Rainy–didacus equivalence continuum: Difference between revisions
Created page with "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..." |
Correct ratio for Rainy comma |
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Revision as of 23:58, 20 September 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 (3136/3125)n ~ 2100875/2097152 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 | ||
| -1 | Mercy | 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 | 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.