352/351
| Interval information |
major gentle comma,
11/13-kleisma
Thulo comma
reduced
(Shannon, [math]\sqrt{nd}[/math])
352/351, the major minthma, major gentle comma or 11/13-kleisma 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, hence the name 11/13-kleisma. This means that when this comma is tempered out, 11/8 plus three perfect fourths octave reduces to 13/8; as such, this comma can be understood as relating harmonics 11 and 13 by a chain of fourths. Second, it is the difference between various tridecimal intervals and their adjacent undecimal intervals such as:
- Between the tridecimal quartertone of 1053/1024 and the undecimal quartertone of 33/32;
- Between 16/13 and 27/22; and
- Between 39/32 and 11/9.
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.
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.