Mercurial comma

Interval information

Ratio	557122275/556583944
Factorization	2-3 × 32 × 52 × 7-2 × 17-5 × 195
Monzo	[-3 2 2 -2 0 0 -5 5⟩
Size in cents	1.67365¢
Name	mercurial comma
Color name	19o517u5rryy-3, quinnosu-abiruyo negative 3rd
FJS name	[math]\displaystyle{ \text{ddd}{-3}^{5,5,19,19,19,19,19}_{7,7,17,17,17,17,17} }[/math]
Special properties	reduced
Tenney height (log2 nd)	58.1054
Weil height (log2 max(n, d))	58.1068
Wilson height (sopfr(nd))	216
Comma size	unnoticeable
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557122275/556583944, or the mercurial comma, is an unnoticeable comma measuring 1.673649 cents. It represents the amount by which five justly tuned 19/17s and two 15/14s exceed an octave, thus being of importance in 5L 2s diatonic scale.

Etymology

As the ratio between these intervals is very close to phi, a diatonic scale consisting of a 19/17 as the tone and 15/14 as the semitone represents a strong harmonic entropy minimum adjacent to the golden meantone sequence. This is particularly well suited to stringed instruments that are normally tuned with slight octave stretches due to the inharmonicity of their partials. Hence, the mercurial comma is named after the chemical element mercury due to its adjacency to gold, since mercury meantone is adjacent to golden meantone.

It is worth noting that per principles of regular temperament theory, a temperament produced by the mercurial comma is a rank-7 19-limit temperament, which would naturally have the name mercurial. Alternately, there is a temperament produced in the 2.3.5.7.17.19 subgroup, which is a rank-5 temperament. This can be easily added to the commas of septimal meantone and 256/255 to produce a rank-2 temperament in the meantone family which supports the mercurial relation.

Temperaments

Tempering this comma out in the full 19-limit produces the rank-19 mercurial temperament.

Subgroup:

Comma list: 557122275/556583944

Mapping:

[⟨	1	1	2	4	3	-3	4	5	],
⟨	0	1	0	1	0	0	0	0	],
⟨	0	0	1	1	0	0	0	0	],
⟨	0	0	0	-5	0	0	0	-2	],
⟨	0	0	0	0	1	0	0	0	]],
⟨	0	0	0	0	1	0	0	0	]],
⟨	0	0	0	0	0	1	0	0	]],
⟨	0	0	0	0	0	0	1	1	]],

mapping generators: ~2, ~3, ~5, ~5415/4046, ~11, ~13, ~17

The interval 5415/4046 is within the range of meantone fourths. From this, 5415/4046 can be derived as the mercurial meantone fourth and respectively its inverse, 8092/5415, mercurial meantone fifth. The comma that separates each from the just versions is 16245/16184.

Despite sharing the name, mercurial temperament is unrelated to the 80th-octave temperament by the same name, though as a curiosity, the 2960edo, more specifically, the 2960dh val, is the unique tuning supporting them both.