Trienstonic clan

The trienstonic clan of temperaments tempers out 28/27, the septimal third-tone or trienstonic comma.

Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharp, blacksmith, and sharptone.

Trienstonic

Period: 1\1

Optimal (POTE) generator: ~3/2 = 721.5586

EDO generators: 3\5, 5\8, 8\13, 11\18, 14\23

Scales (Scala files):

Technical data

Subgroup: 2.3..7

Comma list: 28/27

Mapping: [<1 0 0 -2|, <0 1 0 3|]

Father

See also: Father family § Father

Period: 1\1

Optimal (POTE) generator: ~3/2 = 742.002

EDO generators: 3\5, 5\8, 8\13

Scales (Scala files):

Technical data

Comma list: 16/15, 28/27

Mapping: [⟨1 0 4 -2], ⟨0 1 -1 3]]

Wedgie: ⟨⟨1 -1 3 -4 2 10]]

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [3/2 0 -1/4 1/4⟩, [5/2 0 1/4 -1/4⟩, [5/2 0 -3/4 3/4⟩]

Eigenmonzo subgroup: 2.7/5

• 9-odd-limit

Eigenmonzo subgroup: 2.9/5

Badness: 0.0213

Uncle

Period: 1\1

Optimal (POTE) generator: ~3/2 = 731.1774

EDO generators: 11\18, 14\23

Scales (Scala files):

Technical data

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Wedgie: ⟨⟨1 -6 3 -12 2 24]]

Minimax tuning:

• 7-odd-limit eigenmonzo subgroup: 2.5/3
• 9-odd-limit eigenmonzo subgroup: 2.9/5

Badness: 0.0727

Octokaidecal

Period: 1\2

Optimal (POTE) generator: ~3/2 = 728.874

EDO generators: 5\8, 6\10, 11\18

Scales (Scala files):

Technical data

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Wedgie: ⟨⟨2 6 6 5 4 -2]]

Minimax tuning:

• 7-odd-limit eigenmonzo subgroup: 2.5
• 9-odd-limit eigenmonzo subgroup: 2.5

Badness: 0.0367

11-limit

Period: 1\2

Optimal (POTE) generator: ~3/2 = 732.330

EDO generators: 5\8, 6\10, 11\18

Scales (Scala files):

Technical data

Comma list: 28/27, 50/49, 55/54

Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]

Badness: 0.0302

Opossum

Period: 1\1

Optimal (POTE) generator: ~10/9 = 159.691

EDO generators: 1\7, 1\8, 2\15

Scales (Scala files):

Technical data

Comma list: 28/27, 126/125

Mapping: [⟨1 2 3 4], ⟨0 -3 -5 -9]]

Wedgie: ⟨⟨3 5 9 1 6 7]]

Minimax tuning:

• 7-odd-limit eigenmonzo subgroup: 2.7
• 9-odd-limit eigenmonzo subgroup: 2.7

Badness: 0.0407

11-limit

Period: 1\1

Optimal (POTE) generator: ~10/9 = 159.807

EDO generators: 1\7, 1\8, 2\15

Scales (Scala files):

Technical data

Comma list: 28/27, 55/54, 77/75

Mapping: [⟨1 2 3 4 4], ⟨0 -3 -5 -9 -4]]

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7

Badness: 0.0223

13-limit

Period: 1\1

Optimal (POTE) generator: ~10/9 = 159.805

EDO generators: 1\7, 1\8, 2\15

Scales (Scala files):

Technical data

Comma list: 28/27, 40/39, 55/54, 66/65

Mapping: [⟨1 2 3 4 4 4], ⟨0 -3 -5 -9 -4 -2]]

Minimax tuning:

• 13-odd-limit eigenmonzo subgroup: 2.7
• 15-odd-limit eigenmonzo subgroup: 2.7

Badness: 0.0194

Wallaby

Period: 1\1

Optimal (POTE) generator: ~3/2 = 691.351

EDO generators: 3\5

Scales (Scala files):

Technical data

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Wedgie: ⟨⟨1 -3 3 -7 2 15]]

Badness: 0.0585

Kangaroo

5-limit

Period: 1\1

Optimal (POTE) generator: ~27/20 = 561.195

EDO generators: 9\15

Scales (Scala files):

Technical data

Comma list: 64000/59049

Mapping: [⟨1 0 -3], ⟨0 3 10]]

Badness: 0.2958

7-limit

Period: 1\1

Optimal (POTE) generator: ~7/5 = 560.328

EDO generators: 7\15

Scales (Scala files):

Technical data

Comma list: 28/27, 1029/1000

Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]

Wedgie: ⟨⟨3 10 9 9 6 -7]]

Badness: 0.0779

11-limit

Period: 1\1

Optimal (POTE) generator: ~7/5 = 560.155

EDO generators: 7\15

Scales (Scala files):

Technical data

Comma list: 28/27, 77/75, 245/242

Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]

Badness: 0.0432

13-limit

Period: 1\1

Optimal (POTE) generator: ~7/5 = 559.770

EDO generators: 7\15

Scales (Scala files):

Technical data

Comma list: 28/27, 40/39, 77/75, 147/143

Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]

Badness: 0.0327

Quindecic

Period: 1\15

Optimal (POTE) generator: ~13/8 = 852.924

EDO generators: 21\30, 32\45

Scales (Scala files):

Technical data

Comma list: 28/27, 49/48, 55/54, 77/75

Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]

Badness: 0.0289