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]].
 
== 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).


== Notation ==
== Notation ==
This comma can be represented as a secondary comma for 5C ({{Sagittal|\!}}) in Sagittal.
This comma can be represented as a secondary comma for 5C ({{Sagittal|\!}}) in [[Sagittal]].


== 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]] ({{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.
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.
 
== Etymology ==
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 ==
== See also ==
* [[Small comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:Commas named after their color name]]
[[Category:Commas named after their color name]]
[[Category:Commas named by translating something into another language]]
[[Category:Commas named by translating something into another language]]

Revision as of 05:37, 17 May 2026

Interval information
Ratio 85/84
Factorization 2-2 × 3-1 × 5 × 7-1 × 17
Monzo [-2 -1 1 -1 0 0 1
Size in cents 20.48822¢
Name monk comma
Color name 17ory1, soruyo 1sn,
Soruyo comma
FJS name [math]\displaystyle{ \text{P1}^{5,17}_{7} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 12.8017
Weil norm (log2 max(n, d)) 12.8188
Wilson norm (sopfr(nd)) 36
Comma size small
S-expression S15⋅S16⋅S17
Open this interval in xen-calc

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.

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).

See also