118
	\| 0.3146		\| 0.3146
	\| 5.06		\| 5.06
Line 51:		Line 51:
	\| 540/539, 1375/1372, 4375/4356, 5120/5103		\| 540/539, 1375/1372, 4375/4356, 5120/5103
	\| {{mapping\| 193 306 448 542 668 }}		\| {{mapping\| 193 306 448 542 668 }}
	\| ~~−0~~.2080		\| −0.2080
	\| 0.3408		\| 0.3408
	\| 5.48		\| 5.48
Line 58:		Line 58:
	\| 325/324, 364/363, 540/539, 625/624, 4096/4095		\| 325/324, 364/363, 540/539, 625/624, 4096/4095
	\| {{mapping\| 193 306 448 542 668 714 }}		\| {{mapping\| 193 306 448 542 668 714 }}
	\| ~~−0~~.1216		\| −0.1216
	\| 0.3662		\| 0.3662
	\| 5.89		\| 5.89
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	\| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095		\| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
	\| {{mapping\| 193 306 448 542 668 714 789 }}		\| {{mapping\| 193 306 448 542 668 714 789 }}
	\| ~~−0~~.1302		\| −0.1302
	\| 0.3397		\| 0.3397
	\| 5.46		\| 5.46
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	\| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215		\| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
	\| {{mapping\| 193 306 448 542 668 714 789 820 }}		\| {{mapping\| 193 306 448 542 668 714 789 820 }}
	\| ~~−0~~.1414		\| −0.1414
	\| 0.3191		\| 0.3191
	\| 5.13		\| 5.13
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	\| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528		\| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528
	\| {{mapping\| 193 306 448 542 668 714 789 820 873 }}		\| {{mapping\| 193 306 448 542 668 714 789 820 873 }}
	\| ~~−0~~.1184		\| −0.1184
	\| 0.3078		\| 0.3078
	\| 4.95		\| 4.95
Line 150:		Line 150:
	\| [[Kwai]]		\| [[Kwai]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==

ArrowHead294 at 12:13, 16 November 2024

2024-11-16T12:13:50Z

← Older revision		Revision as of 12:13, 16 November 2024
Line 16:		Line 16:

	== 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
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	\| 0.3078		\| 0.3078
	\| 4.95		\| 4.95
	~~{{comma basis end}~~}		\|}
	* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].		* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].
	* 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo\|212gh]].		* 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo\|212gh]].

	=== 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
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	\| 4/3		\| 4/3
	\| [[Kwai]]		\| [[Kwai]]
	~~{{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

	== Scales ==		== Scales ==

ArrowHead294 at 05:17, 16 November 2024

2024-11-16T05:17:32Z

← Older revision		Revision as of 05:17, 16 November 2024
Line 16:		Line 16:

	== 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\| 306 -193 }}		\| {{monzo\| 306 -193 }}
	\| {{mapping\| 193 306 }}		\| {{mapping\| 193 306 }}
	\| -0.2005		\| −0.2005
	\| 0.2005		\| 0.2005
	\| 3.23		\| 3.23
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	\| 15625/15552, {{monzo\| 50 -33 1 }}		\| 15625/15552, {{monzo\| 50 -33 1 }}
	\| {{mapping\| 193 306 448 }}		\| {{mapping\| 193 306 448 }}
	\| -0.0158		\| −0.0158
	\| 0.3084		\| 0.3084
	\| 4.96		\| 4.96
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	\| 5120/5103, 15625/15552, 16875/16807		\| 5120/5103, 15625/15552, 16875/16807
	\| {{mapping\| 193 306 448 542 }}		\| {{mapping\| 193 306 448 542 }}
	\| -0.1118		\| −0.1118
	\| 0.3146		\| 0.3146
	\| 5.06		\| 5.06
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	\| 540/539, 1375/1372, 4375/4356, 5120/5103		\| 540/539, 1375/1372, 4375/4356, 5120/5103
	\| {{mapping\| 193 306 448 542 668 }}		\| {{mapping\| 193 306 448 542 668 }}
	\| -0.2080		\| −0.2080
	\| 0.3408		\| 0.3408
	\| 5.48		\| 5.48
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	\| 325/324, 364/363, 540/539, 625/624, 4096/4095		\| 325/324, 364/363, 540/539, 625/624, 4096/4095
	\| {{mapping\| 193 306 448 542 668 714 }}		\| {{mapping\| 193 306 448 542 668 714 }}
	\| -0.1216		\| −0.1216
	\| 0.3662		\| 0.3662
	\| 5.89		\| 5.89
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	\| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095		\| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
	\| {{mapping\| 193 306 448 542 668 714 789 }}		\| {{mapping\| 193 306 448 542 668 714 789 }}
	\| -0.1302		\| −0.1302
	\| 0.3397		\| 0.3397
	\| 5.46		\| 5.46
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	\| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215		\| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
	\| {{mapping\| 193 306 448 542 668 714 789 820 }}		\| {{mapping\| 193 306 448 542 668 714 789 820 }}
	\| -0.1414		\| −0.1414
	\| 0.3191		\| 0.3191
	\| 5.13		\| 5.13
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	\| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528		\| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528
	\| {{mapping\| 193 306 448 542 668 714 789 820 873 }}		\| {{mapping\| 193 306 448 542 668 714 789 820 873 }}
	\| -0.1184		\| −0.1184
	\| 0.3078		\| 0.3078
	\| 4.95		\| 4.95
	\|}		{{comma basis end}}
	* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].		* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].
	* 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo\|212gh]].		* 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo\|212gh]].

