Ripple family: Difference between revisions

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== Ripple ==

The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. As one might expect, [[12edo]] ~~is about as accurate as it can be tuned~~.

The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered [[4/3|fourth]] reached at 5 semitones. Interestingly, in optimal tunings, the major third of ~5/4 does not tend to be damaged much sharpwards as one might expect from the equivalence, and is in practice often even flat, so that prime 3 takes on practically the whole damage of the 5-limit equivalence, for which it has the advantage of being the simplest so still having a good chance at psychoacoustic viability. As a result though, the mapping of ~9/8 is often inconsistent, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].

~~Good~~ patent val tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].

Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].

[[Subgroup]]: 2.3.5

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{{Optimal ET sequence|legend=1| 11c, 12, 71b, 83b, 95b, 107bc, 119bc }}

[[Badness]] (~~Smith~~): 0.~~138948~~

Badness (Smith): 0.138948

[[Badness]] (Dirichlet): 3.259

=== Septimal ripple ===

Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~21/16 * ~4/3 = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[126/125]], [[405/392]]

[[Optimal tuning]]s:

* [[CTE]]: ~15/14 = 101.538

: [[error map]]: {{val| 0 -9.643 1.385 9.647 }}

* [[CE]]: ~15/14 = 101.881

: [[error map]]: {{val| 0 -11.361 -1.364 4.837 }}

[[Badness]] (Dirichlet): 1.521

==== 11-limit ====

A notable [[patent val]] tuning of 11-limit ripple not appearing in the optimal ET sequence is [[47edo]].

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[126/125]], [[99/98]], [[45/44]]

[[Optimal tuning]]s:

* [[CTE]]: ~15/14 = 101.538

: [[error map]]: {{val| 0 -11.785, -2.041, 3.651, 13.296 }}

* [[CE]]: ~15/14 = 102.319 (preferred for dual-fifths 11-limit)

: [[error map]]: {{val| 0 -13.551 -4.868 -1.296 6.935 }}

[[Badness]] (Dirichlet): 1.334

== Rip ==

Formerly known as [[#Ripple]], but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.

~~== Septimal ripple ==~~

[[Subgroup]]: 2.3.5.7

@@ Line 2: / Line 2: @@
 == Ripple ==
-The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. As one might expect, [[12edo]] is about as accurate as it can be tuned.
+The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered [[4/3|fourth]] reached at 5 semitones. Interestingly, in optimal tunings, the major third of ~5/4 does not tend to be damaged much sharpwards as one might expect from the equivalence, and is in practice often even flat, so that prime 3 takes on practically the whole damage of the 5-limit equivalence, for which it has the advantage of being the simplest so still having a good chance at psychoacoustic viability. As a result though, the mapping of ~9/8 is often inconsistent, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].
-Good patent val tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].
+Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].
 [[Subgroup]]: 2.3.5
@@ Line 22: / Line 22: @@
 {{Optimal ET sequence|legend=1| 11c, 12, 71b, 83b, 95b, 107bc, 119bc }}
-[[Badness]] (Smith): 0.138948
+Badness (Smith): 0.138948
+[[Badness]] (Dirichlet): 3.259
+=== Septimal ripple ===
+{{ See also | Dual-fifth temperaments }}
+Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~21/16 * ~4/3 = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: [[126/125]], [[405/392]]
+{{Mapping|legend=1| 1 2 3 4 | 0 -5 -8 -14 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~15/14 = 101.538
+: [[error map]]: {{val| 0 -9.643 1.385 9.647 }}
+* [[CE]]: ~15/14 = 101.881
+: [[error map]]: {{val| 0 -11.361 -1.364 4.837 }}
+{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}
+[[Badness]] (Dirichlet): 1.521
+==== 11-limit ====
+A notable [[patent val]] tuning of 11-limit ripple not appearing in the optimal ET sequence is [[47edo]].
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[126/125]], [[99/98]], [[45/44]]
+{{Mapping|legend=1| 1 2 3 4 5 | 0 -5 -8 -14 -18 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~15/14 = 101.538
+: [[error map]]: {{val| 0 -11.785, -2.041, 3.651, 13.296 }}
+* [[CE]]: ~15/14 = 102.319 (preferred for dual-fifths 11-limit)
+: [[error map]]: {{val| 0 -13.551 -4.868 -1.296 6.935 }}
+{{Optimal ET sequence|legend=1| 11cdee, 12, 23de, 35 }}
+[[Badness]] (Dirichlet): 1.334
+== Rip ==
+Formerly known as [[#Ripple]], but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.
-== Septimal ripple ==
 [[Subgroup]]: 2.3.5.7