Dimipent family
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 list: 648/625
Mapping: [⟨4 0 3], ⟨0 1 1]]
- CTE: ~6/5 = 300.000, ~3/2 = 696.983 (~25/24 = 96.983)
- error map: ⟨0.000 -4.972 +10.670]
- POTE: ~6/5 = 300.000, ~3/2 = 699.507 (~25/24 = 99.507)
- error map: ⟨0.000 -2.448 +13.193]
Optimal ET sequence: 4, 8, 12
Badness (Smith): 0.047231
Diminished
Subgroup: 2.3.5.7
Comma list: 36/35, 50/49
Mapping: [⟨4 0 3 5], ⟨0 1 1 1]]
Wedgie: ⟨⟨ 4 4 4 -3 -5 -2 ]]
- CTE: ~6/5 = 300.000, ~3/2 = 691.955 (~21/20 = 91.955)
- error map: ⟨0.000 -10.000 +5.641 +23.129]
- POTE: ~6/5 = 300.000, ~3/2 = 699.523 (~21/20 = 99.523)
- error map: ⟨0.000 -2.432 +13.210 +30.698]
Optimal ET sequence: 4, 8d, 12
Badness (Smith): 0.022401
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]]
Wedgie: ⟨⟨ 4 4 4 0 -3 -5 -14 -2 -14 -14 ]]
Optimal tunings:
- CTE: ~6/5 = 300.000, ~3/2 = 691.955 (~21/20 = 91.955)
- POTE: ~6/5 = 300.000, ~3/2 = 709.109 (~15/14 = 109.109)
Optimal ET sequence: 4, 8d, 12, 32cddee, 44cddeee
Badness (Smith): 0.022132
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:
- CTE: ~6/5 = 300.000, ~3/2 = 691.954 (~21/20 = 91.954)
- POTE: ~6/5 = 300.000, ~3/2 = 713.773 (~15/14 = 113.773)
Optimal ET sequence: 4, 8d, 12f, 20cdef
Badness (Smith): 0.019509
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:
- CTE: ~6/5 = 300.000, ~3/2 = 687.747 (~21/20 = 87.747)
- POTE: ~6/5 = 300.000, ~3/2 = 689.881 (~21/20 = 89.881)
Optimal ET sequence: 12, 28, 40de
Badness (Smith): 0.026574
Cohedim
This temperament 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 11-limit 8bce & 12 temperament, cohedim arguably makes more sense.
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:
- CTE: ~6/5 = 300.000, ~11/7 = 793.423 (~12/11 = 106.577)
- POTE: ~6/5 = 300.000, ~11/7 = 798.421 (~12/11 = 101.679)
Badness (Smith): 0.054965
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:
- CTE: ~6/5 = 300.000, ~11/7 = 791.375 (~12/11 = 108.625)
- POTE: ~6/5 = 300.000, ~11/7 = 797.701 (~12/11 = 102.299)
Optimal ET sequence: 8bcef, 12f
Badness (Smith): 0.041707
Hemidim
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
Wedgie: ⟨⟨ 8 8 4 -6 -16 -13 ]]
- CTE: ~6/5 = 300.000, ~7/4 = 949.540 (~36/35 = 49.540)
- error map: ⟨0.000 -2.875 +12.766 -19.286]
- POTE: ~6/5 = 300.000, ~7/4 = 947.445 (~36/35 = 47.445)
- error map: ⟨0.000 -7.065 +8.576 -21.381]
Optimal ET sequence: 4, …, 20c, 24
Badness (Smith): 0.086378
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:
- CTE: ~6/5 = 300.000, ~7/4 = 949.872 (~36/35 = 49.872)
- POTE: ~6/5 = 300.000, ~7/4 = 948.342 (~36/35 = 48.342)
Optimal ET sequence: 4e, 20ce, 24
Badness (Smith): 0.056576
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:
- CTE: ~6/5 = 300.000, ~7/4 = 949.366 (~36/35 = 49.366)
- POTE: ~6/5 = 300.000, ~7/4 = 947.775 (~36/35 = 47.775)
Badness (Smith): 0.039030
Semidim
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
Wedgie: ⟨⟨ 8 8 16 -6 3 15 ]]
- CTE: ~15/14 = 150.000, ~3/2 = 702.765 (~36/35 = 47.235)
- POTE: ~15/14 = 150.000, ~3/2 = 707.014 (~36/35 = 42.986)
Optimal ET sequence: 8d, 24, 32c
Badness (Smith): 0.107523
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:
- CTE: ~12/11 = 150.000, ~3/2 = 702.662 (~36/35 = 47.338)
- POTE: ~12/11 = 150.000, ~3/2 = 706.645 (~36/35 = 43.355)
Optimal ET sequence: 8d, 24, 32c
Badness (Smith): 0.047598
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:
- CTE: ~12/11 = 150.000, ~3/2 = 701.952 (~36/35 = 48.048)
- POTE: ~12/11 = 150.000, ~3/2 = 707.376 (~36/35 = 42.624)
Optimal ET sequence: 8d, 24, 32cf
Badness (Smith): 0.030597