Diminished family

From Xenharmonic Wiki
Revision as of 13:26, 2 August 2024 by FloraC (talk | contribs) (+CTE tunings. Review generators and optimal ET sequences. Move DKW tuning to the main article.)
Jump to navigation Jump to search

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

Main article: Diminished

Subgroup: 2.3.5

Comma list: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

Optimal tunings:

  • CTE: ~6/5 = 1\4, ~3/2 = 696.9833 (~25/24 = 96.9833)
  • POTE: ~6/5 = 1\4, ~3/2 = 699.507 (~25/24 = 99.507)

Optimal ET sequence: 4, 8, 12

Badness: 0.047231

Diminished

Deutsch
Main article: 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 ]]

Optimal tunings:

  • CTE: ~6/5 = 1\4, ~3/2 = 691.9545 (~21/20 = 91.9545)
  • POTE: ~6/5 = 1\4, ~3/2 = 699.523 (~21/20 = 99.523)

Optimal ET sequence: 4, 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 tunings:

  • CTE: ~6/5 = 1\4, ~3/2 = 691.9546 (~21/20 = 91.9546)
  • POTE: ~6/5 = 1\4, ~3/2 = 709.109 (~15/14 = 109.109)

Optimal ET sequence: 4, 8d, 12, 32cddee, 44cddeee

Badness: 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 = 1\4, ~3/2 = 691.9544 (~21/20 = 91.9544)
  • POTE: ~6/5 = 1\4, ~3/2 = 713.773 (~15/14 = 113.773)

Optimal ET sequence: 4, 8d, 12f, 20cdef

Badness: 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 = 1\4, ~3/2 = 687.7473 (~21/20 = 87.7473)
  • POTE: ~6/5 = 1\4, ~3/2 = 689.881 (~21/20 = 89.881)

Optimal ET sequence: 12, 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 tunings:

  • CTE: ~6/5 = 1\4, ~11/7 = 793.4233 (~12/11 = 106.5767)
  • POTE: ~6/5 = 1\4, ~11/7 = 798.4212 (~12/11 = 101.679)

Optimal ET sequence: 8bce, 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 tunings:

  • CTE: ~6/5 = 1\4, ~11/7 = 791.3751 (~12/11 = 108.6249)
  • POTE: ~6/5 = 1\4, ~11/7 = 797.701 (~12/11 = 102.299)

Optimal ET sequence: 8bcef, 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]]

mapping generators: ~6/5, ~7/4

Wedgie: ⟨⟨ 8 8 4 -6 -16 -13 ]]

Optimal tunings:

  • CTE: ~6/5 = 1\4, ~7/4 = 948.2575 (~36/35 = 48.2575)
  • POTE: ~6/5 = 1\4, ~7/4 = 947.445 (~36/35 = 47.445)

Optimal ET sequence: 4, …, 20c, 24

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 tunings:

  • CTE: ~6/5 = 1\4, ~7/4 = 949.8722 (~36/35 = 49.8722)
  • POTE: ~6/5 = 1\4, ~7/4 = 948.342 (~36/35 = 48.342)

Optimal ET sequence: 4e, 20ce, 24

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 tunings:

  • CTE: ~6/5 = 1\4, ~7/4 = 949.3664 (~36/35 = 49.3664)
  • POTE: ~6/5 = 1\4, ~7/4 = 947.775 (~36/35 = 47.775)

Optimal ET sequence: 4ef, 24

Badness: 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 ]]

Optimal tunings:

  • CTE: ~15/14 = 1\8, ~3/2 = 702.7651 (~36/35 = 47.2349)
  • POTE: ~15/14 = 1\8, ~3/2 = 707.014 (~36/35 = 42.986)

Optimal ET sequence: 8d, 24, 32c

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 tunings:

  • CTE: ~12/11 = 1\8, ~3/2 = 702.6622 (~36/35 = 47.3378)
  • POTE: ~12/11 = 1\8, ~3/2 = 706.645 (~36/35 = 43.355)

Optimal ET sequence: 8d, 24, 32c

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 tunings:

  • CTE: ~12/11 = 1\8, ~3/2 = 701.9520 (~36/35 = 48.0480)
  • POTE: ~12/11 = 1\8, ~3/2 = 707.376 (~36/35 = 42.624)

Optimal ET sequence: 8d, 24, 32cf

Badness: 0.030597

Retrieved from "https://en.xen.wiki/index.php?title=Diminished_family&oldid=149413"