Rainy–didacus equivalence continuum
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 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: [⟨1 0 6], ⟨0 8 -11]]
Optimal tuning (CWE): ~2 = 1\1, ~262144/214375 = 348.289
Supporting EDOs: 789, 31, 410, 379, 1199, 1168, 1957, 348, 1609, 441, 1547, 317, 727, 851
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Todo: get optimal ET sequence and badness |