369edo - Revision history

FloraC: Countritonic -> ragitritonic

2026-05-20T11:12:25Z

Countritonic -> ragitritonic

← Older revision		Revision as of 11:12, 20 May 2026
Line 82:		Line 82:
	\| 588.62		\| 588.62
	\| 128/91		\| 128/91
	\| [[~~Countritonic~~]]		\| [[Ragitritonic]]
	\|-		\|-
	\| 9		\| 9

FloraC: /* Theory */ expand

2026-05-19T09:02:42Z

Theory: expand

← Older revision		Revision as of 09:02, 19 May 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	369edo shares its [[3/2\|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out\|tempers out]] [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit {{nowrap\|130 & 239}} temperament, {{nowrap\|65 & 152}} temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.		369edo shares its [[3/2\|perfect fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp.

	Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.		As an equal temperament, it [[tempering out\|tempers out]] the [[escapade comma]] and the [[ennealimma]] in the 5-limit; [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]], so that it supports [[escapade]] in the [[2.3.5.11 subgroup]] and in fact provides the [[optimal patent val]]. It also provides the optimal patent val for the 11-limit {{nowrap\| 152 & 217 }} temperament (an escapade extension), the {{nowrap\| 130 & 239 }} temperament (a weak escapade extension), and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.

			Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The [[Tenney–Euclidean tuning\|TE-optimal tuning]] of this temperament is [[consistent]] in the 15-integer-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|369\|columns=11}}		{{Harmonics in equal\|369\|columns=11}}

	=== ~~Divisors~~ ===		=== Subsets and supersets ===
	Since 369 factors into primes as {{nowrap\| 3<sup>2</sup> × 41 }}, 369edo has subset edos {{EDOs\| 3, 9, 41, and 123 }}.		Since 369 factors into primes as {{nowrap\| 3<sup>2</sup> × 41 }}, 369edo has subset edos {{EDOs\| 3, 9, 41, and 123 }}.

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

FloraC: Update

2026-05-19T08:53:28Z

Update

← Older revision		Revision as of 08:53, 19 May 2026
Line 11:		Line 11:

	=== Divisors ===		=== Divisors ===
	Since 369 factors into {{~~factorization~~\|~~369~~}}, 369edo has subset edos {{EDOs\| 3, 9, 41, and 123 }}.		Since 369 factors into primes as {{nowrap\| 3<sup>2</sup> × 41 }}, 369edo has subset edos {{EDOs\| 3, 9, 41, and 123 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 19:		Line 19:
	! 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 26:		Line 26:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}		\| {{Monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}
	\| {{~~mapping~~\| 369 585 857 }}		\| {{Mapping\| 369 585 857 }}
	\| −0.1991		\| −0.1991
	\| 0.1409		\| 0.1409
Line 34:		Line 34:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}		\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}
	\| {{~~mapping~~\| 369 585 857 1036 }}		\| {{Mapping\| 369 585 857 1036 }}
	\| −0.1743		\| −0.1743
	\| 0.1294		\| 0.1294
Line 41:		Line 41:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 4000/3993, 4375/4374, 5632/5625		\| 2401/2400, 4000/3993, 4375/4374, 5632/5625
	\| {{~~mapping~~\| 369 585 857 1036 1277 }}		\| {{Mapping\| 369 585 857 1036 1277 }}
	\| −0.2277		\| −0.2277
	\| 0.1576		\| 0.1576
Line 48:		Line 48:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575		\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
	\| {{~~mapping~~\| 369 585 857 1036 1277 1366 }} (369f)		\| {{Mapping\| 369 585 857 1036 1277 1366 }} (369f)
	\| −0.2685		\| −0.2685
	\| 0.1703		\| 0.1703
Line 58:		Line 58:
	\|+ 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 83:		Line 83:
	\|-		\|-
	\| 9		\| 9
	\| 77\369<br />(5\369)		\| 77\369<br>(5\369)
	\| 250.41<br />(16.26)		\| 250.41<br>(16.26)
	\| 140/121<br />(100/99)		\| 140/121<br>(100/99)
	\| [[~~Semiennealimmal~~]]		\| [[Ennealimmapine]]
	\|-		\|-
	\| 9		\| 9
	\| 97\369<br />(15\369)		\| 97\369<br>(15\369)
	\| 315.45<br />(48.78)		\| 315.45<br>(48.78)
	\| 6/5<br />(36/35)		\| 6/5<br>(36/35)
	\| [[Ennealimmal]]		\| [[Ennealimmal]] / enneabiotic
	\|-		\|-
	\| 9		\| 9
	\| 68\369<br />(14\369)		\| 68\369<br>(14\369)
	\| 221.14<br />(45.53)		\| 221.14<br>(45.53)
	\| 25/22<br />(77/75)		\| 25/22<br>(77/75)
	\| [[Quadraennealimmal]]		\| [[Quadraennealimmal]]
	\|-		\|-
	\| 41		\| 41
	\| 55\369<br />(1\369)		\| 55\369<br>(1\369)
	\| 178.86<br />(3.25)		\| 178.86<br>(3.25)
	\| 567/512<br />(352/351)		\| 567/512<br>(352/351)
	\| [[Hemicountercomp]]		\| [[Hemicountercomp]]
	\|}		\|}
	<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:Semiporwellismic]]		[[Category:Semiporwellismic]]

