Keenanismic temperaments

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.

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:

• Selenium (+49/48 or 56/55) → Semaphoresmic family
• Polyhymnia (+81/80) → Didymus rank-3 family
• Orwellian (+99/98) → Orwellismic family
• Keemic (+100/99) → Keemic family
• Zeus (+121/120 or 176/175) → Porwell family and Biyatismic clan
• Starling (+126/125) → Starling family
• Marvel (+225/224) → Marvel family
• Mirwomo (+243/242) → Rastmic rank-3 clan
• Sensamagic (+245/243) → Sensamagic family
• Portent (+441/440) → Gamelismic family
• Augustus (+1232/1215) → Diaschismic rank-3 family
• Agni (+1375/1372) → Breed family
• Akea (+2200/2187) → Hemifamity family
• Zelda (+3388/3375) → Horwell family
• Beyla (+4375/4374) → Ragismic family
• Enlil (+6250/6237) → Kleismic rank-3 family

Considered below are uniwiz and tritiwiz.

Uniwiz

For the 7-limit version, see Miscellaneous 7-limit temperaments #Uniwiz.

Uniwiz tempers out 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. The extension to the 2.3.5.7.11.17 subgroup tempers out 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. The S-expression-based comma list of this extension is {(S17), S33, S34, S35 = S49 × S50, (S99 = S33/S35)}.

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.

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]]

mapping generators: ~99/70, ~16/11, ~5

Optimal tunings:

• WE: ~99/70 = 600.2029 ¢, ~16/11 = 650.9604 ¢, ~5/4 = 385.1616 ¢

error map: ⟨+0.406 +0.169 -0.341 -0.498 -0.655]

• CWE: ~99/70 = 600.0000 ¢, ~16/11 = 650.8311 ¢, ~5/4 = 385.1658 ¢

error map: ⟨0.000 -0.293 -1.148 -1.499 -2.149]

Optimal ET sequence: 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de

Badness (Sintel): 0.636

2.3.5.7.11.17 subgroup

Subgroup: 2.3.5.7.11.17

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

Subgroup-val mapping: [⟨2 1 0 7 8 6], ⟨0 2 0 3 -1 2], ⟨0 0 1 -1 0 0]]

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

Optimal tunings:

• WE: ~99/70 = 600.2365 ¢, ~16/11 = 651.0126 ¢, ~5/4 = 385.1864 ¢
• CWE: ~99/70 = 600.0000 ¢, ~16/11 = 650.8664 ¢, ~5/4 = 385.1951 ¢

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

Badness (Sintel): 0.351

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

Optimal tunings:

• WE: ~2 = 1200.0000 ¢, ~11/10 = 166.1752 ¢, ~5/4 = 384.3796 ¢

error map: ⟨+0.400 +0.320 -1.133 -0.684 +0.438]

• CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0419 ¢, ~5/4 = 384.3625 ¢

error map: ⟨0.000 -0.081 -1.951 -1.718 -0.914]

Optimal ET sequence: 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd

Badness (Sintel): 0.879