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Diminished family: Difference between revisions

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{{Technical data page}}
{{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) }}
{{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
[[Subgroup]]: 2.3.5
Line 16: Line 18:
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 698.2660{{c}} (~25/24 = 98.2660{{c}})
* [[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 }}
: error map: {{val| 0.000 -3.689 +11.952 }}
<!-- * [[CTE]]: ~6/5 = 300.000{{c}}, ~3/2 = 696.983{{c}} (~25/24 = 96.983{{c}})
: [[error map]]: {{val| 0.000 -4.972 +10.670 }}
* [[POTE]]: ~6/5 = 300.000{{c}}, ~3/2 = 699.507{{c}} (~25/24 = 99.507{{c}})
: error map: {{val| 0.000 -2.448 +13.193 }} -->


{{Optimal ET sequence|legend=1| 4, 8, 12 }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}
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[[Badness]] (Sintel): 1.11
[[Badness]] (Sintel): 1.11


== Diminished ==
== Septimal diminished ==
<div style="float: right">[[:de:Verminderte Temperaturen|Deutsch]]</div>
 
{{Main| Diminished (temperament) }}
{{Main| Diminished (temperament) }}


Line 41: Line 37:
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 695.9619{{c}} (~21/20 = 95.9619{{c}})
* [[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 }}
: error map: {{val| 0.000 -5.993 +9.648 +27.136 }}
<!-- * [[CTE]]: ~6/5 = 300.000{{c}}, ~3/2 = 691.955{{c}} (~21/20 = 91.955{{c}})
: [[error map]]: {{val| 0.000 -10.000 +5.641 +23.129 }}
* [[POTE]]: ~6/5 = 300.000{{c}}, ~3/2 = 699.523{{c}} (~21/20 = 99.523{{c}})
: error map: {{val| 0.000 -2.432 +13.210 +30.698 }} -->


{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
Line 60: Line 52:
* WE: ~6/5 = 297.8458{{c}}, ~3/2 = 703.9277{{c}} (~15/14 = 108.2361{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~3/2 = 691.955{{c}} (~21/20 = 91.955{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~3/2 = 709.109{{c}} (~15/14 = 109.109{{c}}) -->


{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}
{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}
Line 77: Line 67:
* WE: ~6/5 = 297.2520{{c}}, ~3/2 = 707.2352{{c}} (~15/14 = 112.7312{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 708.4648{{c}} (~15/14 = 108.4648{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~3/2 = 691.954{{c}} (~21/20 = 91.954{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~3/2 = 713.773{{c}} (~15/14 = 113.773{{c}}) -->


{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}
{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}
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* WE: ~6/5 = 299.6308{{c}}, ~3/2 = 689.0322{{c}} (~21/20 = 89.7707{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 688.9304{{c}} (~21/20 = 88.9304{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~3/2 = 687.747{{c}} (~21/20 = 87.747{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~3/2 = 689.881{{c}} (~21/20 = 89.881{{c}}) -->


{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}
{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}
Line 102: Line 88:


=== Cohedim ===
=== 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
Subgroup: 2.3.5.7.11
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* WE: ~6/5 = 298.7799{{c}}, ~11/7 = 795.0744{{c}} (~12/11 = 101.2653{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 796.0102{{c}} (~12/11 = 103.9898{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~11/7 = 793.423{{c}} (~12/11 = 106.577{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~11/7 = 798.421{{c}} (~12/11 = 101.679{{c}}) -->


{{Optimal ET sequence|legend=0| 8bce, 12 }}
{{Optimal ET sequence|legend=0| 8bce, 12 }}
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* WE: ~6/5 = 298.4646{{c}}, ~11/7 = 793.6185{{c}} (~12/11 = 101.7754{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 794.7323{{c}} (~12/11 = 105.2677{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~11/7 = 791.375{{c}} (~12/11 = 108.625{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~11/7 = 797.701{{c}} (~12/11 = 102.299{{c}}) -->


