Rainy–didacus equivalence continuum: Difference between revisions
Correct ratio for Rainy comma |
m correct equivalence used; smaller ratio is the one that goes to the power of n |
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The '''rainy-didacus continuum''' is the 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 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. | ||
All temperaments in the continuum satisfy ([[ | All temperaments in the continuum satisfy ([[2100875/2097152]])<sup>''n''</sup> ~ ([[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]]. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
Revision as of 00:04, 21 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 (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 | ||
| -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.