136/135: Difference between revisions

Line 8:

== 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, ~~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.

[[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

Line 17:

[[Optimal tuning]]s:

* subgroup CE: 2 = 1\1, ~3/2 = 705.440 ~~(recommended)~~

* [[Tp tuning|subgroup]] [[CEE]]: 2 = 1\1, ~3/2 = 705.440

* [[Tp tuning|subgroup CTE]]: ~2 = 1\1, ~3/2 = 704.1088 ~~(flat of 46edo)~~

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 704.1088

Line 29:

: sval mapping generators: ~2, ~3, ~5

[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}

Line 55:

: sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8

{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*

@@ Line 8: / Line 8: @@
 == 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, 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.
+[[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
@@ Line 17: / Line 17: @@
 [[Optimal tuning]]s:
-* subgroup CE: 2 = 1\1, ~3/2 = 705.440 (recommended)
+* [[Tp tuning|subgroup]] [[CEE]]: 2 = 1\1, ~3/2 = 705.440
-* [[Tp tuning|subgroup CTE]]: ~2 = 1\1, ~3/2 = 704.1088 (flat of 46edo)
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 704.1088
 {{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
@@ Line 29: / Line 29: @@
 : sval mapping generators: ~2, ~3, ~5
-[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544
 {{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}
@@ Line 55: / Line 55: @@
 : sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
-[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8
 {{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*