Tridecapyth comma: Difference between revisions
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=== Tridecapyth === | === Tridecapyth === | ||
{{ See also | No-fives subgroup temperaments#Temperaments with a 2.3.13 gene }} | {{ See also | No-fives subgroup temperaments#Temperaments with a 2.3.13 gene }} | ||
Tempering out this comma in the 2.3.13 subgroup leads tridecapyth, which can be seen as a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized | Tempering out this comma in the 2.3.13 subgroup leads tridecapyth, which can be seen as a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping; for 118edo we get the 118f [[val]] that takes the second-best, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, due to its small note count, [[41edo]] technically uses this mapping too, so that the val sum 41 + 53 = [[94edo]] also uses this mapping, suggesting it's of interest to flatter tunings of [[garibaldi]] with fifths tending close to pure; this corresponds to the extension of garibaldi called [[cassandra]]. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]]. | ||
[[Subgroup]]: 2.3.13 | [[Subgroup]]: 2.3.13 | ||
Revision as of 15:51, 6 June 2026
| Interval information |
Trisatho comma
The tridecapyth comma (monzo: [28 -20 0 0 0 1⟩, ratio: 3489660928/3486784401), also described as the tridecaschisma (after the 2.3.13 temperament), is an unnoticeable comma in 13-limit just intonation which measures roughly 1.43 ¢. It is the interval by which 13/8 exceeds a stack of twenty perfect fifths (3/2) octave reduced, and by which 16/13 falls short of a stack of four Pythagorean limmas (256/243). It is perhaps more easily conceptualized as reaching 13/4 through (9/8)10. In terms of commas, it is the amount by which tridecimal quartertone (1053/1024) is greater than a stack of two Pythagorean commas.
Temperaments
Tridecapyth
Tempering out this comma in the 2.3.13 subgroup leads tridecapyth, which can be seen as a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the 53edo tuning of schismic). Interestingly, the mapping is so accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping; for 118edo we get the 118f val that takes the second-best, flat mapping of prime 13, and the same is true for 171edo where we get the 171f val. However, due to its small note count, 41edo technically uses this mapping too, so that the val sum 41 + 53 = 94edo also uses this mapping, suggesting it's of interest to flatter tunings of garibaldi with fifths tending close to pure; this corresponds to the extension of garibaldi called cassandra. If we add this mapping to 5-limit schismic instead we get tridecaschismic.
Subgroup: 2.3.13
Mapping: [⟨1 1 -8], ⟨0 1 20]]
- mapping generators: ~2, ~3/2
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.025
- error map: ⟨0.000 +0.070 -0.019]
Optimal ET sequence: 12, 29f, 41, 53, 94, 147, 494, 641, 788
Badness (Sintel): 0.211
Etymology
This comma was named by Aura in 2021.
See also
- Tridecimal schisma (disambiguation page)