Trienstonic clan

(Redirected from Wallaby)

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 and 8/7 with 32/27. Trienstonian is at 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:

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 = 1\1, ~3/2 = 717.5172
• POTE: ~2 = 1\1, ~3/2 = 721.5586

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]]

• CTE: ~2 = 1\1, ~3/2 = 727.8550
• POTE: ~2 = 1\1, ~3/2 = 742.002
eigenmonzo (unchanged-interval) basis: 2.7/5
eigenmonzo (unchanged-interval) basis: 2.9/7

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

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 = 1\1, ~3/2 = 731.3937
• POTE: ~2 = 1\1, ~3/2 = 731.177

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 = 1\1, ~3/2 = 691.7571
• POTE: ~2 = 1\1, ~3/2 = 691.351

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 = 1\2, ~3/2 = 726.5480
• POTE: ~27/20 = 1\2, ~3/2 = 729.097

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 = 1\2, ~3/2 = 723.3709
• POTE: ~7/5 = 1\2, ~3/2 = 728.874

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

Parakangaroo

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]]

• CTE: ~2 = 1\1, ~10/7 = 638.8628
• POTE: ~2 = 1\1, ~10/7 = 639.672

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

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

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)

Optimal ET sequence: 15, 30