352/351

Interval information
Ratio	352/351
Factorization	25 × 3-3 × 11 × 13-1
Monzo	[5 -3 0 0 1 -1⟩
Size in cents	4.925278 ¢
Names	major minthma,; major gentle comma
Color name	3u1oM, thuloma
FJS name	[math]\displaystyle{ \text{P1}^{11}_{13} }[/math]
Special properties	superparticular,; reduced
Tenney norm (log2 nd)	16.9148
Weil norm (log2 max(n, d))	16.9189
Wilson norm (sopfr(nd))	43
Comma size	small
	Open this interval in xen-calc

352/351, the major minthma or major gentle comma is a small 13-limit (also 2.3.11.13 subgroup) comma measuring about 4.9 ¢. This comma can be described in a number of ways. First, it is the difference between the tridecimal minor third of 13/11 and the Pythagorean minor third of 32/27. This means that 13/8 plus three perfect fifths differs from 11/8 by exactly this comma up to octave equivalence; as such, this comma can be understood as relating harmonics 11 and 13 by a chain of fifths. Second, it is the difference between various tridecimal intervals and their adjacent undecimal intervals such as between 11/9 and 39/32, between 16/13 and 27/22, and between the tridecimal quartertone of 1053/1024 and the undecimal quartertone of 33/32.

352/351 and 351/350, the ratwolfsma, are extremely close in size and make up a consecutive pair of 13-limit superparticular commas. Their difference is 123201/123200, the chalmersia, the smallest 13-limit superparticular comma; their sum is 176/175, the valinorsma, an 11-limit superparticular comma.

Temperaments

Tempering out this comma leads to a form of major-minthmic temperament (full 13-limit rank-5) or majorminth temperament (2.3.11.13-subgroup rank-3), which enables major-minthmic chords. Mintha was a nymph turned into a mint plant by a goddess whom she got in the way of^[1]^[2]^[3], and major minthmic tempering has the slightly sharp, minty-fresh fifths some people appreciate. Margo Schulter has expressed great fondness for this comma, for example.

It is tempered out in a large number of notable temperaments such as meanpop, srutal, countercata, cassandra, diaschismic, hemififths, amity, rodan, buzzard and higher rank temperaments like parapyth, aberschismic extensions, gentle, akua or valinorsmic.

Notation

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal  and is called the 13/11 kleisma, or 13/11k for short, because the simplest interval it notates is 13/11, as for example in D-F⁠ ⁠. The primary role of  is 5103/5120 (7/5k). The upward version is called 11/13k or 13/11k up and is represented (in a secondary role) by .

History and etymology

This comma was known as the minthma from 2011. The word minthma is a contraction of minor thirds comma into a single word. In 2022, Margo Schulter requested it should be changed to major/greater minthma to distinguish it from 364/363, which would gain the name minor/lesser minthma.

The major gentle comma was also added to this comma in contrast to the minor gentle comma of 364/363. Both commas were essential to define the gentle temperament. Gentle was named for the gentle fifth, a mild version of the superpyth fifth.

References

↑ Dictionary of Greek and Roman Biography and Mythology – William Smith (1849)
↑ Dictionary of Classical Mythology – M. R. Wright (2012)
↑ All About Minthe – Myth of Greek Mythology – Greek Boston (2024)

[1] Dictionary of Greek and Roman Biography and Mythology – William Smith (1849)

[2] Dictionary of Classical Mythology – M. R. Wright (2012)

[3] All About Minthe – Myth of Greek Mythology – Greek Boston (2024)

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