85/84

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.

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
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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

• List of superparticular intervals