136/135: Difference between revisions
Improve readability by preferring verbal descriptions to math expressions. Remove words like "deduce", "therefore" (which deduces which?) in favor of a focus on presenting facts. |
m →Fiventeen: the CTE tuning being flat of 46edo is honestly insane/shows it's not the right thing to use here; i would much prefer to only have CE gives a much more reasonable result (between 17 EDO and 80 EDO) but in case anyone is interested in the CTE tuning i wil leave it here |
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== Temperaments == | == Temperaments == | ||
=== Fiventeen === | === Fiventeen === | ||
[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]) although [[80edo]] might be preferred for | [[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and 80 + 17 = [[97edo]] and 97 + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank 3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, as it correctly finds [[34edo]], then [[80edo]], then 34 + 80 = [[114edo]] and amazingly even 114 + 80 = [[194edo|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and 63 + 80 = [[143edo]] tunings are found in the optimal ET sequence for fiventeen. | ||
[[Subgroup]]: 2.3.17/5 | [[Subgroup]]: 2.3.17/5 | ||
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: sval mapping generators: ~2, ~3 | : sval mapping generators: ~2, ~3 | ||
[[Optimal tuning]] ([[Tp tuning|subgroup CTE]] | [[Optimal tuning]]s: | ||
* subgroup CE: 2 = 1\1, ~3/2 = 705.440 (recommended) | |||
* [[Tp tuning|subgroup CTE]]: ~2 = 1\1, ~3/2 = 704.1088 (flat of 46edo) | |||
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }} | {{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }} | ||