Xenharmonic Wiki

Diminished family: Difference between revisions

← Older edit
VisualWikitext
Update keys
Post-move cleanup
 
(19 intermediate revisions by 5 users not shown)
Line 1: Line 1:
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]].  
{{Technical data page}}
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]].
 
== 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.  


== Dimipent ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 8: Line 13:
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}


[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.507
[[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 }}


{{Optimal ET sequence|legend=1| 4, 8, 12 }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}


[[Badness]]: 0.047231
[[Badness]] (Sintel): 1.11


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 23: Line 32:
{{Mapping|legend=1| 4 0 3 5 | 0 1 1 1 }}
{{Mapping|legend=1| 4 0 3 5 | 0 1 1 1 }}


{{Multival|legend=1|4 4 4 -3 -5 -2}}
[[Optimal tuning]]s:
 
* [[WE]]: ~6/5 = 299.0347{{c}}, ~3/2 = 697.2727{{c}} (~21/20 = 99.2032{{c}})
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.523
: [[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 }}
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}


[[Badness]]: 0.022401
[[Badness]] (Sintel): 0.567


=== 11-limit ===
=== 11-limit ===
Line 38: Line 49:
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109
Optimal tunings:
 
* WE: ~6/5 = 297.8458{{c}}, ~3/2 = 703.9277{{c}} (~15/14 = 108.2361{{c}})
{{Optimal ET sequence|legend=1| 4, 8d, 12, 32cddee, 44cddeee }}
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})


Badness: 0.022132
{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}


Scales: [[diminished12]]
Badness (Sintel): 0.732


==== 13-limit ====
==== 13-limit ====
Line 53: Line 64:
Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}
Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773
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=1| 4, 8d, 12f, 20cdef }}
{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}


Badness: 0.019509
Badness (Sintel): 0.806
 
Scales: [[diminished12]]


=== Demolished ===
=== Demolished ===
Line 68: Line 79:
Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}
Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881
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=1| 12, 28, 40de }}
{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}


Badness: 0.026574
Badness (Smith): 0.879


=== 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 &amp; 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
Line 83: Line 96:
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}


Mapping generators: ~6/5, ~11/7
: mapping generators: ~6/5, ~11/7


Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679
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=1| 8bce, 12 }}
{{Optimal ET sequence|legend=0| 8bce, 12 }}


Badness: 0.054965
Badness (Sintel): 1.82


==== 13-limit ====
==== 13-limit ====
Line 98: Line 113:
Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299
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=1| 8bcef, 12f }}
{{Optimal ET sequence|legend=0| 8bcef, 12f }}


Badness: 0.041707
Badness (Sintel): 1.72


== 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


Line 111: Line 130:
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}


{{Multival|legend=1| 8 8 4 -6 -16 -13 }}
: mapping generators: ~6/5, ~7/4


[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~7/6 = 252.555
[[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 }}


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


[[Badness]]: 0.086378
[[Badness]] (Sintel): 2.19


=== 11-limit ===
=== 11-limit ===
Line 124: 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 tuning (POTE): ~6/5 = 1\4, ~7/6 = 251.658
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}})


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


Badness: 0.056576
Badness (Sintel): 1.87


=== 13-limit ===
=== 13-limit ===
Line 139: Line 164:
Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}
Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.225
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}})
 
{{Optimal ET sequence|legend=0| 4ef, 24 }}
 
Badness (Sintel): 1.61


{{Optimal ET sequence|legend=1| 4ef, 20cef, 24, 52de, 76cdde }}
== Octonion ==
Octonion tempers out 245/243, and may be described as the {{nowrap| 8d & 24 }} temperament. Its ploidacot is octoploid monocot.


Badness: 0.039030
It was formerly known as ''semidim'' but renamed to avoid confusion with another temperament of the same name.  


== Semidim ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 152: Line 183:
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}


{{Multival|legend=1| 8 8 16 -6 3 15 }}
: mapping generators: ~15/14, ~3


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\8, ~3/2 = 707.014
[[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 }}


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


[[Badness]]: 0.107523
[[Badness]] (Sintel): 2.72


=== 11-limit ===
=== 11-limit ===
Line 167: Line 202:
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 706.645
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}})


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


Badness: 0.047598
Badness (Sintel): 1.57


=== 13-limit ===
=== 13-limit ===
Line 180: Line 217:
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 707.376
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}})


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


Badness: 0.030597
Badness (Sintel): 1.26


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dimipent family| ]] <!-- main article -->
[[Category:Pages with mostly numerical content]]
[[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"