Symbiotic comma

Interval information

Ratio	19712/19683
Factorization	28 × 3-9 × 7 × 11
Monzo	[8 -9 0 1 1⟩
Size in cents	2.548841 ¢
Names	symbiotic comma, symbiosis, symbiosma
Color name	s1oz2, Salozo comma
FJS name	[math]\displaystyle{ \text{d2}^{7,11} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	28.5314
Weil norm (log2 max(n, d))	28.5336
Wilson norm (sopfr(nd))	61
Comma size	unnoticeable
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19712/19683, the symbiotic comma, also known as the symbiosis or symbiosma, is an unnoticeable 11-limit (specifically 2.3.7.11 subgroup) comma measuring about 2.55 cents. It is the difference between a Pythagorean apotome (2187/2048) and an undecimal secor (77/72) – the latter of which can be broken down into a quartertone of 33/32 stacked on top of a third-tone of 28/27.

In terms of other commas, it is the difference between a septischisma and an olympia. It is also the sum of an argyria and an olympia, as well as the sum of a quartisma and a nexus comma. In the full 11-limit, it factors into simpler commas as (4375/4374)⋅(5632/5625), and in the 19-limit, (1216/1215)⋅(1540/1539).

Temperaments

Tempering out this comma in the 11-limit results in the rank-4 symbiotic temperament (→ Rank-4 temperament #Symbiotic (19712/19683)), or in the 2.3.7.11 subgroup, the symbiotian temperament. In either case, 28/27 and 33/32 become apotome complements, and 11/8 is identified with an apotome over 9/7, 11/6 with an apotome over 12/7, and 11/9 with an apotome over 8/7. Finally, 11/9 is identified with a limma below 9/7.

Symbiotian

Subgroup: 2.3.7.11

Comma list: 19712/19683

Subgroup-val mapping: [⟨1 0 0 -8], ⟨0 1 0 9], ⟨0 0 1 -1]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

• WE: ~2 = 1199.9290 ¢, ~3/2 = 702.2265 ¢, ~7/4 = 968.8979 ¢

error map: ⟨-0.071 +0.201 -0.070 -0.106]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2440 ¢, ~7/4 = 968.8593 ¢

error map: ⟨0.000 +0.289 +0.033 +0.019]

Optimal ET sequence: 41, 58, 77, 94, 118, 135, 716, 733, 851, 868, 1003, 1138, 1273, 1449, 1584, 1719, 1854

Badness (Sintel): 0.336

Etymology

This comma was named by Aura in 2020, from the fact that it is tempered out in such notable temperaments as vishnu, newt, kwai, supers, guiron and amity, and, from the fact that it also makes a good extension to a number of other temperaments such as cathartic.