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"

2026-03-13T13:32:28Z

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 106:		Line 106:
	\| [[Hemicountercomp]]		\| [[Hemicountercomp]]
	\|}		\|}
	<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:Semiporwellismic]]		[[Category:Semiporwellismic]]

ArrowHead294 at 14:47, 20 February 2025

2025-02-20T14:47:43Z

← Older revision		Revision as of 14:47, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|369~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:24, 15 January 2025

2025-01-15T19:24:45Z

← Older revision		Revision as of 19:24, 15 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~369 = 9 × 41, and~~ 369edo shares its [[3/2\|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out\|tempers out]] [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit 130 & 239 temperament, ~~the~~ 65 & 152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.		369edo shares its [[3/2\|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out\|tempers out]] [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit {{nowrap\|130 & 239}} temperament, {{nowrap\|65 & 152}} temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.

	Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.		Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.
Line 28:		Line 28:
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}
	\| {{mapping\| 369 585 857 }}		\| {{mapping\| 369 585 857 }}
	\| ~~−0~~.1991		\| −0.1991
	\| 0.1409		\| 0.1409
	\| 4.33		\| 4.33
Line 35:		Line 35:
	\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}		\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}
	\| {{mapping\| 369 585 857 1036 }}		\| {{mapping\| 369 585 857 1036 }}
	\| ~~−0~~.1743		\| −0.1743
	\| 0.1294		\| 0.1294
	\| 3.98		\| 3.98
Line 42:		Line 42:
	\| 2401/2400, 4000/3993, 4375/4374, 5632/5625		\| 2401/2400, 4000/3993, 4375/4374, 5632/5625
	\| {{mapping\| 369 585 857 1036 1277 }}		\| {{mapping\| 369 585 857 1036 1277 }}
	\| ~~−0~~.2277		\| −0.2277
	\| 0.1576		\| 0.1576
	\| 4.85		\| 4.85
Line 49:		Line 49:
	\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575		\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
	\| {{mapping\| 369 585 857 1036 1277 1366 }} (369f)		\| {{mapping\| 369 585 857 1036 1277 1366 }} (369f)
	\| ~~−0~~.2685		\| −0.2685
	\| 0.1703		\| 0.1703
	\| 5.24		\| 5.24
Line 106:		Line 106:
	\| [[Hemicountercomp]]		\| [[Hemicountercomp]]
	\|}		\|}
	<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:Semiporwellismic]]		[[Category:Semiporwellismic]]

ArrowHead294: Partial undo

2024-11-16T12:36:50Z

Partial undo

← Older revision		Revision as of 12:36, 16 November 2024
Line 14:		Line 14:

	== 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]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
Line 43:		Line 52:
	\| 0.1703		\| 0.1703
	\| 5.24		\| 5.24
	~~{{comma basis end}~~}		\|}

	=== 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 89:		Line 105:
	\| 567/512<br />(352/351)		\| 567/512<br />(352/351)
	\| [[Hemicountercomp]]		\| [[Hemicountercomp]]
	~~{{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:Semiporwellismic]]		[[Category:Semiporwellismic]]

ArrowHead294 at 04:19, 16 November 2024

2024-11-16T04:19:05Z

← Older revision		Revision as of 04:19, 16 November 2024
Line 14:		Line 14:

	== 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<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}
	\| {{mapping\| 369 585 857 }}		\| {{mapping\| 369 585 857 }}
	\| -0.1991		\| −0.1991
	\| 0.1409		\| 0.1409
	\| 4.33		\| 4.33
Line 34:		Line 26:
	\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}		\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}
	\| {{mapping\| 369 585 857 1036 }}		\| {{mapping\| 369 585 857 1036 }}
	\| -0.1743		\| −0.1743
	\| 0.1294		\| 0.1294
	\| 3.98		\| 3.98
Line 41:		Line 33:
	\| 2401/2400, 4000/3993, 4375/4374, 5632/5625		\| 2401/2400, 4000/3993, 4375/4374, 5632/5625
	\| {{mapping\| 369 585 857 1036 1277 }}		\| {{mapping\| 369 585 857 1036 1277 }}
	\| -0.2277		\| −0.2277
	\| 0.1576		\| 0.1576
	\| 4.85		\| 4.85
Line 48:		Line 40:
	\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575		\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
	\| {{mapping\| 369 585 857 1036 1277 1366 }} (369f)		\| {{mapping\| 369 585 857 1036 1277 1366 }} (369f)
	\| -0.2685		\| −0.2685
	\| 0.1703		\| 0.1703
	\| 5.24		\| 5.24
	\|}		{{comma basis end}}

