Cathartic comma

The cathartic comma (formerly canousma, monzo: [4 -14 3 4⟩, ratio: 4 802 000 / 4 782 969) is a small 7-limit comma measuring about 6.9 cents. It is the amount by which a stack of three instances of 81/70 falls short of 14/9. It is also the interval separating the syntonic comma (81/80) from a stack of two sensamagic commas (245/243).

Interval information

Ratio	4802000/4782969
Factorization	24 × 3-14 × 53 × 74
Monzo	[4 -14 3 4⟩
Size in cents	6.874755¢
Name	carthartic comma
Color name	sz4y33, saquadzo-atriyo 3rd
FJS name	[math]\displaystyle{ \text{d3}^{5,5,5,7,7,7,7} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	44.3847
Weil norm (log2 max(n, d))	44.3904
Wilson norm (sopfr(nd))	93
Comma size	small
Open this interval in xen-calc

Temperaments

Tempering out this comma leads to the rank-3 cathartic temperament, which splits the syntonic comma into two equal parts, each representing 245/243. It also splits 5/4 (classic major third) into two equal parts, each representing 2450/2187~2187/1960. It is similar to hemimean in that regard. Tempering out both equates 245/243 with 126/125 (septimal semicomma) and 225/224 (septimal kleisma) leads to parakleismic.

See Cathartic family for the rank-3 family where it is tempered out. See Cathartic temperaments for a collection of rank-2 temperaments where it is tempered out.

Etymology

This comma was first named by Flora Canou as canousma in 2020 after herself. In 2026, she disavowed that name and came up with the current name as a replacement, from the feeling she associated with the interval 28/27, which had the role of a basic building block in the corresponding temperament.