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 ([[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]]. | 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 62: | Line 68: | ||
| {{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]] | ||
Latest revision as of 23:05, 15 January 2026
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 for the other primes, then 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 Grendel | 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 0 14 13⟩ |
| … | … | … | … |
| ∞ | Rainy | 2100875/2097152 | [-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: [⟨1 0 6], ⟨0 8 -11]]
Optimal tuning (CWE): ~2 = 1\1, ~262144/214375 = 348.289
Optimal ET sequence: 31, 224, 255, 286, 317, 348, 379, 410, 789
Badness (Sintel): 0.0148