Diminished family: Difference between revisions

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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]].  


== 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 9: Line 12:


{{Mapping|legend=1| 4 0 3 | 0 1 1 }}
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}
: mapping generators: ~6/5, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6/5 = 300.000, ~3/2 = 696.983 (~25/24 = 96.983)
* [[WE]]: ~6/5 = 299.6476{{c}}, ~3/2 = 698.6854{{c}} (~25/24 = 99.3903{{c}})
: [[error map]]: {{val| 0.000 -4.972 +10.670 }}
: [[error map]]: {{val| -1.410 -4.679 +9.905 }}
* [[POTE]]: ~6/5 = 300.000, ~3/2 = 699.507 (~25/24 = 99.507)
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 698.2660{{c}} (~25/24 = 98.2660{{c}})
: error map: {{val| 0.000 -2.448 +13.193 }}
: 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]] (Smith): 0.047231
[[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
[[Subgroup]]: 2.3.5.7
Line 30: Line 35:


{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6/5 = 300.000, ~3/2 = 691.955 (~21/20 = 91.955)
* [[WE]]: ~6/5 = 299.0347{{c}}, ~3/2 = 697.2727{{c}} (~21/20 = 99.2032{{c}})
: [[error map]]: {{val| 0.000 -10.000 +5.641 +23.129 }}
: [[error map]]: {{val| -3.861 -8.543 +4.202 +19.759 }}
* [[POTE]]: ~6/5 = 300.000, ~3/2 = 699.523 (~21/20 = 99.523)
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 695.9619{{c}} (~21/20 = 95.9619{{c}})
: error map: {{val| 0.000 -2.432 +13.210 +30.698 }}
: 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]] (Smith): 0.022401
[[Badness]] (Sintel): 0.567


=== 11-limit ===
=== 11-limit ===
Line 49: Line 52:


Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
Wedgie: {{multival| 4 4 4 0 -3 -5 -14 -2 -14 -14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 300.000, ~3/2 = 691.955 (~21/20 = 91.955)
* WE: ~6/5 = 297.8458{{c}}, ~3/2 = 703.9277{{c}} (~15/14 = 108.2361{{c}})
* POTE: ~6/5 = 300.000, ~3/2 = 709.109 (~15/14 = 109.109)
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})


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


Badness (Smith): 0.022132
Badness (Sintel): 0.732


==== 13-limit ====
==== 13-limit ====
Line 68: Line 69:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 300.000, ~3/2 = 691.954 (~21/20 = 91.954)
* WE: ~6/5 = 297.2520{{c}}, ~3/2 = 707.2352{{c}} (~15/14 = 112.7312{{c}})
* POTE: ~6/5 = 300.000, ~3/2 = 713.773 (~15/14 = 113.773)
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 708.4648{{c}} (~15/14 = 108.4648{{c}})


Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12f, 20cdef }}
{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}


Badness (Smith): 0.019509
Badness (Sintel): 0.806


=== Demolished ===
=== Demolished ===
Line 83: Line 84:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 300.000, ~3/2 = 687.747 (~21/20 = 87.747)
* WE: ~6/5 = 299.6308{{c}}, ~3/2 = 689.0322{{c}} (~21/20 = 89.7707{{c}})
* POTE: ~6/5 = 300.000, ~3/2 = 689.881 (~21/20 = 89.881)
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 688.9304{{c}} (~21/20 = 88.9304{{c}})


Optimal ET sequence: {{Optimal ET sequence| 12, 28, 40de }}
{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}


Badness (Smith): 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 102: Line 103:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 300.000, ~11/7 = 793.423 (~12/11 = 106.577)
* WE: ~6/5 = 298.7799{{c}}, ~11/7 = 795.0744{{c}} (~12/11 = 101.2653{{c}})
* POTE: ~6/5 = 300.000, ~11/7 = 798.421 (~12/11 = 101.679)
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 796.0102{{c}} (~12/11 = 103.9898{{c}})


Optimal ET sequence: {{Optimal ET sequence| 8bce, 12 }}
{{Optimal ET sequence|legend=0| 8bce, 12 }}


