Diminished family: Difference between revisions

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 {{Technical data page}}
-The '''dimipent family''' tempers out the major diesis aka diminished comma, [[648/625]], the amount by which four [[6/5]] minor thirds exceed an [[octave]], and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
+The '''diminished family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the major diesis a.k.a. diminished comma, [[648/625]], the amount by which four [[6/5]] minor thirds exceed an [[octave]], and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
-== Dimipent ==
+== Diminished ==
 {{Main| Diminished (temperament) }}
+The [[generator]] of diminished can be taken as a fifth or a semitone, and 12edo, with its excellent fifth, is an obvious tuning, though a flatter fifth might be preferred to go with the flat minor third. Its [[ploidacot]] is tetraploid monocot.
 [[Subgroup]]: 2.3.5
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 {{Mapping|legend=1| 4 0 3 | 0 1 1 }}
+: mapping generators: ~6/5, ~3
 [[Optimal tuning]]s:
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 [[Badness]] (Sintel): 1.11
-== Diminished ==
+== Septimal diminished ==
-<div style="float: right">[[:de:Verminderte Temperaturen|Deutsch]]</div>
+{{Main| Diminished (temperament) }}
-{{Main| Diminished (temperament) }}
+Septimal diminished equates the 300{{c}} minor third to [[7/6]] along with 6/5, and the 600{{c}} tritone to [[7/5]][[~]][[10/7]].
 [[Subgroup]]: 2.3.5.7
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 === Cohedim ===
-This temperament has been documented in Graham Breed's temperament finder as ''hemidim'', the same name as [[Dimipent family #Hemidim|11-limit 4e &amp; 24 and 13-limit 4ef &amp; 24]]. For 11-limit 8bce & 12 temperament, ''cohedim'' arguably makes more sense.
+This extension has been documented in Graham Breed's temperament finder as ''hemidim'', the same name as [[#Hemidim|11-limit 4e & 24 and 13-limit 4ef & 24]]. For the 11-limit 8bce & 12 temperament, ''cohedim'' arguably makes more sense. Its ploidacot is tetraploid alpha-dicot.
 Subgroup: 2.3.5.7.11
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 == Hemidim ==
+Hemidim tempers out 49/48 and may be described as the {{nowrap| 4 & 24 }} temperament. Its ploidcot is tetraploid dicot.
 [[Subgroup]]: 2.3.5.7
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 Comma list: 49/48, 77/75, 243/242
-Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5}}
+Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5 }}
 Optimal tunings:
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 == Octonion ==
-It was formerly known as "semidim" but renamed to avoid confusion with another temperament of the same name.
+Octonion tempers out 245/243, and may be described as the {{nowrap| 8d & 24 }} temperament. Its ploidacot is octoploid monocot.
+It was formerly known as ''semidim'' but renamed to avoid confusion with another temperament of the same name.
 [[Subgroup]]: 2.3.5.7
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 Badness (Sintel): 1.26
+== Subgroup extensions ==
+Diminished works on 17- and 19-limit subgroups very well, where the 1/4-octave period stands in for ~6/5, ~19/16, and ~20/17.
+=== Diminished (2.3.5.17 subgroup) ===
+[[Subgroup]]: 2.3.5.17
+[[Comma list]]: 51/50, 289/288
+{{Mapping|legend=2| 4 0 3 10 | 0 1 1 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~6/5 = 299.899{{c}}, ~3/2 = 698.251{{c}} (~17/16 = 98.453{{c}})
+* [[CWE]]: ~6/5 = 300.000{{c}}, ~3/2 = 698.137{{c}} (~17/16 = 98.137{{c}})
+{{Optimal ET sequence|legend=1| 4, 8, 12 }}
+[[Badness]] (Sintel): 0.509
+=== Diminished (2.3.5.19 subgroup) ===
+[[Subgroup]]: 2.3.5.19
+[[Comma list]]: 96/95, 513/500
+{{Mapping|legend=2| 4 0 3 17 | 0 1 1 0 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~6/5 = 299.744{{c}}, ~3/2 = 698.206{{c}} (~20/19 = 98.718{{c}})
+* [[CWE]]: ~6/5 = 300.000{{c}}, ~3/2 = 698.195{{c}} (~20/19 = 98.195{{c}})
+{{Optimal ET sequence|legend=1| 4, 8, 12 }}
+[[Badness]] (Sintel): 0.681
+[[Category:Diminished family| ]] <!-- main article -->
+[[Category:Diminished| ]] <!-- key article -->
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Dimipent family| ]] <!-- main article -->
-[[Category:Diminished| ]] <!-- key article -->
-[[Category:Rank 2]]