Trienstonic clan
The trienstonic clan of rank-2 temperaments tempers out 28/27, the septimal third-tone or trienstonic comma.
Adding 16/15 to 28/27 leads to father, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 35/32 gives wallaby. Other members of the clan discussed elsewhere are:
- Sharptone (+21/20) → Meantone family
- Sharp (+25/24) → Dicot family
- Inflated (+128/125) → Augmented family
- Opossum (+126/125) → Porcupine family
- Blacksmith (+49/48) → Limmic temperaments
Trienstonian
Subgroup: 2.3.7
Comma list: 28/27
Sval mapping: [⟨1 0 -2], ⟨0 1 3]]
- mapping generators: ~2, ~3
Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]
Optimal ET sequence: 2d, 3d, 5
Father
- Main article: Father
- See also: Father family #Father
Subgroup: 2.3.5.7
Comma list: 16/15, 28/27
Mapping: [⟨1 0 4 -2], ⟨0 1 -1 3]]
Wedgie: ⟨⟨1 -1 3 -4 2 10]]
- 7-odd-limit: ~3/2 = [1/2 0 -1/4 1/4⟩
- 9-odd-limit: ~3/2 = [1 -2 0 1⟩ = 14/9
Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd
Badness: 0.021312
11-limit
Subgroup: 2.3.5.7.11
Comma list: 16/15, 22/21, 28/27
Mapping: [⟨1 0 4 -2 -3], ⟨0 1 -1 3 4]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 732.2094
- POTE: ~2 = 1\1, ~3/2 = 747.156
Optimal ET sequence: 2de, 3de, 5, 8d
Badness: 0.020589
Uncle
Subgroup: 2.3.5.7
Comma list: 28/27, 256/245
Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]
Wedgie: ⟨⟨1 -6 3 -12 2 24]]
- 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5
Optimal ET sequence: 5, 13d, 18, 23bc, 41bbcd
Badness: 0.072653
Wallaby
- See also: Pelogic family
Subgroup: 2.3.5.7
Comma list: 28/27, 35/32
Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]
Wedgie: ⟨⟨1 -3 3 -7 2 15]]
Optimal ET sequence: 2d, 5c, 7d, 19ccdd
Badness: 0.058468
Octokaidecal
5-limit (supersharp)
Subgroup: 2.3.5
Comma list: 800/729
Mapping: [⟨2 0 -5], ⟨0 1 3]]
- mapping generators: ~27/20, ~3
Optimal ET sequence: 8, 10, 18, 28b
Badness: 0.122848
7-limit
Subgroup: 2.3.5.7
Comma list: 28/27, 50/49
Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]
Wedgie: ⟨⟨2 6 6 5 4 -2]]
Optimal ET sequence: 8d, 10, 18, 28b
Badness: 0.036747
11-limit
Subgroup: 2.3.5.7.11
Comma list: 28/27, 50/49, 55/54
Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]
Optimal tunings:
- CTE: ~7/5 = 1\2, ~3/2 = 723.3709
- POTE: ~7/5 = 1\2, ~3/2 = 732.330
Optimal ET sequence: 8d, 10, 18e
Badness: 0.030235
Kangaroo
- For the 5-limit version of this temperament, see High badness temperaments #Kangaroo.
Subgroup: 2.3.5.7
Comma list: 28/27, 1029/1000
Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]
- mapping generators: ~2, ~10/7
Wedgie: ⟨⟨3 10 9 9 6 -7]]
Optimal ET sequence: 2cd, …, 13cd, 15
Badness: 0.077857
11-limit
Subgroup: 2.3.5.7.11
Comma list: 28/27, 77/75, 245/242
Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/7 = 639.0363
- POTE: ~2 = 1\1, ~10/7 = 639.845
Optimal ET sequence: 15
Badness: 0.043195
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 66/65, 147/143
Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/7 = 638.7168
- POTE: ~2 = 1\1, ~10/7 = 640.230
Optimal ET sequence: 15
Badness: 0.032653
Quindecic
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 49/48, 55/54, 77/75
Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]
- mapping generators: ~22/21, ~13
Optimal tunings:
- CTE: ~22/21 = 1\15, ~13/8 = 840.5277 (~40/39 = 39.4723)
- POTE: ~22/21 = 1\15, ~13/8 = 852.924 (~40/39 = 27.076)
Badness: 0.028944