Badness (Smith): 0.054965
Badness (Sintel): 1.82


==== 13-limit ====
==== 13-limit ====
Line 117: Line 118:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 300.000, ~11/7 = 791.375 (~12/11 = 108.625)
* WE: ~6/5 = 298.4646{{c}}, ~11/7 = 793.6185{{c}} (~12/11 = 101.7754{{c}})
* POTE: ~6/5 = 300.000, ~11/7 = 797.701 (~12/11 = 102.299)
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 794.7323{{c}} (~12/11 = 105.2677{{c}})


Optimal ET sequence: {{Optimal ET sequence| 8bcef, 12f }}
{{Optimal ET sequence|legend=0| 8bcef, 12f }}


Badness (Smith): 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 132: Line 135:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6/5 = 300.000, ~7/4 = 949.540 (~36/35 = 49.540)
* [[WE]]: ~6/5 = 300.5053{{c}}, ~7/4 = 949.0409{{c}} (~36/35 = 47.5250{{c}})
: [[error map]]: {{val| 0.000 -2.875 +12.766 -19.286 }}
: [[error map]]: {{val| +2.021 -3.873 +13.284 -15.743 }}
* [[POTE]]: ~6/5 = 300.000, ~7/4 = 947.445 (~36/35 = 47.445)
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~7/4 = 948.2575{{c}} (~36/35 = 48.2575{{c}})
: error map: {{val| 0.000 -7.065 +8.576 -21.381 }}
: error map: {{val| 0.000 -5.440 +10.201 -20.568 }}


{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}


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


=== 11-limit ===
=== 11-limit ===
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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:  
* CTE: ~6/5 = 300.000, ~7/4 = 949.872 (~36/35 = 49.872)
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.6958{{c}} (~36/35 = 48.4112{{c}})
* POTE: ~6/5 = 300.000, ~7/4 = 948.342 (~36/35 = 48.342)
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.9065{{c}} (~36/35 = 48.9065{{c}})


Optimal ET sequence: {{Optimal ET sequence| 4e, 20ce, 24 }}
{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}


Badness (Smith): 0.056576
Badness (Sintel): 1.87


=== 13-limit ===
=== 13-limit ===
Line 168: Line 169:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 300.000, ~7/4 = 949.366 (~36/35 = 49.366)
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.2440{{c}} (~36/35 = 47.8487{{c}})
* POTE: ~6/5 = 300.000, ~7/4 = 947.775 (~36/35 = 47.775)
* 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: {{Optimal ET sequence| 4ef, 24 }}
== Octonion ==
Octonion tempers out 245/243, and may be described as the {{nowrap| 8d & 24 }} temperament. Its ploidacot is octoploid monocot.


Badness (Smith): 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 183: Line 188:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 150.000, ~3/2 = 702.765 (~36/35 = 47.235)
* [[WE]]: ~15/14 = 149.6673{{c}}, ~3/2 = 705.4455{{c}} (~36/35 = 42.8910{{c}})
* [[POTE]]: ~15/14 = 150.000, ~3/2 = 707.014 (~36/35 = 42.986)
: [[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 }}
{{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }}


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


=== 11-limit ===
=== 11-limit ===
Line 202: Line 207:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 150.000, ~3/2 = 702.662 (~36/35 = 47.338)
* WE: ~12/11 = 149.7102{{c}}, ~3/2 = 705.2799{{c}} (~36/35 = 43.2712{{c}})
* POTE: ~12/11 = 150.000, ~3/2 = 706.645 (~36/35 = 43.355)
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 704.9285{{c}} (~36/35 = 45.0715{{c}})


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


Badness (Smith): 0.047598
Badness (Sintel): 1.57


=== 13-limit ===
=== 13-limit ===
Line 217: Line 222:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 150.000, ~3/2 = 701.952 (~36/35 = 48.048)
* WE: ~12/11 = 149.6311{{c}}, ~3/2 = 705.6367{{c}} (~36/35 = 42.5188{{c}})
* POTE: ~12/11 = 150.000, ~3/2 = 707.376 (~36/35 = 42.624)
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 705.2777{{c}} (~36/35 = 44.7223{{c}})
 
