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]].
 
== 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]]s:
: mapping generators: ~6/5, ~3
* [[POTE]]: ~6/5 = 1\4, ~3/2 = 699.507
 
* [[DKW theory|DKW]]: ~6/5 = 1\4, ~3/2 = 690.29
[[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
 
== Septimal diminished ==
{{Main| Diminished (temperament) }}


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 25: Line 36:
{{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 40: Line 53:
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}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})


{{Optimal ET sequence|legend=1| 4, 8d, 12, 32cddee, 44cddeee }}
{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}


Badness: 0.022132
Badness (Sintel): 0.732
 
Scales: [[diminished12]]


==== 13-limit ====
==== 13-limit ====
Line 55: Line 68:
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 70: Line 83:
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 85: Line 100:
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 100: Line 117:
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 113: Line 134:
{{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 126: Line 151:
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 141: Line 168:
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 }}


{{Optimal ET sequence|legend=1| 4ef, 20cef, 24, 52de, 76cdde }}
Badness (Sintel): 1.61


Badness: 0.039030
== 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.  


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


Line 154: Line 187:
{{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 169: Line 206:
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 182: Line 221:
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


== 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|legend=1| 4, 8, 12 }}
[[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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