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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 ==
Subgroup: 2.3.5
{{Main| Diminished (temperament) }}


[[Comma]]: 648/625
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.


[[POTE tuning|POTE generator]]: ~3/2 = 699.507
[[Subgroup]]: 2.3.5


[[Map]]: [<4 0 3|, <0 1 1|]
[[Comma list]]: 648/625


{{Vals|legend=1| 4, 8, 12 }}
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}


[[Badness]]: 0.047231
[[Optimal tuning]]s:  
* [[WE]]: ~6/5 = 299.6476{{c}}, ~3/2 = 698.6854{{c}} (~25/24 = 99.3903{{c}})
: [[error map]]: {{val| -1.410 -4.679 +9.905 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 698.2660{{c}} (~25/24 = 98.2660{{c}})
: error map: {{val| 0.000 -3.689 +11.952 }}


=Diminished=
{{Optimal ET sequence|legend=1| 4, 8, 12 }}
{{see also|Jubilismic clan}}


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 1.11
 
== Septimal diminished ==
{{Main| Diminished (temperament) }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 50/49
[[Comma list]]: 36/35, 50/49


[[POTE tuning|POTE generator]]: ~3/2 = 699.523
{{Mapping|legend=1| 4 0 3 5 | 0 1 1 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 299.0347{{c}}, ~3/2 = 697.2727{{c}} (~21/20 = 99.2032{{c}})
: [[error map]]: {{val| -3.861 -8.543 +4.202 +19.759 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 695.9619{{c}} (~21/20 = 95.9619{{c}})
: error map: {{val| 0.000 -5.993 +9.648 +27.136 }}
 
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
 
[[Badness]] (Sintel): 0.567
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 50/49, 56/55
 
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
 
Optimal tunings:
* WE: ~6/5 = 297.8458{{c}}, ~3/2 = 703.9277{{c}} (~15/14 = 108.2361{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})
 
{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}
 
Badness (Sintel): 0.732
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 40/39, 50/49, 66/65
 
Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}
 
Optimal tunings:
* WE: ~6/5 = 297.2520{{c}}, ~3/2 = 707.2352{{c}} (~15/14 = 112.7312{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 708.4648{{c}} (~15/14 = 108.4648{{c}})
 
{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}
 
Badness (Sintel): 0.806
 
=== Demolished ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 50/49
 
Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}
 
Optimal tunings:
* WE: ~6/5 = 299.6308{{c}}, ~3/2 = 689.0322{{c}} (~21/20 = 89.7707{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 688.9304{{c}} (~21/20 = 88.9304{{c}})
 
{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}


[[Map]]: [<4 0 3 5|, <0 1 1 1|]
Badness (Smith): 0.879


Wedgie: <<4 4 4 -3 -5 -2||
=== Cohedim ===
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.


{{Vals|legend=1| 4, 8d, 12 }}
Subgroup: 2.3.5.7.11


[[Badness]]: 0.022401
Comma list: 36/35, 50/49, 125/121


=Hemidim=
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
Subgroup: 2.3.5.7
 
: mapping generators: ~6/5, ~11/7
 
Optimal tunings:
* WE: ~6/5 = 298.7799{{c}}, ~11/7 = 795.0744{{c}} (~12/11 = 101.2653{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 796.0102{{c}} (~12/11 = 103.9898{{c}})
 
{{Optimal ET sequence|legend=0| 8bce, 12 }}
 
Badness (Sintel): 1.82
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 50/49, 66/65, 125/121
 
Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
 
Optimal tunings:
* WE: ~6/5 = 298.4646{{c}}, ~11/7 = 793.6185{{c}} (~12/11 = 101.7754{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 794.7323{{c}} (~12/11 = 105.2677{{c}})
 
{{Optimal ET sequence|legend=0| 8bcef, 12f }}
 
Badness (Sintel): 1.72
 
== 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


[[Comma list]]: 49/48, 648/625
[[Comma list]]: 49/48, 648/625


[[POTE tuning|POTE generator]]: ~8/7 = 252.555
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}


