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. | 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. | ||
=Dimipent= | == Dimipent == | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 15: | Line 15: | ||
[[Badness]]: 0.047231 | [[Badness]]: 0.047231 | ||
=Diminished= | == Diminished == | ||
{{see also|Jubilismic clan #Diminished}} | {{see also|Jubilismic clan #Diminished}} | ||
| Line 32: | Line 32: | ||
[[Badness]]: 0.022401 | [[Badness]]: 0.022401 | ||
=Hemidim= | == Hemidim == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 47: | Line 47: | ||
[[Badness]]: 0.0864 | [[Badness]]: 0.0864 | ||
==11-limit== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 60: | Line 60: | ||
Badness: 0.0566 | Badness: 0.0566 | ||
==13-limit== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 73: | Line 73: | ||
Badness: 0.039 | Badness: 0.039 | ||
=Semidim= | == Semidim == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 88: | Line 88: | ||
[[Badness]]: 0.1075 | [[Badness]]: 0.1075 | ||
==11-limit== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 101: | Line 101: | ||
Badness: 0.0476 | Badness: 0.0476 | ||
==13-limit== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 21:27, 1 June 2021
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.
Dimipent
Subgroup: 2.3.5
Comma: 648/625
POTE generator: ~3/2 = 699.507
Map: [<4 0 3|, <0 1 1|]
Optimal ET sequence: 4, 8, 12
Badness: 0.047231
Diminished
Subgroup: 2.3.5.7
Comma list: 36/35, 50/49
POTE generator: ~3/2 = 699.523
Map: [<4 0 3 5|, <0 1 1 1|]
Wedgie: <<4 4 4 -3 -5 -2||
Optimal ET sequence: 4, 8d, 12
Badness: 0.022401
Hemidim
Subgroup: 2.3.5.7
Comma list: 49/48, 648/625
POTE generator: ~8/7 = 252.555
Map: [<4 0 3 8|, <0 2 2 1|]
Wedgie: <<8 8 4 -6 -16 -13||
Optimal ET sequence: 24, 52d, 76cd
Badness: 0.0864
11-limit
Subgroup: 2.3.5.7.11
Comma list: 49/48, 77/75, 243/242
POTE generator: ~8/7 = 251.658
Map: [<4 0 3 8 -2|, <0 2 2 1 5|]
Optimal ET sequence: 24, 76cde
Badness: 0.0566
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 66/65, 77/75, 648/625
POTE generator: ~8/7 = 252.225
Map: [<4 0 3 8 -2 -1|, <0 2 2 1 5 5|]
Optimal ET sequence: 24, 52de, 76cde
Badness: 0.039
Semidim
Subgroup: 2.3.5.7
Comma list: 245/243, 392/375
POTE generator: ~3/2 = 707.014
Map: [<8 0 6 -3|, <0 1 1 2|]
Wedgie: <<8 8 16 -6 3 15||
Optimal ET sequence: 24, 32c, 56c
Badness: 0.1075
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 245/243
POTE generator: ~3/2 = 706.645
Map: [<8 0 6 -3 15|, <0 1 1 2 1|]
Optimal ET sequence: 24, 32c, 56c
Badness: 0.0476
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 66/65, 77/75, 507/500
POTE generator: ~3/2 = 707.376
Map: [<8 0 6 -3 15 17|, <0 1 1 2 1 1|]
Optimal ET sequence: 24, 32cf, 56cf
Badness: 0.0306