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}
 
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: {{Optimal ET sequence| 8d, 24, 32cf }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}


Badness (Smith): 0.030597
[[Badness]] (Sintel): 0.681


[[Category:Diminished family| ]] <!-- main article -->
[[Category:Diminished| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dimipent family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Diminished| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 11:55, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The diminished family of temperaments 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 article: 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

Comma list: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

mapping generators: ~6/5, ~3

Optimal tunings:

  • WE: ~6/5 = 299.6476 ¢, ~3/2 = 698.6854 ¢ (~25/24 = 99.3903 ¢)
error map: ⟨-1.410 -4.679 +9.905]
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 698.2660 ¢ (~25/24 = 98.2660 ¢)
error map: ⟨0.000 -3.689 +11.952]

Optimal ET sequence: 4, 8, 12

Badness (Sintel): 1.11

Septimal diminished

Main article: Diminished (temperament)

Septimal diminished equates the 300 ¢ minor third to 7/6 along with 6/5, and the 600 ¢ tritone to 7/5~10/7.

Subgroup: 2.3.5.7

Comma list: 36/35, 50/49

Mapping: [⟨4 0 3 5], ⟨0 1 1 1]]

Optimal tunings:

  • WE: ~6/5 = 299.0347 ¢, ~3/2 = 697.2727 ¢ (~21/20 = 99.2032 ¢)
error map: ⟨-3.861 -8.543 +4.202 +19.759]
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 695.9619 ¢ (~21/20 = 95.9619 ¢)
error map: ⟨0.000 -5.993 +9.648 +27.136]

Optimal ET sequence: 4, 8d, 12

Badness (Sintel): 0.567

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 56/55

Mapping: [⟨4 0 3 5 14], ⟨0 1 1 1 0]]

Optimal tunings:

  • WE: ~6/5 = 297.8458 ¢, ~3/2 = 703.9277 ¢ (~15/14 = 108.2361 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 703.5558 ¢ (~15/14 = 103.5558 ¢)

Optimal ET sequence: 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: [⟨4 0 3 5 14 15], ⟨0 1 1 1 0 0]]

Optimal tunings:

  • WE: ~6/5 = 297.2520 ¢, ~3/2 = 707.2352 ¢ (~15/14 = 112.7312 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 708.4648 ¢ (~15/14 = 108.4648 ¢)

Optimal ET sequence: 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: [⟨4 0 3 5 -5], ⟨0 1 1 1 3]]

Optimal tunings:

  • WE: ~6/5 = 299.6308 ¢, ~3/2 = 689.0322 ¢ (~21/20 = 89.7707 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 688.9304 ¢ (~21/20 = 88.9304 ¢)

Optimal ET sequence: 4e, 8dee, 12, 28

Badness (Smith): 0.879

Cohedim

This extension has been documented in Graham Breed's temperament finder as hemidim, the same name as 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

Comma list: 36/35, 50/49, 125/121

Mapping: [⟨4 1 4 6 6], ⟨0 2 2 2 3]]

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

Optimal tunings:

  • WE: ~6/5 = 298.7799 ¢, ~11/7 = 795.0744 ¢ (~12/11 = 101.2653 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~11/7 = 796.0102 ¢ (~12/11 = 103.9898 ¢)

Optimal ET sequence: 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: [⟨4 1 4 6 6 7], ⟨0 2 2 2 3 3]]

Optimal tunings:

  • WE: ~6/5 = 298.4646 ¢, ~11/7 = 793.6185 ¢ (~12/11 = 101.7754 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~11/7 = 794.7323 ¢ (~12/11 = 105.2677 ¢)

Optimal ET sequence: 8bcef, 12f

Badness (Sintel): 1.72

Hemidim

Hemidim tempers out 49/48 and may be described as the 4 & 24 temperament. Its ploidcot is tetraploid dicot.

