Trienstonic clan
The trienstonic clan of rank-2 temperaments tempers out 28/27, the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, 9/8 with 7/6, 8/7 with 32/27, and 4/3 with 9/7. Trienstonian is close to the edge of what can sensibly be called a temperament at all; in other words, it is an exotemperament.
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]]
- CTE: ~2 = 1200.000, ~3/2 = 717.517
- error map: ⟨0.000 +15.562 -16.274]
- POTE: ~2 = 1200.000, ~3/2 = 721.559
- error map: ⟨0.000 +19.604 -4.150]
Optimal ET sequence: 2d, 3d, 5
Badness (Smith): 0.00685
Father
See Father family #Septimal father.
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 ]]
- CTE: ~2 = 1200.000, ~3/2 = 731.394
- error map: ⟨0.000 +30.268 +20.350 +27.842]
- POTE: ~2 = 1200.000, ~3/2 = 731.177
- error map: ⟨0.000 +29.222 +26.622 +24.706]
- 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 (Smith): 0.072653
Wallaby
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 ]]
- CTE: ~2 = 1200.000, ~3/2 = 691.757
- error map: ⟨0.000 -8.515 -66.635 -88.505]
- POTE: ~2 = 1200.000, ~3/2 = 691.351
- error map: ⟨0.000 -10.604 -60.368 -94.772]
Optimal ET sequence: 2d, 5c, 7d, 19ccdd
Badness (Smith): 0.058468
Octokaidecal
The 5-limit restriction of octokaidecal is supersharp, which tempers out 800/729, the difference between the 27/20 wolf fourth and the 40/27 wolf fifth, splitting the octave into two 27/20~40/27 semioctaves. It generally requires a very sharp fifth, even sharper than 3\5, as a generator. This means that five steps from the Zarlino generator sequence starting with 6/5 are tempered to one and a half octaves. The only reasonable 7-limit extension adds 28/27 and 50/49 to the comma list, taking advantage of the existing semioctave.
5-limit (supersharp)
Subgroup: 2.3.5
Comma list: 800/729
Mapping: [⟨2 0 -5], ⟨0 1 3]]
- mapping generators: ~27/20, ~3
- CTE: ~27/20 = 600.000, ~3/2 = 723.608 (~10/9 = 123.608)
- error map: ⟨0.000 +21.653 -15.490]
- POTE: ~27/20 = 600.000, ~3/2 = 729.097 (~10/9 = 129.097)
- error map: ⟨0.000 +27.142 +0.976]
Optimal ET sequence: 8, 10, 18, 28b
Badness (Smith): 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 ]]
- CTE: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
- error map: ⟨0.000 +21.416 -16.201 +1.287]
- POTE: ~7/5 = 600.000, ~3/2 = 728.874 (~15/14 = 128.874)
- error map: ⟨0.000 +26.919 +0.307 +17.795]
Optimal ET sequence: 8d, 10, 18, 28b
Badness (Smith): 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 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
- POTE: ~7/5 = 600.000, ~3/2 = 732.330 (~15/14 = 132.330)
Optimal ET sequence: 8d, 10, 18e
Badness (Smith): 0.030235
Parakangaroo
- For the 5-limit version of this temperament, see High badness temperaments #Kangaroo.
This temperament used to be known as 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 ]]
- CTE: ~2 = 1200.000, ~10/7 = 638.863
- error map: ⟨0.000 +14.633 +2.314 -19.061]
- POTE: ~2 = 1200.000, ~10/7 = 639.672
- error map: ⟨0.000 +17.060 +10.404 -11.780]
Optimal ET sequence: 2cd, …, 13cd, 15
Badness (Smith): 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 = 1200.000, ~10/7 = 639.036
- POTE: ~2 = 1200.000, ~10/7 = 639.845
Optimal ET sequence: 15
Badness (Smith): 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 = 1200.000, ~10/7 = 638.717
- POTE: ~2 = 1200.000, ~10/7 = 640.230
Optimal ET sequence: 15
Badness (Smith): 0.032653