{{Optimal ET sequence|legend=0| 8bcef, 12f }}
{{Optimal ET sequence|legend=0| 8bcef, 12f }}
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== Hemidim ==
== 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
[[Subgroup]]: 2.3.5.7


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* [[CWE]]: ~6/5 = 300.0000{{c}}, ~7/4 = 948.2575{{c}} (~36/35 = 48.2575{{c}})
* [[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 }}
: error map: {{val| 0.000 -5.440 +10.201 -20.568 }}
<!-- * [[CTE]]: ~6/5 = 300.000{{c}}, ~7/4 = 949.540{{c}} (~36/35 = 49.540{{c}})
: [[error map]]: {{val| 0.000 -2.875 +12.766 -19.286 }}
* [[POTE]]: ~6/5 = 300.000{{c}}, ~7/4 = 947.445{{c}} (~36/35 = 47.445{{c}})
: error map: {{val| 0.000 -7.065 +8.576 -21.381 }} -->


{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}
Line 167: Line 147:
Comma list: 49/48, 77/75, 243/242
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:  
Optimal tunings:  
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.6958{{c}} (~36/35 = 48.4112{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.9065{{c}} (~36/35 = 48.9065{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~7/4 = 949.872{{c}} (~36/35 = 49.872{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~7/4 = 948.342{{c}} (~36/35 = 48.342{{c}}) -->


{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}
{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}
Line 189: Line 167:
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.2440{{c}} (~36/35 = 47.8487{{c}})
* 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}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.3581{{c}} (~36/35 = 48.3581{{c}})
<!-- * CTE: ~6/5 = 300.000{{c}}, ~7/4 = 949.366{{c}} (~36/35 = 49.366{{c}})
* POTE: ~6/5 = 300.000{{c}}, ~7/4 = 947.775{{c}} (~36/35 = 47.775{{c}}) -->


{{Optimal ET sequence|legend=0| 4ef, 24 }}
{{Optimal ET sequence|legend=0| 4ef, 24 }}
Line 197: Line 173:


== Octonion ==
== 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
[[Subgroup]]: 2.3.5.7
Line 212: Line 190:
* [[CWE]]: ~15/14 = 150.0000{{c}}, ~3/2 = 704.9636{{c}} (~36/35 = 45.0364{{c}})
* [[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 }}
: error map: {{val| 0.000 +3.008 +18.650 -8.899 }}
<!-- * [[CTE]]: ~15/14 = 150.000{{c}}, ~3/2 = 702.765{{c}} (~36/35 = 47.235{{c}})
* [[POTE]]: ~15/14 = 150.000{{c}}, ~3/2 = 707.014{{c}} (~36/35 = 42.986{{c}}) -->


{{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }}
{{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }}
Line 229: Line 205:
* WE: ~12/11 = 149.7102{{c}}, ~3/2 = 705.2799{{c}} (~36/35 = 43.2712{{c}})
* 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}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 704.9285{{c}} (~36/35 = 45.0715{{c}})
<!-- * CTE: ~12/11 = 150.000{{c}}, ~3/2 = 702.662{{c}} (~36/35 = 47.338{{c}})
* POTE: ~12/11 = 150.000{{c}}, ~3/2 = 706.645{{c}} (~36/35 = 43.355{{c}}) -->


{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32c }}
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32c }}
Line 246: Line 220:
* WE: ~12/11 = 149.6311{{c}}, ~3/2 = 705.6367{{c}} (~36/35 = 42.5188{{c}})
* 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}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 705.2777{{c}} (~36/35 = 44.7223{{c}})
<!-- * CTE: ~12/11 = 150.000{{c}}, ~3/2 = 701.952{{c}} (~36/35 = 48.048{{c}})
* POTE: ~12/11 = 150.000{{c}}, ~3/2 = 707.376{{c}} (~36/35 = 42.624{{c}}) -->


{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}
Line 255: Line 227:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with mostly numerical content]]
[[Category:Dimipent family| ]] <!-- main article -->
[[Category:Diminished family| ]] <!-- main article -->
[[Category:Diminished| ]] <!-- key article -->
[[Category:Diminished| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Diminished_family"