Rainy–didacus equivalence continuum: Difference between revisions

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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]].

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|-

| 0.5

| 2.5.7 ~~restriction* of~~ [[~~Ostara~~]]

| 2.5.7 [[Mirkwai clan #Grendel|Grendel]]

| 8589934592/8544921875

| {{monzo| 33 0 -13 -1 }}

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| 2

| Exodia (2.5.7 [[Meantone family#Mohajira|Mohajira]])

| 281484423828125/281474976710656

| [[Exodia comma|281484423828125/281474976710656]]

| {{monzo| -48 0 11 8 }}

|-

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| 31 & 612

| 591363588909912109375/590295810358705651712

| {{monzo| -69 14 13 }}

| {{monzo| -69 0 14 13 }}

|-

| …

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| {{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 ==

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[[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): 0.0148

[[Category:31edo]]

[[Category:Equivalence continua]]

@@ Line 1: / Line 1: @@
-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]].
@@ Line 34: / Line 34: @@
 |-
 | 0.5
-| 2.5.7 restriction* of [[Ostara]]
+| 2.5.7 [[Mirkwai clan #Grendel|Grendel]]
 | 8589934592/8544921875
 | {{monzo| 33 0 -13 -1 }}
@@ Line 50: / Line 50: @@
 | 2
 | Exodia (2.5.7 [[Meantone family#Mohajira|Mohajira]])
-| 281484423828125/281474976710656
+| [[Exodia comma|281484423828125/281474976710656]]
 | {{monzo| -48 0 11 8 }}
 |-
@@ Line 56: / Line 56: @@
 | 31 & 612
 | 591363588909912109375/590295810358705651712
-| {{monzo| -69 14 13 }}
+| {{monzo| -69 0 14 13 }}
 |-
+| …
 | …
 | …
@@ Line 67: / Line 68: @@
 | {{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 ==
@@ Line 81: / Line 81: @@
 [[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): 0.0148
 [[Category:31edo]]
 [[Category:Equivalence continua]]