328edo - 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"

2026-03-13T13:32:26Z

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 100:		Line 100:
	\| [[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:Hemiwürschmidt]]		[[Category:Hemiwürschmidt]]
	[[Category:Semiporwell]]		[[Category:Semiporwell]]

ArrowHead294 at 14:41, 20 February 2025

2025-02-20T14:41:49Z

← Older revision		Revision as of 14:41, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|328~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:43, 15 January 2025

2025-01-15T19:43:43Z

← Older revision		Revision as of 19:43, 15 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 328 factors into ~~2<sup>3</sup> × 41~~, 328edo has subset edos {{EDOs\| 2, 4, 8, 41, 82, and 164 }}.		Since 328 factors into {{Factorisation\|328}}, 328edo has subset edos {{EDOs\| 2, 4, 8, 41, 82, and 164 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 26:		Line 26:
	\| 2401/2400, 3136/3125, 589824/588245		\| 2401/2400, 3136/3125, 589824/588245
	\| {{mapping\| 328 520 762 921 }}		\| {{mapping\| 328 520 762 921 }}
	\| ~~−0~~.298		\| −0.298
	\| 0.229		\| 0.229
	\| 6.27		\| 6.27
Line 33:		Line 33:
	\| 2401/2400, 3136/3125, 9801/9800, 19712/19683		\| 2401/2400, 3136/3125, 9801/9800, 19712/19683
	\| {{mapping\| 328 520 762 921 1135 }}		\| {{mapping\| 328 520 762 921 1135 }}
	\| ~~−0~~.303		\| −0.303
	\| 0.205		\| 0.205
	\| 5.61		\| 5.61
Line 40:		Line 40:
	\| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647		\| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
	\| {{mapping\| 328 520 762 921 1135 1214 }}		\| {{mapping\| 328 520 762 921 1135 1214 }}
	\| ~~−0~~.295		\| −0.295
	\| 0.188		\| 0.188
	\| 5.15		\| 5.15
Line 47:		Line 47:
	\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125		\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
	\| {{mapping\| 328 520 762 921 1135 1214 1341 }}		\| {{mapping\| 328 520 762 921 1135 1214 1341 }}
	\| ~~−0~~.293		\| −0.293
	\| 0.174		\| 0.174
	\| 4.77		\| 4.77
Line 100:		Line 100:
	\| [[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:Hemiwürschmidt]]		[[Category:Hemiwürschmidt]]
	[[Category:Semiporwell]]		[[Category:Semiporwell]]

ArrowHead294: Partial undo

2024-11-16T12:37:56Z

Partial undo

← Older revision		Revision as of 12:37, 16 November 2024
Line 12:		Line 12:

	== 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]] (%)
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
Line 41:		Line 50:
	\| 0.174		\| 0.174
	\| 4.77		\| 4.77
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	Note: 5-limit temperaments supported by 164et are not listed.		Note: 5-limit temperaments supported by 164et are not listed.

	{{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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	\| 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:Hemiwürschmidt]]		[[Category:Hemiwürschmidt]]
	[[Category:Semiporwell]]		[[Category:Semiporwell]]

ArrowHead294 at 04:15, 16 November 2024

2024-11-16T04:15:45Z

← Older revision		Revision as of 04:15, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! 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]] (%)~~
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 3136/3125, 589824/588245		\| 2401/2400, 3136/3125, 589824/588245
	\| {{mapping\| 328 520 762 921 }}		\| {{mapping\| 328 520 762 921 }}
	\| -0.298		\| −0.298
	\| 0.229		\| 0.229
	\| 6.27		\| 6.27
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	\| 2401/2400, 3136/3125, 9801/9800, 19712/19683		\| 2401/2400, 3136/3125, 9801/9800, 19712/19683
	\| {{mapping\| 328 520 762 921 1135 }}		\| {{mapping\| 328 520 762 921 1135 }}
	\| -0.303		\| −0.303
	\| 0.205		\| 0.205
	\| 5.61		\| 5.61
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	\| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647		\| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
	\| {{mapping\| 328 520 762 921 1135 1214 }}		\| {{mapping\| 328 520 762 921 1135 1214 }}
	\| -0.295		\| −0.295
	\| 0.188		\| 0.188
	\| 5.15		\| 5.15
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	\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125		\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
	\| {{mapping\| 328 520 762 921 1135 1214 1341 }}		\| {{mapping\| 328 520 762 921 1135 1214 1341 }}
	\| -0.293		\| −0.293
	\| 0.174		\| 0.174
	\| 4.77		\| 4.77
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	Note: 5-limit temperaments supported by 164et are not listed.		Note: 5-limit temperaments supported by 164et are not listed.

