85/84: Difference between revisions

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== Temperaments ==

It can be [[tempering out|tempered out]] in the [[17-limit]], leading to a rank-6 temperament, or in the 2.3.5.7.17-subgroup, leading to a rank-4 temperament. However, 85/84 factors as ([[225/224]])⋅([[256/255]])⋅([[289/288]]), i.e. ({{S|15}})⋅({{S|16}})⋅({{S|17}}), so that it is natural to temper out all of them, leading to the 2.3.5.7.17 version of [[pajara]] temperament, which also tempers out [[50/49]]. From there we can see that 50/49 = ([[99/98]])⋅([[100/99]]), and tempering both out leads to the 2.3.5.7.11.17-subgroup version of pajara. We find a factorization of 85/84 as ([[169/168]])⋅([[170/169]]), leading to an unnamed full 17-limit weak extension (34d & 44) that splits the fifth into two ~[[16/13]]'s. However, splitting the fifth doubles the [[generator complexity]] of every interval, greatly reducing the temperament's practicality.

It can be [[tempering out|tempered out]] in the [[17-limit]], leading to a rank-6 temperament, or in the 2.3.5.7.17-subgroup, leading to a rank-4 temperament. However, 85/84 factors as ([[225/224]])⋅([[256/255]])⋅([[289/288]]), i.e. {{S|15}}⋅{{S|16}}⋅{{S|17}}, so that it is natural to temper out all of them, leading to the 2.3.5.7.17 version of [[pajara]] temperament, which also tempers out [[50/49]]. From there we can see that 50/49 = ([[99/98]])⋅([[100/99]]), and tempering both out leads to the 2.3.5.7.11.17-subgroup version of pajara. We find a factorization of 85/84 as ([[169/168]])⋅([[170/169]]), leading to an unnamed full 17-limit weak extension (34d & 44) that splits the fifth into two ~[[16/13]]'s. However, splitting the fifth doubles the [[generator complexity]] of every interval, greatly reducing the temperament's practicality.

== Etymology ==

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 == Temperaments ==
-It can be [[tempering out|tempered out]] in the [[17-limit]], leading to a rank-6 temperament, or in the 2.3.5.7.17-subgroup, leading to a rank-4 temperament. However, 85/84 factors as ([[225/224]])⋅([[256/255]])⋅([[289/288]]), i.e. ({{S|15}})⋅({{S|16}})⋅({{S|17}}), so that it is natural to temper out all of them, leading to the 2.3.5.7.17 version of [[pajara]] temperament, which also tempers out [[50/49]]. From there we can see that 50/49 = ([[99/98]])⋅([[100/99]]), and tempering both out leads to the 2.3.5.7.11.17-subgroup version of pajara. We find a factorization of 85/84 as ([[169/168]])⋅([[170/169]]), leading to an unnamed full 17-limit weak extension (34d & 44) that splits the fifth into two ~[[16/13]]'s. However, splitting the fifth doubles the [[generator complexity]] of every interval, greatly reducing the temperament's practicality.
+It can be [[tempering out|tempered out]] in the [[17-limit]], leading to a rank-6 temperament, or in the 2.3.5.7.17-subgroup, leading to a rank-4 temperament. However, 85/84 factors as ([[225/224]])⋅([[256/255]])⋅([[289/288]]), i.e. {{S|15}}⋅{{S|16}}⋅{{S|17}}, so that it is natural to temper out all of them, leading to the 2.3.5.7.17 version of [[pajara]] temperament, which also tempers out [[50/49]]. From there we can see that 50/49 = ([[99/98]])⋅([[100/99]]), and tempering both out leads to the 2.3.5.7.11.17-subgroup version of pajara. We find a factorization of 85/84 as ([[169/168]])⋅([[170/169]]), leading to an unnamed full 17-limit weak extension (34d & 44) that splits the fifth into two ~[[16/13]]'s. However, splitting the fifth doubles the [[generator complexity]] of every interval, greatly reducing the temperament's practicality.
 == Etymology ==