Diminished family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled
4,516 edits
mNo edit summary
Post-move cleanup
 
(28 intermediate revisions by 8 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]].  


== Dimipent ==
== Diminished ==
Subgroup: 2.3.5
{{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


[[Comma]]: 648/625
[[Comma list]]: 648/625


[[Mapping]]: [{{Val|4 0 3}}, {{Val|0 1 1}}]
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}


[[POTE generator]]: ~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 }}


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


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


== Diminished ==
== Septimal diminished ==
{{see also|Jubilismic clan #Diminished}}
{{Main| Diminished (temperament) }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|4 0 3 5}}, {{val|0 1 1 1}}]
{{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 }}
 
Badness (Smith): 0.879
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 50/49, 125/121
 
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
 
: 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 }}


{{Multival|legend=1|4 4 4 -3 -5 -2}}
Badness (Sintel): 1.82


[[POTE generator]]: ~3/2 = 699.523
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 4, 8d, 12 }}
Comma list: 36/35, 50/49, 66/65, 125/121


[[Badness]]: 0.022401
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 ==
Subgroup: 2.3.5.7
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


[[Mapping]]: [{{val|4 0 3 8}}, {{val|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


[[POTE generator]]: ~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 }}


{{Val list|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 51: Line 147:
Comma list: 49/48, 77/75, 243/242
Comma list: 49/48, 77/75, 243/242


Mapping: [{{val|4 0 3 8 -2}}, {{val|0 2 2 1 5}}]
Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5 }}


POTE generator: ~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}})


Vals: {{Val list| 4e, 20ce, 24, 76cdde }}
{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}


Badness: 0.056576
Badness (Sintel): 1.87


=== 13-limit ===
=== 13-limit ===
Line 64: Line 162:
Comma list: 49/48, 66/65, 77/75, 243/242
Comma list: 49/48, 66/65, 77/75, 243/242


Mapping: [{{val|4 0 3 8 -2 -1}}, {{val|0 2 2 1 5 5}}]
Mapping: {{mapping| 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}})
 
{{Optimal ET sequence|legend=0| 4ef, 24 }}


POTE generator: ~7/6 = 252.225
Badness (Sintel): 1.61


Vals: {{Val list| 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


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


[[Mapping]]: [{{val|8 0 6 -3}}, {{val|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


[[POTE generator]]: ~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 }}


{{Vals|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 92: Line 200:
Comma list: 56/55, 77/75, 245/243
Comma list: 56/55, 77/75, 245/243


Mapping: [{{val|8 0 6 -3 15}}, {{val|0 1 1 2 1}}]
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}


POTE generator: ~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}})


Vals: {{Val list| 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 105: Line 215:
Comma list: 56/55, 66/65, 77/75, 507/500
Comma list: 56/55, 66/65, 77/75, 507/500


Mapping: [{{val|8 0 6 -3 15 17}}, {{val|0 1 1 2 1 1}}]
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}


POTE generator: ~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}})


Vals: {{Val list| 8d, 24, 32cf, 56cf }}
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}


Badness: 0.030597
Badness (Sintel): 1.26


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

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]]

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)

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

Retrieved from "https://en.xen.wiki/index.php?title=Diminished_family&oldid=207883"