Luna and hemithirds - Revision history

FloraC: Cleanup on infobox

2026-02-09T09:21:14Z

Cleanup on infobox

← Older revision		Revision as of 09:21, 9 February 2026
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	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Hemithirds		\| Title = Hemithirds
	\| Subgroups = 2.3.5.7		\| Subgroups = 2.3.5.7
	\| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)		\| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)
	\| Edo join 1 = 31 \| Edo join 2 = 87		\| Edo join 1 = 31 \| Edo join 2 = 87
	\| ~~Generator~~ = 28/25 \| ~~Generator~~ tuning = 193.2 \| Optimization method = CWE		\| Mapping = 1; -15 2 5
			\| Generators = 28/25 \| Generators tuning = 193.2 \| Optimization method = CWE
	\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]		\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]
	~~\| Mapping = 1; -15 2 5~~
	\| Pergen = (P8, ccP4/15)		\| Pergen = (P8, ccP4/15)
	\| Color name = Latrizo & Zozoquinguti		\| Color name = Latrizo & Zozoquinguti
	\| Odd limit 1 = 7 \| Mistuning 1 = 2.5 \| Complexity 1 = 56		\| Odd limit 1 = 7 \| Mistuning 1 = 2.5 \| Complexity 1 = 56
	\| Odd limit 2 = (7-limit) 25 \| Mistuning 2 = 3.6 \| Complexity 2 = 87		\| Odd limit 2 = 7-limit 25 \| Mistuning 2 = 3.6 \| Complexity 2 = 87
	}}		}}

	The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, which is defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus in the range between [[25edo]] and [[31edo]] tuning, by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.		The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, which is defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus in the range between [[25edo]] and [[31edo]] tuning, by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.

Overthink: 56edo is rather obscure and less accurate than 31; 87edo is better

2025-12-09T00:37:46Z

56edo is rather obscure and less accurate than 31; 87edo is better

← Older revision		Revision as of 00:37, 9 December 2025
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	\| Subgroups = 2.3.5.7		\| Subgroups = 2.3.5.7
	\| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)		\| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)
	\| Edo join 1 = 31 \| Edo join 2 = 56		\| Edo join 1 = 31 \| Edo join 2 = 87
	\| Generator = 28/25 \| Generator tuning = 193.2 \| Optimization method = CWE		\| Generator = 28/25 \| Generator tuning = 193.2 \| Optimization method = CWE
	\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]		\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]

Lériendil: /* Tuning spectrum */

2025-07-15T05:17:09Z

Tuning spectrum

← Older revision		Revision as of 05:17, 15 July 2025
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	=== Tuning spectrum ===		=== Tuning spectrum ===
			{{see also\|Didacus #Tuning spectrum}}

	Vals are for 13-limit hemithirds and 7-limit lunatic in their respective ranges.		Vals are for 13-limit hemithirds and 7-limit lunatic in their respective ranges.

Lériendil: removed most 77-odd intervals

2025-07-14T18:13:19Z

removed most 77-odd intervals

← Older revision		Revision as of 18:13, 14 July 2025
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	\| 386.5		\| 386.5
	\| '''5/4'''		\| '''5/4'''
	\| ~~96/77~~		\|
	\|-		\|-
	\| 3		\| 3
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	\| 773.0		\| 773.0
	\| '''25/16'''		\| '''25/16'''
	\| 39/25~~, 120/77~~		\| 39/25
	\|-		\|-
	\| 5		\| 5
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	\| 112.1		\| 112.1
	\| '''16/15'''		\| '''16/15'''
	\| ~~77/72~~		\|
	\|-		\|-
	\| 14		\| 14
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	\| 71.5		\| 71.5
	\| 25/24		\| 25/24
	\| 26/25~~, 80/77~~		\| 26/25
	\|-		\|-
	\| 20		\| 20

Lériendil: followed suggestion of reducing decimal places in this spot

2025-06-09T22:49:18Z

followed suggestion of reducing decimal places in this spot

← Older revision		Revision as of 22:49, 9 June 2025
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	\| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)		\| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)
	\| Edo join 1 = 31 \| Edo join 2 = 56		\| Edo join 1 = 31 \| Edo join 2 = 56
	\| Generator = 28/25 \| Generator tuning = 193.~~239~~ \| Optimization method = CWE		\| Generator = 28/25 \| Generator tuning = 193.2 \| Optimization method = CWE
	\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]		\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]
	\| Mapping = 1; -15 2 5		\| Mapping = 1; -15 2 5

Lériendil at 15:19, 9 June 2025

2025-06-09T15:19:29Z

← Older revision		Revision as of 15:19, 9 June 2025
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	}}		}}

	The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, ~~in the range between [[25edo]] and [[31edo]] tuning,~~ defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.		The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, which is defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus in the range between [[25edo]] and [[31edo]] tuning, by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.

	'''Luna''' is a restriction of hemithirds to the [[5-limit]] that is a [[microtemperament]], supported by such high-precision tuning systems as [[323edo]] and [[441edo]]; another notable tuning of luna is [[1000edo]]. It can further be re-extended to the 7-limit in the form of [[lunatic]] by adding [[4375/4374]] to the comma list, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).		'''Luna''' is a restriction of hemithirds to the [[5-limit]] that is a [[microtemperament]], supported by such high-precision tuning systems as [[323edo]] and [[441edo]]; another notable tuning of luna is [[1000edo]]. It can further be re-extended to the 7-limit in the form of [[lunatic]] by adding [[4375/4374]] to the comma list, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).

	See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.		See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.

	== Intervals ==		== Intervals ==

Lériendil: don't think we need 3/7

2025-05-01T12:48:23Z

don't think we need 3/7

← Older revision		Revision as of 12:48, 1 May 2025
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	\| 193.5484		\| 193.5484
	\| '''Upper bound of 9- to 15-odd-limit diamond monotone'''		\| '''Upper bound of 9- to 15-odd-limit diamond monotone'''
	\|-
	\|
	~~\| [[35/32]]~~
	~~\| 193.5914~~
	~~\| 3/7-comma didacus~~
	\|-		\|-
	\|		\|

FloraC: - parent category

2025-04-28T14:20:39Z

- parent category

← Older revision		Revision as of 14:20, 28 April 2025
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	\|}		\|}

	~~[[Category:Temperaments]]~~
	[[Category:Hemithirds\| ]] <!-- main article -->		[[Category:Hemithirds\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]

Lériendil at 16:34, 26 April 2025

2025-04-26T16:34:27Z

← Older revision		Revision as of 16:34, 26 April 2025
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	~~{{Breadcrumb\|Didacus}}~~

	{{Infobox Regtemp		{{Infobox Regtemp
	\| Title = Hemithirds		\| Title = Hemithirds

Lériendil: added 40/21 eigenmonzo

2025-04-26T16:30:34Z

added 40/21 eigenmonzo

← Older revision		Revision as of 16:30, 26 April 2025
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	\| [[32/21]]		\| [[32/21]]
	\| 192.9219		\| 192.9219
			\|
			\|-
			\|
			\| [[40/21]]
			\| 192.9601
	\|		\|
	\|-		\|-
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	\| 194.5946		\| 194.5946
	\| 37beff val, '''upper bound of 7-odd-limit diamond monotone'''		\| 37beff val, '''upper bound of 7-odd-limit diamond monotone'''
			\|-
			\|
			\| [[28/25]]
			\| 196.1985
			\| Untempered didacus
	\|}		\|}
	<nowiki />* Besides the octave		<nowiki />* Besides the octave