Rainy–didacus equivalence continuum: Difference between revisions
m Lériendil moved page Rainy-didacus continuum to Rainy-didacus equivalence continuum: consistency in article titles |
documented exodia |
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| 2.5.7 [[Meantone family#Mohajira|Mohajira]] | | Exodia (2.5.7 [[Meantone family#Mohajira|Mohajira]]) | ||
| 281484423828125/281474976710656 | | 281484423828125/281474976710656 | ||
| {{monzo| -48 0 11 8 }} | | {{monzo| -48 0 11 8 }} | ||
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<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>*</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 == | |||
=== Exodia === | |||
Exodia is the 2.5.7 subgroup restriction of [[mohajira]], but is a true microtemperament, supported among others by [[789edo]], an extremely strong system 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 | |||
{{todo|get optimal ET sequence and badness|inline=1}} | |||
[[Category:31edo]] | [[Category:31edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 04:58, 24 October 2024
The rainy-didacus continuum is the continuum of 2.5.7 subgroup temperaments which equate a number of rainy commas 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)n ~ (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.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -2 | 2.5.7 Mothra | 69206436005/68719476736 | [-36 0 1 12⟩ |
| -1 | Mercy (2.5.7 Miracle) | 823543/819200 | [-15 0 -2 7⟩ |
| -0.5 | 2.5.7 Myna | 40353607/40000000 | [-9 0 -7 9⟩ |
| 0 | Didacus | 3136/3125 | [6 0 -5 2⟩ |
| 0.5 | 2.5.7 restriction* of Ostara | 8589934592/8544921875 | [33 0 -13 -1⟩ |
| 1 | Vorwell | 134217728/133984375 | [27 0 -8 -3⟩ |
| 1.5 | 31 & 494 | 37778931862957161709568/37714514598846435546875 | [75 0 -19 -11⟩ |
| 2 | Exodia (2.5.7 Mohajira) | 281484423828125/281474976710656 | [-48 0 11 8⟩ |
| 3 | 31 & 612 | 591363588909912109375/590295810358705651712 | [-69 14 13⟩ |
| … | … | … | |
| ∞ | Rainy | 2100875/2097152 | [-21 0 3 5⟩ |
* 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
Exodia
Exodia is the 2.5.7 subgroup restriction of mohajira, but is a true microtemperament, supported among others by 789edo, an extremely strong system in this subgroup.
Subgroup: 2.5.7
Comma list: 281484423828125/281474976710656
Mapping: [⟨1 0 6], ⟨0 8 -11]]
Optimal tuning (CWE): ~2 = 1\1, ~262144/214375 = 348.289
|
Todo: get optimal ET sequence and badness |