Rainy–didacus equivalence continuum - Revision history

Overthink: clarify

2026-01-15T23:05:43Z

clarify

← Older revision		Revision as of 23:05, 15 January 2026
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	The '''rainy–didacus continuum''' is the [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[rainy comma]]s with the [[didacus comma]] ([[3136/3125]]), and thus is the continuum of all 2.5.7 subgroup temperaments supported by [[31edo]], which tempers both and thus tempers all linear combinations of them. If one wants to use all of these simultaneously but wants more accurate tuning than [[31edo]], [[31st-octave temperaments]] extending [[birds]] may be interesting.		The '''rainy–didacus continuum''' is the [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[rainy comma]]s with the [[didacus comma]] ([[3136/3125]]), and thus is the continuum of all 2.5.7 subgroup temperaments supported by [[31edo]], which tempers both and thus tempers all linear combinations of them. If one wants to use all of these simultaneously but wants more accurate tuning than [[31edo]] for the other primes, then [[31st-octave temperaments]] extending [[birds]] may be interesting.

	All temperaments in the continuum satisfy {{nowrap\|([[2100875/2097152]])<sup>''n''</sup> ~ ([[3136/3125]])}} for some rational value of ''n''. The just value of ''n'' is approximately 1.981... so that {{nowrap\|''n'' {{=}} 2}} is especially close to the [[JIP]].		All temperaments in the continuum satisfy {{nowrap\|([[2100875/2097152]])<sup>''n''</sup> ~ ([[3136/3125]])}} for some rational value of ''n''. The just value of ''n'' is approximately 1.981... so that {{nowrap\|''n'' {{=}} 2}} is especially close to the [[JIP]].

Xenllium at 10:37, 15 January 2026

2026-01-15T10:37:51Z

← Older revision		Revision as of 10:37, 15 January 2026
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	\| {{monzo\| -69 0 14 13 }}		\| {{monzo\| -69 0 14 13 }}
	\|-		\|-
			\| …
	\| …		\| …
	\| …		\| …
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	{{Optimal ET sequence\|legend=1\|31, 224, 255, 286, 317, 348, 379, 410, 789}}		{{Optimal ET sequence\|legend=1\|31, 224, 255, 286, 317, 348, 379, 410, 789}}

	[[Badness]] (Sintel): ?		[[Badness]] (Sintel): 0.0148

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Xenllium at 11:46, 12 January 2026

2026-01-12T11:46:37Z

← Older revision		Revision as of 11:46, 12 January 2026
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	\|-		\|-
	\| 0.5		\| 0.5
	\| 2.5.7 ~~restriction* of~~ [[~~Ostara~~]]		\| 2.5.7 [[Mirkwai clan #Grendel\|Grendel]]
	\| 8589934592/8544921875		\| 8589934592/8544921875
	\| {{monzo\| 33 0 -13 -1 }}		\| {{monzo\| 33 0 -13 -1 }}
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	\| 2		\| 2
	\| Exodia (2.5.7 [[Meantone family#Mohajira\|Mohajira]])		\| Exodia (2.5.7 [[Meantone family#Mohajira\|Mohajira]])
	\| 281484423828125/281474976710656		\| [[Exodia comma\|281484423828125/281474976710656]]
	\| {{monzo\| -48 0 11 8 }}		\| {{monzo\| -48 0 11 8 }}
	\|-		\|-
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	\| 31 & 612		\| 31 & 612
	\| 591363588909912109375/590295810358705651712		\| 591363588909912109375/590295810358705651712
	\| {{monzo\| -69 14 13 }}		\| {{monzo\| -69 0 14 13 }}
	\|-		\|-
	\| …		\| …
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	\| {{monzo\| -21 0 3 5 }}		\| {{monzo\| -21 0 3 5 }}
	\|}		\|}
	~~<nowiki />* Note that [[ostara]] is contorted in the 2.5.7 subgroup, hence why this is not merely "2.5.7 ostara"; by contrast, neither "2.5.7 [[myna]]" nor "2.5.7 [[mohajira]]" are contorted.~~

	== Temperaments ==		== Temperaments ==
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	[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~262144/214375 = 348.289		[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~262144/214375 = 348.289

	~~Supporting EDOs:~~ {{~~EDOs~~\|~~789,~~ 31, ~~410~~, ~~379~~, ~~1199~~, ~~1168, 1957~~, 348, ~~1609~~, ~~441~~, ~~1547, 317, 727, 851~~}}		{{Optimal ET sequence\|legend=1\|31, 224, 255, 286, 317, 348, 379, 410, 789}}

	~~{{Todo\|inline=1\|complete temperament data\|comment=get optimal ET sequence and badness}}~~		[[Badness]] (Sintel): ?

