248edo - 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"

2026-03-13T13:32:24Z

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
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	\| [[Birds]]		\| [[Birds]]
	\|}		\|}
	<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:Bischismic]]		[[Category:Bischismic]]
	[[Category:Essence]]		[[Category:Essence]]

Lériendil: Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

2025-04-17T02:29:23Z

Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

← Older revision		Revision as of 02:29, 17 April 2025
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	\| 498.39		\| 498.39
	\| 4/3		\| 4/3
	\| [[Helmholtz temperament\|Helmholtz]]		\| [[Helmholtz (temperament)\|Helmholtz]]
	\|-		\|-
	\| 2		\| 2

Nick Vuci: Edited Helmholtz redirect

2025-04-16T15:38:58Z

Edited Helmholtz redirect

← Older revision		Revision as of 15:38, 16 April 2025
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	\| 498.39		\| 498.39
	\| 4/3		\| 4/3
	\| [[Helmholtz]]		\| [[Helmholtz temperament\|Helmholtz]]
	\|-		\|-
	\| 2		\| 2

Recentlymaterialized: /* Theory */

2025-04-11T18:25:24Z

Theory

← Older revision		Revision as of 18:25, 11 April 2025
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	== Theory ==		== Theory ==
	248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]].		248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]]. Similarly, using the lower-error 248[[Wart notation\|h]] val, the mappings of all its [[2.5.7_subgroup\|no-3]] harmonics up to [[23-limit\|28]] are multiples of 2 and derived from [[124edo]].

	It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.		It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.

ArrowHead294 at 14:25, 20 February 2025

2025-02-20T14:25:25Z

← Older revision		Revision as of 14:25, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|248~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:54, 15 January 2025

2025-01-15T18:54:55Z

← Older revision		Revision as of 18:54, 15 January 2025
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	== Theory ==		== Theory ==
	~~248 = 8 × 31, and~~ 248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]].		248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]].

	It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.		It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.
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	\| [[Birds]]		\| [[Birds]]
	\|}		\|}
	<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:Bischismic]]		[[Category:Bischismic]]
	[[Category:Essence]]		[[Category:Essence]]

ArrowHead294: Partial undo

2024-11-16T12:32:22Z

Partial undo

← Older revision		Revision as of 12:32, 16 November 2024
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	== 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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	\| 0.275		\| 0.275
	\| 5.69		\| 5.69
	~~{{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
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	\| 4/3<br />(385/384)		\| 4/3<br />(385/384)
	\| [[Birds]]		\| [[Birds]]
	~~{{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:Bischismic]]		[[Category:Bischismic]]
	[[Category:Essence]]		[[Category:Essence]]

ArrowHead294 at 04:29, 16 November 2024

2024-11-16T04:29:55Z

← Older revision		Revision as of 04:29, 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		\| 2.3
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	\| 0.275		\| 0.275
	\| 5.69		\| 5.69
	\|}		{{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
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	\|-		\|-
	\| 2		\| 2
	\| 77\248<br>(47\248)		\| 77\248<br />(47\248)
	\| 372.58<br>(227.42)		\| 372.58<br />(227.42)
	\| 26/21<br>(154/135)		\| 26/21<br />(154/135)
	\| [[Essence]]		\| [[Essence]]
	\|-		\|-
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	\|-		\|-
	\| 8		\| 8
	\| 117\248<br>(7\248)		\| 117\248<br />(7\248)
	\| 566.13<br>(33.87)		\| 566.13<br />(33.87)
	\| 104/75<br>(49/48)		\| 104/75<br />(49/48)
	\| [[Octowerck]]		\| [[Octowerck]]
	\|-		\|-
	\| 31		\| 31
	\| 103\248<br>(1\248)		\| 103\248<br />(1\248)
	\| 498.39<br>(4.84)		\| 498.39<br />(4.84)
	\| 4/3<br>(385/384)		\| 4/3<br />(385/384)
	\| [[Birds]]		\| [[Birds]]
	\|}		{{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:Bischismic]]		[[Category:Bischismic]]
	[[Category:Essence]]		[[Category:Essence]]

Lériendil at 04:25, 21 September 2024

2024-09-21T04:25:50Z

← Older revision		Revision as of 04:25, 21 September 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	248 = 8 × 31, and 248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]].		248 = 8 × 31, and 248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]].

	It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.		It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.

FloraC: Expand on theory and misc. cleanup

2024-03-22T15:17:38Z

Expand on theory and misc. cleanup

← Older revision		Revision as of 15:17, 22 March 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~248et~~ tempers out [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]].		248 = 8 × 31, and 248edo shares the mapping of [[harmonic]]s [[5/1\|5]] and [[7/1\|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]].

	It [[support]]s [[~~Schismatic family #Bischismic\|~~bischismic ~~temperament~~]], providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for [[~~Varunismic temperaments #Essence\|~~essence ~~temperament~~]]. It is notable for its combination of precise intonation with an abundance of essentially tempered chords~~. 248 has divisors 2, 4, 8, 31, 62, and 124~~.		It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|248\|}}		{{Harmonics in equal\|248\|}}

			=== Subsets and supersets ===
			Since 248 factors into {{factorization\|248}}, 248edo has subset edos {{EDOs\| 2, 4, 8, 31, 62, and 124 }}.

	== 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\|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]] (¢)
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	\| 2.3		\| 2.3
	\| {{monzo\| 287 -181 }}		\| {{monzo\| 287 -181 }}
	\| [{{~~val~~\| 248 393 }}]		\| {{mapping\| 248 393 }}
	\| +0.108		\| +0.108
	\| 0.108		\| 0.108
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	\| 2.3.5		\| 2.3.5
	\| 32805/32768, {{monzo\| 12 32 -27 }}		\| 32805/32768, {{monzo\| 12 32 -27 }}
	\| [{{~~val~~\| 248 393 576 }}]		\| {{mapping\| 248 393 576 }}
	\| -0.041		\| -0.041
	\| 0.228		\| 0.228
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	\| 2.3.5.7		\| 2.3.5.7
	\| 3136/3125, 32805/32768, 420175/419904		\| 3136/3125, 32805/32768, 420175/419904
	\| [{{~~val~~\| 248 393 576 696 }}]		\| {{mapping\| 248 393 576 696 }}
	\| +0.066		\| +0.066
	\| 0.270		\| 0.270
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 441/440, 3136/3125, 8019/8000, 41503/41472		\| 441/440, 3136/3125, 8019/8000, 41503/41472
	\| [{{~~val~~\| 248 393 576 696 858 }}]		\| {{mapping\| 248 393 576 696 858 }}
	\| +0.036		\| +0.036
	\| 0.249		\| 0.249
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 441/440, 729/728, 847/845, 1001/1000, 3136/3125		\| 441/440, 729/728, 847/845, 1001/1000, 3136/3125
	\| [{{~~val~~\| 248 393 576 696 858 918 }}]		\| {{mapping\| 248 393 576 696 858 918 }}
	\| +0.079		\| +0.079
	\| 0.275		\| 0.275
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	{\| 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*
	! Cents~~<br>(reduced)~~		! Cents*
	! Associated<br>~~ratio~~		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[Birds]]		\| [[Birds]]
	\|}		\|}
			<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:Bischismic]]		[[Category:Bischismic]]
	[[Category:Essence]]		[[Category:Essence]]
	~~[[Category:Quartismic]]~~