Dimipent family

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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.

Dimipent

Subgroup: 2.3.5

Comma list: 648/625

Mapping[4 0 3], 0 1 1]]

Optimal tunings:

  • POTE: ~6/5 = 1\4, ~3/2 = 699.507
  • DKW: ~6/5 = 1\4, ~3/2 = 690.289

Optimal ET sequence4, 8, 12

Badness: 0.047231

Diminished

Deutsch

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]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 699.523

Optimal ET sequence4, 8d, 12

Badness: 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]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109

Optimal ET sequence4, 8d, 12, 32cddee, 44cddeee

Badness: 0.022132

Scales: diminished12

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 tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773

Optimal ET sequence4, 8d, 12f, 20cdef

Badness: 0.019509

Scales: diminished12

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 tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881

Optimal ET sequence12, 28, 40de

Badness: 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 tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679

Optimal ET sequence8bce, 12

Badness: 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 tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299

Optimal ET sequence8bcef, 12f

Badness: 0.041707

Hemidim

Subgroup: 2.3.5.7

Comma list: 49/48, 648/625

Mapping[4 0 3 8], 0 2 2 1]]

Wedgie⟨⟨8 8 4 -6 -16 -13]]

Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.555

Optimal ET sequence4, 20c, 24, 52d, 76cdd

Badness: 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 tuning (POTE): ~6/5 = 1\4, ~7/6 = 251.658

Optimal ET sequence4e, 20ce, 24, 76cdde

Badness: 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 tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.225

Optimal ET sequence4ef, 20cef, 24, 52de, 76cdde

Badness: 0.039030

Semidim

Subgroup: 2.3.5.7

Comma list: 245/243, 392/375

Mapping[8 0 6 -3], 0 1 1 2]]

Wedgie⟨⟨8 8 16 -6 3 15]]

Optimal tuning (POTE): ~15/14 = 1\8, ~3/2 = 707.014

Optimal ET sequence8d, 24, 32c, 56c

Badness: 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 tuning (POTE): ~12/11 = 1\8, ~3/2 = 706.645

Optimal ET sequence8d, 24, 32c, 56c

Badness: 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 tuning (POTE): ~12/11 = 1\8, ~3/2 = 707.376

Optimal ET sequence8d, 24, 32cf, 56cf

Badness: 0.030597