Trienstonic clan

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The trienstonic clan of rank-2 temperaments are low-complexity, high-error temperaments that temper 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 be sensibly called a temperament at all; in other words, it is an exotemperament.

Adding 16/15 to 28/27 leads to father, 256/245 gives uncle, 50/49 gives octokaidecal, and 35/32 gives wallaby. Other members of the clan discussed elsewhere are:

• Sharptone (+21/20) → Meantone family
• Wallaby (+35/32) → Very low accuracy temperaments
• Sharpie (+25/24) → Dicot family
• Mite (+27/25) → Bug 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 tunings:

• 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

Main article: Father

See Father family #Septimal father.

Uncle

For the 5-limit version, see Syntonic–diatonic equivalence continuum.

Subgroup: 2.3.5.7

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Optimal tunings:

• 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]

Minimax tuning:

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

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

Badness (Smith): 0.072653

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

Optimal tunings:

• 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]]

Optimal tunings:

• 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]

Minimax tuning:

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

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

Optimal tunings:

• 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