	=== 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
Line 81:		Line 67:
	\|-		\|-
	\| 9		\| 9
	\| 77\369<br>(5\369)		\| 77\369<br />(5\369)
	\| 250.41<br>(16.26)		\| 250.41<br />(16.26)
	\| 140/121<br>(100/99)		\| 140/121<br />(100/99)
	\| [[Semiennealimmal]]		\| [[Semiennealimmal]]
	\|-		\|-
	\| 9		\| 9
	\| 97\369<br>(15\369)		\| 97\369<br />(15\369)
	\| 315.45<br>(48.78)		\| 315.45<br />(48.78)
	\| 6/5<br>(36/35)		\| 6/5<br />(36/35)
	\| [[Ennealimmal]]		\| [[Ennealimmal]]
	\|-		\|-
	\| 9		\| 9
	\| 68\369<br>(14\369)		\| 68\369<br />(14\369)
	\| 221.14<br>(45.53)		\| 221.14<br />(45.53)
	\| 25/22<br>(77/75)		\| 25/22<br />(77/75)
	\| [[Quadraennealimmal]]		\| [[Quadraennealimmal]]
	\|-		\|-
	\| 41		\| 41
	\| 55\369<br>(1\369)		\| 55\369<br />(1\369)
	\| 178.86<br>(3.25)		\| 178.86<br />(3.25)
	\| 567/512<br>(352/351)		\| 567/512<br />(352/351)
	\| [[Hemicountercomp]]		\| [[Hemicountercomp]]
	\|}		{{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:Semiporwellismic]]		[[Category:Semiporwellismic]]

FloraC: Cleanup; clarify the title row of the rank-2 temp table; -redundant categories

2023-11-09T14:54:25Z

Cleanup; clarify the title row of the rank-2 temp table; -redundant categories

← Older revision		Revision as of 14:54, 9 November 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	369 = 9 × 41, and it shares ~~the~~ fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It tempers out [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit 130 & 239 temperament, the 65 & 152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as the no-7 subgroup version ~~of it~~.		369 = 9 × 41, and 369edo shares its [[3/2\|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out\|tempers out]] [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit 130 & 239 temperament, the 65 & 152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.

	Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.		Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.
Line 11:		Line 11:

	=== Divisors ===		=== Divisors ===
	Since 369 factors into ~~3<sup>2</sup> × 41~~, 369edo has subset edos {{EDOs\| 3, 9, 41, and 123 }}.		Since 369 factors into {{factorization\|369}}, 369edo has subset edos {{EDOs\| 3, 9, 41, and 123 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 26:		Line 26:
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| 1 -27 18 }}
	\| [{{~~val~~\| 369 585 857 }}]		\| {{mapping\| 369 585 857 }}
	\| -0.1991		\| -0.1991
	\| 0.1409		\| 0.1409
Line 33:		Line 33:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}		\| 2401/2400, 4375/4374, {{monzo\| 32 -7 -9 }}
	\| [{{~~val~~\| 369 585 857 1036 }}]		\| {{mapping\| 369 585 857 1036 }}
	\| -0.1743		\| -0.1743
	\| 0.1294		\| 0.1294
Line 40:		Line 40:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 4000/3993, 4375/4374, 5632/5625		\| 2401/2400, 4000/3993, 4375/4374, 5632/5625
	\| [{{~~val~~\| 369 585 857 1036 1277 }}]		\| {{mapping\| 369 585 857 1036 1277 }}
	\| -0.2277		\| -0.2277
	\| 0.1576		\| 0.1576
Line 47:		Line 47:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575		\| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
	\| [{{~~val~~\| 369 585 857 1036 1277 1366 }}] (369f)		\| {{mapping\| 369 585 857 1036 1277 1366 }} (369f)
	\| -0.2685		\| -0.2685
	\| 0.1703		\| 0.1703
Line 57:		Line 57:
	\|+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*
	! Temperaments		! Temperaments
	\|-		\|-
Line 104:		Line 104:
	\| [[Hemicountercomp]]		\| [[Hemicountercomp]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

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

FloraC: +link to countritonic

2023-09-06T09:40:38Z

+link to countritonic

← Older revision		Revision as of 09:40, 6 September 2023
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	== Theory ==		== Theory ==
	369 = 9 × 41, and it shares the fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It tempers out [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit 130&239 temperament, the 65&152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as the no-7 subgroup version of it.		369 = 9 × 41, and it shares the fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It tempers out [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit 130 & 239 temperament, the 65 & 152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as the no-7 subgroup version of it.

	Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.		Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.
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	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \| [[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list\|Comma List]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br>8ve Stretch (¢)		! rowspan="2" \| Optimal<br>8ve Stretch (¢)
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	\| 864/625		\| 864/625
	\| [[Tritriple]] (5-limit)		\| [[Tritriple]] (5-limit)
			\|-
			\| 1
			\| 181\369
			\| 588.62
			\| 128/91
			\| [[Countritonic]]
	\|-		\|-
	\| 9		\| 9