	=== 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*
	~~! Temperament~~
	\|-		\|-
	\| 1		\| 1
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	\| 4/3		\| 4/3
	\| [[Kwai]]		\| [[Kwai]]
	\|}		{{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}}

	== Scales ==		== Scales ==

FloraC: +data for the 23-limit

2024-04-16T15:50:00Z

+data for the 23-limit

← Older revision		Revision as of 15:50, 16 April 2024
Line 5:		Line 5:
	193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.		193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.

	As an equal temperament, 193et [[tempering out\|tempers out]] the [[15625/15552\|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[529/528]] in the 23-limit.		As an equal temperament, 193et [[tempering out\|tempers out]] the [[15625/15552\|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[460/459]], [[507/506]], [[529/528]] in the 23-limit.

	It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.		It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.
Line 74:		Line 74:
	\| 0.3191		\| 0.3191
	\| 5.13		\| 5.13
			\|-
			\| 2.3.5.7.11.13.17.19.23
			\| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528
			\| {{mapping\| 193 306 448 542 668 714 789 820 873 }}
			\| -0.1184
			\| 0.3078
			\| 4.95
	\|}		\|}
	* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].		* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].

FloraC: Expand

2024-04-16T14:28:05Z

Expand

← Older revision		Revision as of 14:28, 16 April 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.		193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.

			As an equal temperament, 193et [[tempering out\|tempers out]] the [[15625/15552\|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[529/528]] in the 23-limit.

			It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.

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

FloraC: Cleanup

2024-04-16T14:11:04Z

Cleanup

← Older revision		Revision as of 14:11, 16 April 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[~~Swetismic temperaments #Minos\|~~minos]] and [[Mirkwai family #Indra\|vish]] temperaments.		193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra\|vish]] temperaments.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|193~~\|columns=11~~}}		{{Harmonics in equal\|193}}

	=== ~~Miscellaneous properties~~ ===		=== Subsets and supersets ===
	193edo is the 44th [[prime edo]].		193edo is the 44th [[prime edo]].

Line 24:		Line 24:
	\| 2.3		\| 2.3
	\| {{monzo\| 306 -193 }}		\| {{monzo\| 306 -193 }}
	\| [{{~~val~~\| 193 306 }}]		\| {{mapping\| 193 306 }}
	\| -0.2005		\| -0.2005
	\| 0.2005		\| 0.2005
Line 30:		Line 30:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| 15625/15552, {{monzo\|50 -33 1}}		\| 15625/15552, {{monzo\| 50 -33 1 }}
	\| [{{~~val~~\| 193 306 448 }}]		\| {{mapping\| 193 306 448 }}
	\| -0.0158		\| -0.0158
	\| 0.3084		\| 0.3084
Line 38:		Line 38:
	\| 2.3.5.7		\| 2.3.5.7
	\| 5120/5103, 15625/15552, 16875/16807		\| 5120/5103, 15625/15552, 16875/16807
	\| [{{~~val~~\| 193 306 448 542 }}]		\| {{mapping\| 193 306 448 542 }}
	\| -0.1118		\| -0.1118
	\| 0.3146		\| 0.3146
Line 45:		Line 45:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 540/539, 1375/1372, 4375/4356, 5120/5103		\| 540/539, 1375/1372, 4375/4356, 5120/5103
	\| [{{~~val~~\| 193 306 448 542 668 }}]		\| {{mapping\| 193 306 448 542 668 }}
	\| -0.2080		\| -0.2080
	\| 0.3408		\| 0.3408
Line 52:		Line 52:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 325/324, 364/363, 540/539, 625/624, 4096/4095		\| 325/324, 364/363, 540/539, 625/624, 4096/4095
	\| [{{~~val~~\| 193 306 448 542 668 714 }}]		\| {{mapping\| 193 306 448 542 668 714 }}
	\| -0.1216		\| -0.1216
	\| 0.3662		\| 0.3662
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095		\| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
	\| [{{~~val~~\| 193 306 448 542 668 714 789 }}]		\| {{mapping\| 193 306 448 542 668 714 789 }}
	\| -0.1302		\| -0.1302
	\| 0.3397		\| 0.3397
Line 66:		Line 66:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215		\| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
	\| [{{~~val~~\| 193 306 448 542 668 714 789 820 }}]		\| {{mapping\| 193 306 448 542 668 714 789 820 }}
	\| -0.1414		\| -0.1414
	\| 0.3191		\| 0.3191
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	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperament		! Temperament
	\|-		\|-
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	\| [[Kwai]]		\| [[Kwai]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Scales ==		== Scales ==
	*Approximation of sqrt (π): '''159\193''' (988.60104 cents), and of φ: '''134\193''' (833.16062 cents), both inside in the [[7L 2s\|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9		* Approximation of sqrt (π): '''159\193''' (988.60104 cents), and of φ: '''134\193''' (833.16062 cents), both inside in the [[7L 2s\|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9

	~~[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->~~
	~~[[Category:Prime EDO]]~~
	[[Category:Sqrtphi]]		[[Category:Sqrtphi]]

FloraC: /* Regular temperament properties */ +note on accuracy in the 19- and 23-limit

2022-10-22T08:22:45Z

Regular temperament properties: +note on accuracy in the 19- and 23-limit

← Older revision		Revision as of 08:22, 22 October 2022
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	\| 5.13		\| 5.13
	\|}		\|}
			* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo\|190g]] and followed by [[217edo\|217]].
			* 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo\|212gh]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===