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 12: Line 14:


[[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 ==
<div style="float: right">[[:de:Verminderte Temperaturen|Deutsch]]</div>


== Septimal diminished ==
{{Main| Diminished (temperament) }}
{{Main| Diminished (temperament) }}


Line 33: Line 33:


[[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 50: Line 50:


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|legend=0| 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 65: Line 65:


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|legend=0| 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 80: Line 80:


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|legend=0| 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 99: Line 99:


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


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


==== 13-limit ====
==== 13-limit ====
Line 114: Line 114:


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|legend=0| 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


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


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


=== 13-limit ===
=== 13-limit ===
Line 163: Line 165:


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 }}
{{Optimal ET sequence|legend=0| 4ef, 24 }}


Badness (Smith): 0.039030
Badness (Sintel): 1.61


== 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 182: Line 186:


[[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 ===
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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|legend=0| 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 212: Line 218:


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


Badness (Smith): 0.030597
Badness (Sintel): 1.26


[[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]]

Latest revision as of 12:13, 21 August 2025

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

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]]

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 sequence4, 8, 12

Badness (Sintel): 1.11

Septimal diminished

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 sequence4, 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 sequence4, …, 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 sequence8d, 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