	{~~\| 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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	\|-		\|-
	\| 2		\| 2
	\| 111\328<br>(53\328)		\| 111\328<br />(53\328)
	\| 406.10<br>(193.90)		\| 406.10<br />(193.90)
	\| 495/392<br>(28/25)		\| 495/392<br />(28/25)
	\| [[Semihemiwürschmidt]]		\| [[Semihemiwürschmidt]]
	\|-		\|-
	\| 8		\| 8
	\| 136\328<br>(13\328)		\| 136\328<br />(13\328)
	\| 497.56<br>(47.56)		\| 497.56<br />(47.56)
	\| 4/3<br>(36/35)		\| 4/3<br />(36/35)
	\| [[Twilight]]		\| [[Twilight]]
	\|-		\|-
	\| 41		\| 41
	\| 49\328<br>(1\328)		\| 49\328<br />(1\328)
	\| 179.27<br>(3.66)		\| 179.27<br />(3.66)
	\| 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:Hemiwürschmidt]]		[[Category:Hemiwürschmidt]]
	[[Category:Semiporwell]]		[[Category:Semiporwell]]

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

2024-01-22T08:15:49Z

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

← Older revision		Revision as of 08:15, 22 January 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	328edo is [[enfactoring\|enfactored]] in the 5-limit, with the same tuning as [[164edo]], but the approximation of higher ~~harmonics~~ are much improved. It has a sharp tendency, with ~~[[harmonic]]s~~ 3 through 17 all tuned sharp. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].		328edo is [[enfactoring\|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|328\|intervals=prime\|columns=11}}		{{Harmonics in equal\|328\|intervals=prime\|columns=11}}

	=== ~~Divisors~~ ===		=== Subsets and supersets ===
	Since 328 factors into 2<sup>3</sup> × 41, it has subset edos {{EDOs\| 2, 4, 8, 41, 82, and 164 }}.		Since 328 factors into 2<sup>3</sup> × 41, 328edo has subset edos {{EDOs\| 2, 4, 8, 41, 82, and 164 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 24:		Line 24:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 3136/3125, 589824/588245		\| 2401/2400, 3136/3125, 589824/588245
	\| [{{~~val~~\| 328 520 762 921 }}]		\| {{mapping\| 328 520 762 921 }}
	\| -0.298		\| -0.298
	\| 0.229		\| 0.229
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3136/3125, 9801/9800, 19712/19683		\| 2401/2400, 3136/3125, 9801/9800, 19712/19683
	\| [{{~~val~~\| 328 520 762 921 1135 }}]		\| {{mapping\| 328 520 762 921 1135 }}
	\| -0.303		\| -0.303
	\| 0.205		\| 0.205
Line 38:		Line 38:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647		\| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
	\| [{{~~val~~\| 328 520 762 921 1135 1214 }}]		\| {{mapping\| 328 520 762 921 1135 1214 }}
	\| -0.295		\| -0.295
	\| 0.188		\| 0.188
Line 45:		Line 45:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125		\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
	\| [{{~~val~~\| 328 520 762 921 1135 1214 1341 }}]		\| {{mapping\| 328 520 762 921 1135 1214 1341 }}
	\| -0.293		\| -0.293
	\| 0.174		\| 0.174
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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*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[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:Hemiwürschmidt]]		[[Category:Hemiwürschmidt]]
	[[Category:Semiporwell]]		[[Category:Semiporwell]]