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Overthink: /* Temperaments */ comment on todo

2025-11-19T16:58:27Z

Temperaments: comment on todo

← Older revision		Revision as of 16:58, 19 November 2025
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	Supporting EDOs: {{EDOs\|789, 31, 410, 379, 1199, 1168, 1957, 348, 1609, 441, 1547, 317, 727, 851}}		Supporting EDOs: {{EDOs\|789, 31, 410, 379, 1199, 1168, 1957, 348, 1609, 441, 1547, 317, 727, 851}}

	{{Todo\|inline=1\|complete temperament data}}		{{Todo\|inline=1\|complete temperament data\|comment=get optimal ET sequence and badness}}

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

FloraC: Text replacement - "{{todo|get optimal ET sequence and badness|inline=1}}" to "{{Todo|inline=1|complete temperament data}}"

2025-10-24T10:30:18Z

Text replacement - "{{todo|get optimal ET sequence and badness|inline=1}}" to "{{Todo|inline=1|complete temperament data}}"

← Older revision		Revision as of 10:30, 24 October 2025
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	Supporting EDOs: {{EDOs\|789, 31, 410, 379, 1199, 1168, 1957, 348, 1609, 441, 1547, 317, 727, 851}}		Supporting EDOs: {{EDOs\|789, 31, 410, 379, 1199, 1168, 1957, 348, 1609, 441, 1547, 317, 727, 851}}

	{{~~todo\|get optimal ET sequence and badness~~\|inline=1}}		{{Todo\|inline=1\|complete temperament data}}

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

ArrowHead294 at 15:10, 18 December 2024

2024-12-18T15:10:14Z

← Older revision		Revision as of 15:10, 18 December 2024
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	The '''rainy–didacus continuum''' is the continuum of [[2.5.7 subgroup]] temperaments which equate a number of [[rainy comma]]s with the [[didacus comma]] ([[3136/3125]]), and thus is the continuum of all 2.5.7 subgroup temperaments supported by [[31edo]], which tempers both and thus tempers all linear combinations of them. If one wants to use all of these simultaneously but wants more accurate tuning than [[31edo]], [[31st-octave temperaments]] extending [[birds]] may be interesting.		The '''rainy–didacus continuum''' is the [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[rainy comma]]s with the [[didacus comma]] ([[3136/3125]]), and thus is the continuum of all 2.5.7 subgroup temperaments supported by [[31edo]], which tempers both and thus tempers all linear combinations of them. If one wants to use all of these simultaneously but wants more accurate tuning than [[31edo]], [[31st-octave temperaments]] extending [[birds]] may be interesting.

	All temperaments in the continuum satisfy {{nowrap\|([[2100875/2097152]])<sup>''n''</sup> ~ ([[3136/3125]])}} for some rational value of ''n''. The just value of ''n'' is approximately 1.981... so that {{nowrap\|''n'' {{=}} 2}} is especially close to the [[JIP]].		All temperaments in the continuum satisfy {{nowrap\|([[2100875/2097152]])<sup>''n''</sup> ~ ([[3136/3125]])}} for some rational value of ''n''. The just value of ''n'' is approximately 1.981... so that {{nowrap\|''n'' {{=}} 2}} is especially close to the [[JIP]].

