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
Blackwood (+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:

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

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:

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.