Diminished family: Difference between revisions
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The ''' | {{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. | |||
[[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]]s: | [[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]]: | [[Badness]] (Sintel): 1.11 | ||
== | == Septimal diminished == | ||
{{Main| Diminished (temperament) }} | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Line 25: | Line 32: | ||
{{Mapping|legend=1| 4 0 3 5 | 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 }} | {{Optimal ET sequence|legend=1| 4, 8d, 12 }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.567 | ||
=== 11-limit === | === 11-limit === | ||
Line 40: | 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 | 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 ==== | ==== 13-limit ==== | ||
Line 55: | 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 | 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= | {{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }} | ||
Badness: 0. | Badness (Sintel): 0.806 | ||
=== Demolished === | === Demolished === | ||
Line 70: | 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 | 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= | {{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }} | ||
Badness: 0. | Badness (Smith): 0.879 | ||
=== Cohedim === | === Cohedim === | ||
This | 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 85: | 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 | |||
Optimal | 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= | {{Optimal ET sequence|legend=0| 8bce, 12 }} | ||
Badness: | Badness (Sintel): 1.82 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Line 100: | 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 | 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= | {{Optimal ET sequence|legend=0| 8bcef, 12f }} | ||
Badness: | 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 113: | Line 130: | ||
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }} | {{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }} | ||
: mapping generators: ~6/5, ~7/4 | |||
[[Optimal tuning]] ([[ | [[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 | {{Optimal ET sequence|legend=1| 4, …, 20c, 24 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.19 | ||
=== 11-limit === | === 11-limit === | ||
Line 126: | 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 | 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= | {{Optimal ET sequence|legend=0| 4e, 20ce, 24 }} | ||
Badness: | Badness (Sintel): 1.87 | ||
=== 13-limit === | === 13-limit === | ||
Line 141: | 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 | 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 | |||
{{ | == Octonion == | ||
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 154: | Line 183: | ||
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }} | {{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }} | ||
: mapping generators: ~15/14, ~3 | |||
[[Optimal tuning]] ([[ | [[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 | {{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.72 | ||
=== 11-limit === | === 11-limit === | ||
Line 169: | 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 | 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= | {{Optimal ET sequence|legend=0| 8d, 16d, 24, 32c }} | ||
Badness: | Badness (Sintel): 1.57 | ||
=== 13-limit === | === 13-limit === | ||
Line 182: | 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 | 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= | {{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }} | ||
Badness: | Badness (Sintel): 1.26 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[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]] |
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]]
- 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
Subgroup: 2.3.5.7
Comma list: 36/35, 50/49
Mapping: [⟨4 0 3 5], ⟨0 1 1 1]]
- 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 ¢)
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
- 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 ¢)
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
- 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