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| == Temperaments == | | == Temperaments == |
| === Tridecapyth ===
| | [[Tempering out]] this comma in the 2.3.13 subgroup leads to the [[tridecapyth]] temperament. |
| {{ See also | No-fives subgroup temperaments#Temperaments with a 2.3.13 gene }}
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| 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]]. | |
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| [[Subgroup]]: 2.3.13
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| {{mapping|legend=1| 1 1 -8 | 0 1 20 }}
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| : mapping generators: ~2, ~3/2
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| [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.025
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| : [[error map]]: {{val| 0.000 +0.070 -0.019 }}
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| {{Optimal ET sequence|legend=1| 12, 29f, 41, 53, 94, 147, 494, 641, 788 }}
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| [[Badness]] (Sintel): 0.211
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| == Etymology == | | == Etymology == |