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.

Trienstonian

This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. 5edo is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the 7-odd-limit, leading to an oneirotonic scale or otherwise a diatonic scale with negative small steps.

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:

• WE: ~2 = 1196.254 ¢, ~3/2 = 719.306 ¢

error map: ⟨-3.746 +13.604 -14.655]

• CWE: ~2 = 1200.000 ¢, ~3/2 = 719.606 ¢

error map: ⟨0.000 +17.651 -10.007]

Optimal ET sequence: 2d, 3d, 5

Badness (Sintel): 0.235

Overview to extensions

Adding 16/15 to 28/27 leads to father, 21/20 gives sharptone, 256/245 gives uncle, and 35/32 gives wallaby. These all use the same generators as trienstonian.

50/49 gives octokaidecal with a semi-octave period. 25/24 gives sharpie; 27/25 gives mite. Those split the generator in two. 1029/1000 gives parakangaroo; 126/125 gives opossum. Those split the generator in three. 128/125 gives inflated with a 1/3-octave period. Finally, 49/48 gives blackwood, with a 1/5-octave period.

Members of the clan discussed elsewhere are:

• 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
• Blackwood (+49/48) → Limmic temperaments

Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.

Father

Main article: Father

See Father family #Septimal father.

Sharptone

See Meantone family #Sharptone.

Uncle

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

Uncle tempers out 256/245, mapping the interval class of 5 to -6 generator steps, as a major 2-step in oneirotonic or a diminished fifth in diatonic.

Subgroup: 2.3.5.7

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Optimal tunings:

• WE: ~2 = 1190.224 ¢, ~3/2 = 725.221 ¢

error map: ⟨-9.776 +13.490 +3.707 -2.939]

• CWE: ~2 = 1200.000 ¢, ~3/2 = 731.394 ¢

error map: ⟨0.000 +29.439 +25.324 +25.355]

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 (Sintel): 1.84

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:

• WE: ~27/20 = 596.986 ¢, ~3/2 = 725.434 ¢ (~10/9 = 128.448 ¢)

error map: ⟨-6.029 +17.450 -13.027]

• CWE: ~27/20 = 600.000 ¢, ~3/2 = 726.548 ¢ (~10/9 = 126.548 ¢)

error map: ⟨0.000 +24.593 -6.670]

Optimal ET sequence: 8, 10, 18, 28b

Badness (Sintel): 2.88

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Optimal tunings:

• WE: ~7/5 = 596.984 ¢, ~3/2 = 725.210 ¢ (~15/14 = 128.226 ¢)

error map: ⟨-6.031 +17.224 -13.699 +0.774]

• CWE: ~7/5 = 600.000 ¢, ~3/2 = 726.319 ¢ (~15/14 = 126.319 ¢)

error map: ⟨0.000 +24.364 -7.358 +10.130]

Minimax tuning:

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

Optimal ET sequence: 8d, 10, 18, 28b

Badness (Sintel): 0.930

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:

• WE: ~7/5 = 595.139 ¢, ~3/2 = 726.397 ¢ (~15/14 = 131.258 ¢)
• CWE: ~7/5 = 600.000 ¢, ~3/2 = 729.485 ¢ (~15/14 = 129.485 ¢)

Optimal ET sequence: 8d, 10, 18e

Badness (Sintel): 1.00

Parakangaroo

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Kangaroo.

This temperament used to be known as kangaroo, but was decanonicalized in 2024 in favor of a more accurate extension. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 15edo shows us an obvious tuning.

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:

• WE: ~2 = 596.984 ¢, ~10/7 = 638.135 ¢

error map: ⟨-2.883 +12.450 +3.685 -19.845]

• CWE: ~2 = 1200.000 ¢, ~10/7 = 639.302 ¢

error map: ⟨0.000 +15.952 +6.710 -15.104]

Optimal ET sequence: 2cd, …, 13cd, 15

Badness (Sintel): 1.97

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:

• WE: ~2 = 1196.971 ¢, ~10/7 = 638.230 ¢
• CWE: ~2 = 1200.000 ¢, ~10/7 = 639.480 ¢

Optimal ET sequence: 15

Badness (Sintel): 1.43

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:

• WE: ~2 = 1194.720 ¢, ~10/7 = 637.413 ¢
• CWE: ~2 = 1200.000 ¢, ~10/7 = 639.609 ¢

Optimal ET sequence: 15

Badness (Sintel): 1.35