136/135: Difference between revisions

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== Temperaments ==

=== 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]]), 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, finding [[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.

[[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, finding [[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

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 == Temperaments ==
 === 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]]), 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, finding [[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.
+[[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, finding [[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