[[Map]]: [<4 0 3 8|, <0 2 2 1|]
: mapping generators: ~6/5, ~7/4


Wedgie: <<8 8 4 -6 -16 -13||
[[Optimal tuning]]s:  
* [[WE]]: ~6/5 = 300.5053{{c}}, ~7/4 = 949.0409{{c}} (~36/35 = 47.5250{{c}})
: [[error map]]: {{val| +2.021 -3.873 +13.284 -15.743 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~7/4 = 948.2575{{c}} (~36/35 = 48.2575{{c}})
: error map: {{val| 0.000 -5.440 +10.201 -20.568 }}


{{Vals|legend=1| 24, 52d, 76cd }}
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}


[[Badness]]: 0.0864
[[Badness]] (Sintel): 2.19


==11-limit==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 77/75, 243/242
Comma list: 49/48, 77/75, 243/242


POTE generator: ~8/7 = 251.658
Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5 }}


Map: [<4 0 3 8 -2|, <0 2 2 1 5|]
Optimal tunings:  
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.6958{{c}} (~36/35 = 48.4112{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.9065{{c}} (~36/35 = 48.9065{{c}})


{{Vals|legend=1| 24, 76cde }}
{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}


Badness: 0.0566
Badness (Sintel): 1.87


==13-limit==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 66/65, 77/75, 648/625
Comma list: 49/48, 66/65, 77/75, 243/242


POTE generator: ~8/7 = 252.225
Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}


Map: [<4 0 3 8 -2 -1|, <0 2 2 1 5 5|]
Optimal tunings:  
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.2440{{c}} (~36/35 = 47.8487{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.3581{{c}} (~36/35 = 48.3581{{c}})


{{Vals|legend=1| 24, 52de, 76cde }}
{{Optimal ET sequence|legend=0| 4ef, 24 }}


Badness: 0.039
Badness (Sintel): 1.61


=Semidim=
== Octonion ==
Subgroup: 2.3.5.7
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


[[Comma list]]: 245/243, 392/375
[[Comma list]]: 245/243, 392/375


[[POTE tuning|POTE generator]]: ~3/2 = 707.014
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}


[[Map]]: [<8 0 6 -3|, <0 1 1 2|]
: mapping generators: ~15/14, ~3


Wedgie: <<8 8 16 -6 3 15||
[[Optimal tuning]]s:  
* [[WE]]: ~15/14 = 149.6673{{c}}, ~3/2 = 705.4455{{c}} (~36/35 = 42.8910{{c}})
: [[error map]]: {{val| -2.662 +0.828 +14.474 -12.260 }}
* [[CWE]]: ~15/14 = 150.0000{{c}}, ~3/2 = 704.9636{{c}} (~36/35 = 45.0364{{c}})
: error map: {{val| 0.000 +3.008 +18.650 -8.899 }}


{{Vals|legend=1| 24, 32c, 56c }}
{{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }}


[[Badness]]: 0.1075
[[Badness]] (Sintel): 2.72


==11-limit==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 245/243
Comma list: 56/55, 77/75, 245/243


POTE generator: ~3/2 = 706.645
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}


Map: [<8 0 6 -3 15|, <0 1 1 2 1|]
Optimal tunings:  
* WE: ~12/11 = 149.7102{{c}}, ~3/2 = 705.2799{{c}} (~36/35 = 43.2712{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 704.9285{{c}} (~36/35 = 45.0715{{c}})


{{Vals|legend=1| 24, 32c, 56c }}
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32c }}


Badness: 0.0476
Badness (Sintel): 1.57


==13-limit==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 66/65, 77/75, 507/500
Comma list: 56/55, 66/65, 77/75, 507/500


POTE generator: ~3/2 = 707.376
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
 
Map: [<8 0 6 -3 15 17|, <0 1 1 2 1 1|]


{{Vals|legend=1| 24, 32cf, 56cf }}
Optimal tunings:
* WE: ~12/11 = 149.6311{{c}}, ~3/2 = 705.6367{{c}} (~36/35 = 42.5188{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 705.2777{{c}} (~36/35 = 44.7223{{c}})


Badness: 0.0306
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}


[[Category:Theory]]
Badness (Sintel): 1.26
[[Category:Temperament family]]
[[Category:Diminished]]
[[Category:Rank-2]]


[[Category:Todo:review]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Diminished family| ]] <!-- main article -->
[[Category:Diminished| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Diminished_family"