Keenanismic temperaments: Difference between revisions

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

* ''[[Selenium]]'', {49/48, 56/55} → [[Semiphore family #Selenium|Semiphore family]]

* ''[[Polyhymnia]]'', {81/80, 385/384} → [[Didymus rank three family #Polyhymnia|Didymus rank-3 family]]

* ''[[Orwellian]]'', {99/98, 385/384} → [[Orwellismic family #Orwellian|Orwellismic family]]

* ''[[Supermagic]]'', {100/99, 385/384} → [[Keemic family #Undecimal supermagic|Keemic family]]

* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]]

* ''[[Starling]]'', {126/125, 385/384} → [[Starling family #Undecimal starling|Starling family]]

* [[Marvel]], {225/224, 385/384} → [[Marvel family #Undecimal marvel (unimarv)|Marvel family]]

* ''[[Mirwomo]]'', {243/242, 385/384} → [[Rastmic rank three clan #Mirwomo|Rastmic rank-3 clan]]

* [[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|Diaschismic rank-3 family]]

* ''[[Agni]]'', {385/384, 1375/1372} → [[Breed family #Agni|Breed family]]

* ''[[Beyla]]'', {385/384, 4375/4374} → [[Ragismic family #Beyla|Ragismic family]]

* ''[[Enlil]]'', {385/384, 6250/6237} → [[Kleismic rank three family #Enlil|Kleismic rank-3 family]]

Considered below are ''uniwiz'' and ''tritiwiz''.  

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]].  

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

[[Mapping]]: [{{val| 2 1 0 7 8 }}, {{val| 0 2 0 3 -1 }}, {{val| 0 0 1 -1 0 }}]

{{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

[[Badness]]: 0.529 × 10^-3

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 ET sequence|legend=1| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}

Badness: 0.340 × 10^-3

Subgroup: 2.3.5.7.11

[[Mapping]]: [{{val| 1 2 0 8 1 }}, {{val| 0 -3 0 -4 1 }}, {{val| 0 0 1 -2 1 }}]

 

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

 

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

[[Badness]]: 0.732 × 10^-3