FloraC: Expand theory

2022-12-22T13:41:48Z

Expand theory

← Older revision		Revision as of 13:41, 22 December 2022
Line 3:		Line 3:

	== Theory ==		== Theory ==
	328edo is [[enfactoring\|enfactored]] in the 5-limit, with the same tuning as [[164edo]]. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].		328edo is [[enfactoring\|enfactored]] in the 5-limit, with the same tuning as [[164edo]], but the approximation of higher harmonics are much improved. It has a sharp tendency, with [[harmonic]]s 3 through 17 all tuned sharp. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|328\|columns=11}}		{{Harmonics in equal\|328\|intervals=prime\|columns=11}}

	=== Divisors ===		=== Divisors ===

FloraC: Cleanup and update

2022-12-22T13:37:35Z

Cleanup and update

← Older revision		Revision as of 13:37, 22 December 2022
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~The '''~~328 equal divisions of the octave''' ('''328edo'''), or the '''328(-tone) equal temperament''' ('''328tet''', '''328et''') when viewed from a [[regular temperament]] perspective, divides the octave into 328 [[equal]] parts of about 3.66 [[cent]]s each.		{{EDO intro\|328}}

	== Theory ==		== Theory ==
	328edo is [[enfactoring\|enfactored]] in the 5-limit, with the same tuning as [[164edo]]. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].		328edo is [[enfactoring\|enfactored]] in the 5-limit, with the same tuning as [[164edo]]. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].

	328 ~~factors into 2<sup>3</sup> × 41, with subset edos 2, 4, 8, 41, 82, and 164.~~		=== Prime harmonics ===
			{{Harmonics in equal\|328\|columns=11}}

	=== ~~Prime harmonics~~ ===		=== Divisors ===
	{{~~Primes in edo~~\|~~328~~}}		Since 328 factors into 2<sup>3</sup> × 41, it has subset edos {{EDOs\| 2, 4, 8, 41, 82, and 164 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
	{\| 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]]
	! 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
	\|-		\|-
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
Line 51:		Line 52:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
			Note: 5-limit temperaments supported by 164et are not listed.

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per ~~octave~~		! Periods<br>per 8ve
	! Generator<br>(~~reduced~~)		! Generator<br>(Reduced)
	! Cents<br>(~~reduced~~)		! Cents<br>(Reduced)
	! Associated<br>~~ratio~~		! Associated<br>Ratio
	! Temperaments		! Temperaments
	\|-		\|-
Line 93:		Line 96:
	\| 179.27<br>(3.66)		\| 179.27<br>(3.66)
	\| 567/512<br>(352/351)		\| 567/512<br>(352/351)
	\| [[~~Hemicounterpyth~~]]		\| [[Hemicountercomp]]
	\|}		\|}

Plumtree: Infobox ET now computes most parameters automatically

2022-10-04T18:56:12Z

Infobox ET now computes most parameters automatically

← Older revision		Revision as of 18:56, 4 October 2022
Line 1:		Line 1:
	{{Infobox ET		{{Infobox ET}}
	~~\| Prime factorization = 2<sup>3</sup> × 41~~
	~~\| Step size = 3.65854¢~~
	~~\| Fifth = 192\328 (702.44¢) (→ [[41edo\|24\41]])~~
	~~\| Semitones = 32:24 (117.07¢ : 87.80¢)~~
	~~\| Consistency = 13~~
	}}
	The '''328 equal divisions of the octave''' ('''328edo'''), or the '''328(-tone) equal temperament''' ('''328tet''', '''328et''') when viewed from a [[regular temperament]] perspective, divides the octave into 328 [[equal]] parts of about 3.66 [[cent]]s each.		The '''328 equal divisions of the octave''' ('''328edo'''), or the '''328(-tone) equal temperament''' ('''328tet''', '''328et''') when viewed from a [[regular temperament]] perspective, divides the octave into 328 [[equal]] parts of about 3.66 [[cent]]s each.

Fredg999 category edits: Sort key

2022-07-03T19:19:30Z

Sort key

← Older revision		Revision as of 19:19, 3 July 2022
Line 102:		Line 102:
	\|}		\|}

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