Subgroup: 2.3.5.7

Comma list: 49/48, 648/625

Mapping: [⟨4 0 3 8], ⟨0 2 2 1]]

mapping generators: ~6/5, ~7/4

Optimal tunings:

  • WE: ~6/5 = 300.5053 ¢, ~7/4 = 949.0409 ¢ (~36/35 = 47.5250 ¢)
error map: ⟨+2.021 -3.873 +13.284 -15.743]
  • CWE: ~6/5 = 300.0000 ¢, ~7/4 = 948.2575 ¢ (~36/35 = 48.2575 ¢)
error map: ⟨0.000 -5.440 +10.201 -20.568]

Optimal ET sequence: 4, …, 20c, 24

Badness (Sintel): 2.19

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 0 3 8 -2], ⟨0 2 2 1 5]]

Optimal tunings:

  • WE: ~6/5 = 300.4282 ¢, ~7/4 = 949.6958 ¢ (~36/35 = 48.4112 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~7/4 = 948.9065 ¢ (~36/35 = 48.9065 ¢)

Optimal ET sequence: 4e, 20ce, 24

Badness (Sintel): 1.87

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨4 0 3 8 -2 -1], ⟨0 2 2 1 5 5]]

Optimal tunings:

  • WE: ~6/5 = 300.4282 ¢, ~7/4 = 949.2440 ¢ (~36/35 = 47.8487 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~7/4 = 948.3581 ¢ (~36/35 = 48.3581 ¢)

Optimal ET sequence: 4ef, 24

Badness (Sintel): 1.61

Octonion

Octonion tempers out 245/243, and may be described as the 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

Mapping: [⟨8 0 6 -3], ⟨0 1 1 2]]

mapping generators: ~15/14, ~3

Optimal tunings:

  • WE: ~15/14 = 149.6673 ¢, ~3/2 = 705.4455 ¢ (~36/35 = 42.8910 ¢)
error map: ⟨-2.662 +0.828 +14.474 -12.260]
  • CWE: ~15/14 = 150.0000 ¢, ~3/2 = 704.9636 ¢ (~36/35 = 45.0364 ¢)
error map: ⟨0.000 +3.008 +18.650 -8.899]

Optimal ET sequence: 8d, 16d, 24, 32c

Badness (Sintel): 2.72

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨8 0 6 -3 15], ⟨0 1 1 2 1]]

Optimal tunings:

  • WE: ~12/11 = 149.7102 ¢, ~3/2 = 705.2799 ¢ (~36/35 = 43.2712 ¢)
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 704.9285 ¢ (~36/35 = 45.0715 ¢)

Optimal ET sequence: 8d, 16d, 24, 32c

Badness (Sintel): 1.57

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨8 0 6 -3 15 17], ⟨0 1 1 2 1 1]]

Optimal tunings:

  • WE: ~12/11 = 149.6311 ¢, ~3/2 = 705.6367 ¢ (~36/35 = 42.5188 ¢)
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 705.2777 ¢ (~36/35 = 44.7223 ¢)

Optimal ET sequence: 8d, 16d, 24, 32cf

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

Subgroup-val mapping: [⟨4 0 3 10], ⟨0 1 1 1]]

Optimal tunings:

  • WE: ~6/5 = 299.899 ¢, ~3/2 = 698.251 ¢ (~17/16 = 98.453 ¢)
  • CWE: ~6/5 = 300.000 ¢, ~3/2 = 698.137 ¢ (~17/16 = 98.137 ¢)

Optimal ET sequence: 4, 8, 12

Badness (Sintel): 0.509

Diminished (2.3.5.19 subgroup)

Subgroup: 2.3.5.19

Comma list: 96/95, 513/500

Subgroup-val mapping: [⟨4 0 3 17], ⟨0 1 1 0]]

Optimal tunings:

  • WE: ~6/5 = 299.744 ¢, ~3/2 = 698.206 ¢ (~20/19 = 98.718 ¢)
  • CWE: ~6/5 = 300.000 ¢, ~3/2 = 698.195 ¢ (~20/19 = 98.195 ¢)

Optimal ET sequence: 4, 8, 12

Badness (Sintel): 0.681

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