ArrowHead294 at 14:55, 18 December 2024

2024-12-18T14:55:33Z

← Older revision		Revision as of 14:55, 18 December 2024
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	! Monzo		! Monzo
	\|-		\|-
	\| -2		\| −2
	\| 2.5.7 [[Mothra]]		\| 2.5.7 [[Mothra]]
	\| 69206436005/68719476736		\| 69206436005/68719476736
	\| {{monzo\| -36 0 1 12 }}		\| {{monzo\| -36 0 1 12 }}
	\|-		\|-
	\| -1		\| −1
	\| [[Mercy]] (2.5.7 [[Miracle]])		\| [[Mercy]] (2.5.7 [[Miracle]])
	\| [[823543/819200]]		\| [[823543/819200]]
	\| {{monzo\| -15 0 -2 7 }}		\| {{monzo\| -15 0 -2 7 }}
	\|-		\|-
	\| -0.5		\| −0.5
	\| 2.5.7 [[Starling temperaments#Myna\|Myna]]		\| 2.5.7 [[Starling temperaments#Myna\|Myna]]
	\| 40353607/40000000		\| 40353607/40000000

ArrowHead294 at 14:55, 18 December 2024

2024-12-18T14:55:10Z

← Older revision		Revision as of 14:55, 18 December 2024
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	The '''~~rainy-didacus~~ continuum''' is the continuum of [[2.5.7 subgroup]] temperaments which equate a number of [[rainy comma]]s with the [[didacus comma]] ([[3136/3125]]), and thus is the continuum of all 2.5.7 subgroup temperaments supported by [[31edo]], which tempers both and thus tempers all linear combinations of them. If one wants to use all of these simultaneously but wants more accurate tuning than [[31edo]], [[31st-octave temperaments]] extending [[birds]] may be interesting.		The '''rainy–didacus continuum''' is the continuum of [[2.5.7 subgroup]] temperaments which equate a number of [[rainy comma]]s with the [[didacus comma]] ([[3136/3125]]), and thus is the continuum of all 2.5.7 subgroup temperaments supported by [[31edo]], which tempers both and thus tempers all linear combinations of them. If one wants to use all of these simultaneously but wants more accurate tuning than [[31edo]], [[31st-octave temperaments]] extending [[birds]] may be interesting.

	All temperaments in the continuum satisfy ([[2100875/2097152]])<sup>''n''</sup> ~ ([[3136/3125]]) for some rational value of ''n''. The just value of ''n'' is approximately 1.981... so that ''n'' = 2 is especially close to the [[JIP]].		All temperaments in the continuum satisfy {{nowrap\|([[2100875/2097152]])<sup>''n''</sup> ~ ([[3136/3125]])}} for some rational value of ''n''. The just value of ''n'' is approximately 1.981... so that {{nowrap\|''n'' {{=}} 2}} is especially close to the [[JIP]].

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
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	\| {{monzo\| -21 0 3 5 }}		\| {{monzo\| -21 0 3 5 }}
	\|}		\|}
	<nowiki>*~~</nowiki> note~~ that [[ostara]] is contorted in the 2.5.7 subgroup, hence why this is not merely "2.5.7 ostara"; by contrast, neither "2.5.7 [[myna]]" nor "2.5.7 [[mohajira]]" are contorted.		<nowiki />* Note that [[ostara]] is contorted in the 2.5.7 subgroup, hence why this is not merely "2.5.7 ostara"; by contrast, neither "2.5.7 [[myna]]" nor "2.5.7 [[mohajira]]" are contorted.

	== Temperaments ==		== Temperaments ==

ArrowHead294: ArrowHead294 moved page Rainy-didacus equivalence continuum to Rainy–didacus equivalence continuum: The dash in titles like these should be an en dash, not a hyphen-minus, since "Rainy" does not modify "didacus"

2024-12-18T14:54:28Z

ArrowHead294 moved page Rainy-didacus equivalence continuum to Rainy–didacus equivalence continuum: The dash in titles like these should be an en dash, not a hyphen-minus, since "Rainy" does not modify "didacus"

← Older revision	Revision as of 14:54, 18 December 2024
(No difference)

Lériendil: /* Temperaments */

2024-10-24T05:27:26Z

Temperaments

← Older revision		Revision as of 05:27, 24 October 2024
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	== Temperaments ==		== Temperaments ==
	=== Exodia ===		=== Exodia ===
	Exodia is the 2.5.7 subgroup restriction of [[mohajira]], but is a true microtemperament, supported among others by [[789edo]], [[1957edo]], and [[5902edo]], extremely strong systems in this subgroup.		Exodia is the 2.5.7 subgroup restriction of [[mohajira]], but unlike mohajira, is a true microtemperament, supported among others by [[789edo]], [[1957edo]], and [[5902edo]], extremely strong systems in this subgroup.

	[[Subgroup]]: 2.5.7		[[Subgroup]]: 2.5.7