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, 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]]

Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 721.5586

Optimal ET sequence: 2d, 3d, 5, 78bb

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 742.002

Minimax tuning:

• 7-odd-limit: ~3/2 = [1/2 0 -1/4 1/4⟩

eigenmonzo (unchanged-interval) basis: 2.7/5

• 9-odd-limit: ~3/2 = [1 -2 0 1⟩ = 14/9

eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd

Badness: 0.021312

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 731.177

Minimax tuning:

• 7-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.5/3
• 9-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5

Optimal ET sequence: 5, 13d, 18, 23bc, 41bcd

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 tuning (POTE): ~2 = 1\1, ~3/2 = 691.351

Optimal ET sequence: 2d, 5c, 19ccdd

Badness: 0.058468

Octokaidecal

5-limit (supersharp)

Subgroup: 2.3.5

Comma list: 800/729

Mapping: [⟨2 0 -5], ⟨0 1 3]]

Optimal tuning (POTE): ~27/20 = 1\2, ~3/2 = 729.097

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 728.874

Minimax tuning:

• 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.5
• 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.5

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 tuning (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]]

Wedgie: ⟨⟨ 3 10 9 9 6 -7 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 560.328

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 tuning (POTE): ~2 = 1\1, ~7/5 = 560.155

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 tuning (POTE): ~2 = 1\1, ~7/5 = 559.770

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]]

Optimal tuning (POTE): ~22/21 = 1\15, ~13/8 = 852.924

Optimal ET sequence: 15, 30, 45bc

Badness: 0.028944