85/84: Difference between revisions
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== Temperaments == | == 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. | 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 == | == Etymology == | ||
Latest revision as of 07:36, 19 May 2026
| Interval information |
Soruyo comma
reduced
85/84, the monk comma, is a small 17-limit superparticular comma measuring about 20.5 cents. It is the difference between 6/5 and 17/14, as well as 7/5 and 17/12.
Notation
This comma can be represented as a secondary comma for 5C () in Sagittal.
Temperaments
It can be 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. S15⋅S16⋅S17, 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
The word monk for this comma was introduced by Xenllium in 2023. It was translated from Japanese word 僧侶 (そうりょ, sōryo), a play of its color name, soruyo (as resembling pronunciation).