Keenanismic temperaments: Difference between revisions
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This is a collection of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out]] the keenanisma, [[385/384]]. For the rank-4 keenanismic temperament, see [[Keenanismic family]]. | This is a collection of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out]] the keenanisma, [[385/384]]. For the [[rank-4]] keenanismic temperament, see [[Keenanismic family]]. | ||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
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Uniwiz tempers out 1500625/1492992 = {{monzo| -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 = {{monzo| -17 2 0 0 4 }}, in the 11-limit. The [[extension]] to the 2.3.5.7.11.13.17 [[subgroup]] tempers out two no-13's 17-limit [[PIFE comma]]s, [[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 {([[289/288|S17]],) [[1089/1088|S33]], [[1156/1155|S34]], [[1225/1224|S35 = S49*S50]](, [[9801/9800|S99 = S33/S35]])}. | Uniwiz tempers out 1500625/1492992 = {{monzo| -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 = {{monzo| -17 2 0 0 4 }}, in the 11-limit. The [[extension]] to the 2.3.5.7.11.13.17 [[subgroup]] tempers out two no-13's 17-limit [[PIFE comma]]s, [[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 {([[289/288|S17]],) [[1089/1088|S33]], [[1156/1155|S34]], [[1225/1224|S35 = S49*S50]](, [[9801/9800|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 | 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 [[quartertone]]s. It can be viewed as the temperament described by the three quartertone-size equal temperaments: [[22edo|22]], [[24edo|24]] and [[26edo|26]]. In the 11-limit it shares the [[optimal patent val]], [[284edo|284]], with [[keenanismic]]. | ||
=== 7-limit === | === 7-limit === | ||
Revision as of 06:29, 24 December 2024
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) → Semiphore family
- Polyhymnia (+81/80) → Didymus rank-3 family
- Orwellian (+99/98) → Orwellismic family
- Supermagic (+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
- Zelda (+3388/3375) → Horwell family
- Beyla (+4375/4374) → Ragismic family
- Enlil (+6250/6237) → Kleismic rank-3 family
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. The extension to the 2.3.5.7.11.13.17 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.
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): ~1225/864 = 1\2, ~35/24 = 651.1974, ~5/4 = 385.6847
Optimal ET sequence: 22, 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): ~99/70 = 1\2, ~16/11 = 651.1082, ~5/4 = 385.5778
Optimal ET sequence: 22, 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): ~17/12 = 1\2, ~16/11 = 651.2344, ~5/4 = 385.7289
Optimal ET 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): ~2 = 1\1, ~11/10 = 165.8258, ~5/4 = 384.6711
Optimal ET sequence: 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd
Badness: 0.732 × 10-3