Rainy–didacus equivalence continuum: Difference between revisions
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The ''' | 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 ([[ | 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'' | ||
| Line 13: | Line 13: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| -1 | | −2 | ||
| [[Mercy]] | | 2.5.7 [[Mothra]] | ||
| 69206436005/68719476736 | |||
| {{monzo| -36 0 1 12 }} | |||
|- | |||
| −1 | |||
| [[Mercy]] (2.5.7 [[Miracle]]) | |||
| [[823543/819200]] | | [[823543/819200]] | ||
| {{monzo| -15 0 -2 7 }} | | {{monzo| -15 0 -2 7 }} | ||
|- | |- | ||
| | | −0.5 | ||
| 2.5.7 [[Starling temperaments#Myna|Myna]] | | 2.5.7 [[Starling temperaments#Myna|Myna]] | ||
| | | 40353607/40000000 | ||
| {{monzo| -9 0 -7 9 }} | | {{monzo| -9 0 -7 9 }} | ||
|- | |- | ||
| Line 29: | Line 34: | ||
|- | |- | ||
| 0.5 | | 0.5 | ||
| 2.5.7 | | 2.5.7 [[Mirkwai clan #Grendel|Grendel]] | ||
| 8589934592/8544921875 | | 8589934592/8544921875 | ||
| {{monzo| 33 0 -13 -1 }} | | {{monzo| 33 0 -13 -1 }} | ||
| Line 44: | Line 49: | ||
|- | |- | ||
| 2 | | 2 | ||
| 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 }} | ||
|- | |- | ||
| Line 51: | Line 56: | ||
| 31 & 612 | | 31 & 612 | ||
| 591363588909912109375/590295810358705651712 | | 591363588909912109375/590295810358705651712 | ||
| {{monzo| -69 14 13 }} | | {{monzo| -69 0 14 13 }} | ||
|- | |- | ||
| … | |||
| … | | … | ||
| … | | … | ||
| Line 59: | Line 65: | ||
| ∞ | | ∞ | ||
| [[Rainy]] | | [[Rainy]] | ||
| [[ | | [[2100875/2097152]] | ||
| {{monzo| -21 0 3 5 }} | | {{monzo| -21 0 3 5 }} | ||
|} | |} | ||
== Temperaments == | |||
=== Exodia === | |||
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 | |||
[[Comma list]]: 281484423828125/281474976710656 | |||
[[Mapping]]: [{{val| 1 0 6 }}, {{val| 0 8 -11 }}] | |||
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~262144/214375 = 348.289 | |||
{{Optimal ET sequence|legend=1|31, 224, 255, 286, 317, 348, 379, 410, 789}} | |||
[[Badness]] (Sintel): 0.0148 | |||
[[Category:31edo]] | [[Category:31edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||