Keenanismic temperaments: Difference between revisions
Move the 7-limit parent to a dedicated collection page |
Restore mapping generators. Cleanup |
||
| Line 25: | Line 25: | ||
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Uniwiz]].'' | : ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Uniwiz]].'' | ||
Uniwiz tempers out | Uniwiz tempers out 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 2.3.5.7.11.17 [[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 [[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]]. | 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]]. | ||
| Line 34: | Line 34: | ||
{{Mapping|legend=1| 2 1 0 7 8 | 0 2 0 3 -1 | 0 0 1 -1 0 }} | {{Mapping|legend=1| 2 1 0 7 8 | 0 2 0 3 -1 | 0 0 1 -1 0 }} | ||
: mapping generators: ~99/70, ~16/11, ~5 | |||
[[Optimal tuning]] ([[CTE]]): ~99/70 = | [[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000{{c}}, ~16/11 = 651.1082{{c}}, ~5/4 = 385.5778{{c}} | ||
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }} | {{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }} | ||
[[Badness]]: 0.529 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.529 × 10<sup>-3</sup> | ||
=== 2.3.5.7.11.17 subgroup === | === 2.3.5.7.11.17 subgroup === | ||
| Line 46: | Line 47: | ||
Comma list: 289/288, 385/384, 561/560 | Comma list: 289/288, 385/384, 561/560 | ||
Subgroup-val mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }} | |||
Gencom mapping: {{mapping| 2 1 0 7 8 0 6 | 0 2 0 3 -1 0 2 | 0 0 1 -1 0 0 0 }} | |||
{{ | Optimal tuning (CTE): ~17/12 = 600.0000{{c}}, ~16/11 = 651.2344{{c}}, ~5/4 = 385.7289{{c}} | ||
Badness: 0.340 × 10<sup>-3</sup> | {{Optimal ET sequence|legend=0| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }} | ||
Badness (Smith): 0.340 × 10<sup>-3</sup> | |||
== Tritiwiz == | == Tritiwiz == | ||
| Line 60: | Line 63: | ||
{{Mapping|legend=1| 1 2 0 8 1 | 0 -3 0 -4 1 | 0 0 1 -2 1 }} | {{Mapping|legend=1| 1 2 0 8 1 | 0 -3 0 -4 1 | 0 0 1 -2 1 }} | ||
: mapping generators: ~2, ~11/10, ~5 | : mapping generators: ~2, ~11/10, ~5 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~11/10 = 165.8258{{c}}, ~5/4 = 384.6711{{c}} | ||
{{Optimal ET sequence|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }} | {{Optimal ET sequence|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }} | ||
[[Badness]]: 0.732 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.732 × 10<sup>-3</sup> | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||