Keenanismic temperaments: Difference between revisions
→Uniwiz: added no-13's 17-limit extension |
Cleanup; update the links; start replacing POTE by CTE |
||
| Line 2: | Line 2: | ||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
* ''[[Selenium]]'', {49/48, 56/55} → [[Semiphore family #Selenium]] | * ''[[Selenium]]'', {49/48, 56/55} → [[Semiphore family #Selenium|Semiphore family]] | ||
* ''[[Polyhymnia]]'', {81/80, 385/384} → [[Didymus rank three family #Polyhymnia]] | * ''[[Polyhymnia]]'', {81/80, 385/384} → [[Didymus rank three family #Polyhymnia|Didymus rank-3 family]] | ||
* ''[[Orwellian]]'', {99/98, 385/384} → [[Orwellismic family #Orwellian]] | * ''[[Orwellian]]'', {99/98, 385/384} → [[Orwellismic family #Orwellian|Orwellismic family]] | ||
* ''[[Supermagic]]'', {100/99, 385/384} → [[Keemic family #Undecimal supermagic]] | * ''[[Supermagic]]'', {100/99, 385/384} → [[Keemic family #Undecimal supermagic|Keemic family]] | ||
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus]] | * [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]] | ||
* ''[[Starling]]'', {126/125, 385/384} → [[Starling family #Undecimal starling]] | * ''[[Starling]]'', {126/125, 385/384} → [[Starling family #Undecimal starling|Starling family]] | ||
* [[Marvel]], {225/224, 385/384} → [[Marvel family #Undecimal marvel (unimarv)]] | * [[Marvel]], {225/224, 385/384} → [[Marvel family #Undecimal marvel (unimarv)|Marvel family]] | ||
* ''[[Mirwomo]]'', {243/242, 385/384} → [[Rastmic | * ''[[Mirwomo]]'', {243/242, 385/384} → [[Rastmic rank three clan #Mirwomo|Rastmic rank-3 clan]] | ||
* [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic]] | * [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]] | ||
* | * [[Portent]], {385/384, 441/440} → [[Gamelismic family #Portent|Gamelismic family]] | ||
* ''[[Augustus]]'', {385/384, 1232/1215} → [[Diaschismic rank three family #Augustus]] | * ''[[Augustus]]'', {385/384, 1232/1215} → [[Diaschismic rank three family #Augustus|Diaschismic rank-3 family]] | ||
* ''[[Agni]]'', {385/384, 1375/1372} → [[Breed family #Agni]] | * ''[[Agni]]'', {385/384, 1375/1372} → [[Breed family #Agni|Breed family]] | ||
* ''[[Beyla]]'', {385/384, 4375/4374} → [[Ragismic family #Beyla]] | * ''[[Beyla]]'', {385/384, 4375/4374} → [[Ragismic family #Beyla|Ragismic family]] | ||
* ''[[Enlil]]'', {385/384, 6250/6237} → [[Kleismic rank three family #Enlil]] | * ''[[Enlil]]'', {385/384, 6250/6237} → [[Kleismic rank three family #Enlil|Kleismic rank-3 family]] | ||
Considered below are ''uniwiz'' and ''tritiwiz''. | Considered below are ''uniwiz'' and ''tritiwiz''. | ||
== Uniwiz == | == Uniwiz == | ||
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. In the 17-limit it tempers two no-13's 17-limit [[PIFE | 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. In the 17-limit it tempers 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. | ||
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: [[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 quartertones. 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 === | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 1500625/1492992 | [[Comma list]]: 1500625/1492992 | ||
| Line 35: | Line 35: | ||
{{Multival|legend=1|rank=3| 4 -4 -6 11 }} | {{Multival|legend=1|rank=3| 4 -4 -6 11 }} | ||
[[ | Optimal tuning ([[CTE]]): ~35/24 = 651.1974, ~5/4 = 385.6847 | ||
{{Val list|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }} | {{Val list|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }} | ||
| Line 42: | Line 42: | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 385/384, 9801/9800 | [[Comma list]]: 385/384, 9801/9800 | ||
| Line 50: | Line 50: | ||
{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }} | {{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }} | ||
[[ | Optimal tuning ([[CTE]]): ~16/11 = 651.1082, ~5/4 = 385.5778 | ||
{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }} | {{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }} | ||
| Line 56: | Line 56: | ||
[[Badness]]: 0.529 × 10<sup>-3</sup> | [[Badness]]: 0.529 × 10<sup>-3</sup> | ||
=== | === 2.3.5.7.11.17 subgroup === | ||
Subgroup 2.3.5.7.11.17 | Subgroup: 2.3.5.7.11.17 | ||
Comma list: 289/288, 385/384, 561/560 | |||
[ | Sval mapping: [{{val| 2 1 0 7 8 6 }}, {{val| 0 2 0 3 -1 2 }}, {{val| 0 0 1 -1 0 0 }}] | ||
Optimal tuning (CTE): ~16/11 = 651.2344, ~5/4 = 385.7289 | |||
Optimal GPV sequence: {{Val list| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }} | |||
Badness: 0.340 × 10<sup>-3</sup> | |||
== Tritiwiz == | == Tritiwiz == | ||
| Line 84: | Line 80: | ||
{{Multival|legend=1|rank=3| 3 -6 3 -4 1 2 16 5 -26 -12 }} | {{Multival|legend=1|rank=3| 3 -6 3 -4 1 2 16 5 -26 -12 }} | ||
[[ | Optimal tuning ([[CTE]]): ~11/10 = 165.8258, ~5/4 = 384.6711 | ||
{{Val list|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }} | {{Val list|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }} | ||