Symbiotic comma: Difference between revisions
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'''19712/19683''', the '''symbiotic comma''', also known as the '''symbiosis''' or '''symbiosma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup]]) [[comma]] measuring about 2.55 [[cent]]s. It is the difference between | '''19712/19683''', the '''symbiotic comma''', also known as the '''symbiosis''' or '''symbiosma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup]]) [[comma]] measuring about 2.55 [[cent]]s. It is the difference between a [[2187/2048|Pythagorean apotome (2187/2048)]] and an [[77/72|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 == | == Temperaments == | ||
Revision as of 12:05, 23 October 2026
| Interval information |
symbiosis,
symbiosma
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, it identifies 11/8 by an apotome over 9/7, 11/6 by an apotome over 12/7, and 11/9 by an apotome over 8/7. Finally, it identifies 11/9 by 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
- 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.