85/84: Difference between revisions

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'''85/84''', the '''monk comma''', is a [[17-limit]] [[superparticular ~~interval~~]]. It is the difference between [[6/5]] and [[17/14]], as well as [[7/5]] and [[17/12]].

'''85/84''', the '''monk comma''', is a [[small comma|small]] [[17-limit]] [[superparticular]] [[comma]] measuring about 20.5 [[cent]]s. 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 ({{Sagittal|\!}}) in [[Sagittal]].

== 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.

== Etymology ==

The word ''monk'' for this comma was introduced by [[~~User:Xenllium|~~Xenllium]] in 2023. It was translated from Japanese word 僧侶 (そうりょ, sōryo), a play of its [[~~Color~~ notation|color name]], ''soruyo'' (as resembling pronunciation).

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 [[Kite's color notation|color name]], ''soruyo'' (as resembling pronunciation).

== See also ==

* [[Small comma]]

* [[List of superparticular intervals]]

[[Category:Commas named after their color name]]

[[Category:Commas named by translating something into another language]]

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-'''85/84''', the '''monk comma''', is a [[17-limit]] [[superparticular interval]]. It is the difference between [[6/5]] and [[17/14]], as well as [[7/5]] and [[17/12]].
+'''85/84''', the '''monk comma''', is a [[small comma|small]] [[17-limit]] [[superparticular]] [[comma]] measuring about 20.5 [[cent]]s. 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 ({{Sagittal|\!}}) in [[Sagittal]].
+== 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.
 == Etymology ==
-The word ''monk'' for this comma was introduced by [[User:Xenllium|Xenllium]] in 2023. It was translated from Japanese word <span style="font-family:san-serif" lang="ja">僧侶</span> (<span style="font-family:san-serif" lang="ja">そうりょ</span>, sōryo), a play of its [[Color notation|color name]], ''soruyo'' (as resembling pronunciation).
+The word ''monk'' for this comma was introduced by [[Xenllium]] in 2023. It was translated from Japanese word <span style="font-family:san-serif" lang="ja">僧侶</span> (<span style="font-family:san-serif" lang="ja">そうりょ</span>, sōryo), a play of its [[Kite's color notation|color name]], ''soruyo'' (as resembling pronunciation).
 == See also ==
-* [[Small comma]]
 * [[List of superparticular intervals]]
 [[Category:Commas named after their color name]]
 [[Category:Commas named by translating something into another language]]