Keenanismic temperaments

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This is a collection of rank-3 temperaments tempering out the keenanisma, 385/384. For the rank-4 keenanismic temperament, see Keenanismic family.

Temperaments discussed elsewhere include:

Considered below are uniwiz and tritiwiz.

Uniwiz

Uniwiz tempers out 1500625/1492992 = [-11 -6 4 4 in the 7-limit, and the keenanisma, 385/384, the kalisma, 9801/9800, as well as the Alpharabian comma, 131769/131072 = [-17 2 0 0 4, in the 11-limit. In the 17-limit it tempers two no-13's 17-limit PIFE commas, 561/560 and 595/594, along with their product, 289/288, thus also equating the half-octave with 17/12~24/17.

It features the split of the whole tone into four equal parts, each representing 36/35~35/34~34/33~33/32. Therefore, it is most interesting to those who would like to make extensive use of quartertones. It can be viewed as the temperament described by the three quartertone-size equal temperaments: 22, 24 and 26. In the 11-limit it shares the optimal patent val, 284, with keenanismic.

7-limit

Subgroup: 2.3.5.7

Comma list: 1500625/1492992

Mapping: [2 1 0 7], 0 2 0 3], 0 0 1 -1]]

Mapping generators: ~1225/864, ~35/24, ~5

Wedgie⟨⟨⟨4 -4 -6 11]]]

Optimal tuning (CTE): ~35/24 = 651.1974, ~5/4 = 385.6847

Optimal GPV sequence22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd

Badness: 0.7042 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800

Mapping: [2 1 0 7 8], 0 2 0 3 -1], 0 0 1 -1 0]]

Wedgie⟨⟨⟨4 -4 0 -6 2 -2 11 17 -17 -31]]]

Optimal tuning (CTE): ~16/11 = 651.1082, ~5/4 = 385.5778

Optimal GPV sequence22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de

Badness: 0.529 × 10-3

2.3.5.7.11.17 subgroup

Subgroup: 2.3.5.7.11.17

Comma list: 289/288, 385/384, 561/560

Sval mapping: [2 1 0 7 8 6], 0 2 0 3 -1 2], 0 0 1 -1 0 0]]

Optimal tuning (CTE): ~16/11 = 651.2344, ~5/4 = 385.7289

Optimal GPV sequence: 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg

Badness: 0.340 × 10-3

Tritiwiz

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993

Mapping: [1 2 0 8 1], 0 -3 0 -4 1], 0 0 1 -2 1]]

Mapping generators: ~2, ~11/10, ~5

Wedgie⟨⟨⟨3 -6 3 -4 1 2 16 5 -26 -12]]]

Optimal tuning (CTE): ~11/10 = 165.8258, ~5/4 = 384.6711

Optimal GPV sequence15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd

Badness: 0.732 × 10-3