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. The ploidacot of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered 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 sometimes 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 very flat, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].

The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. The [[ploidacot]] of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered 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 sometimes 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 very flat, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].

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

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[[Badness]] (Sintel): 3.26

=== Overview to extensions ===

The second comma of the comma list defines which 7-limit family member we are looking at:

* Septimal ripple adds [[126/125]];

* Rip adds [[36/35]];

Both use the same nominal generator as ripple.

For weak extensions, we have hemiripple and cohemiripple. Hemiripple adds [[49/48]], spliting the semitone generator in two. Cohemiripple adds [[245/243]], spliting the [[octave complement]] of the semitone generator in two.

== Septimal ripple ==

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== Hemiripple ==

Hemiripple tempers out 49/48 and splits the semitone generator in two for ~36/35. Its ploidacot is omega-decacot.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~36/35

[[Optimal tuning]]s:

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== Cohemiripple ==

Cohemiripple tempers out 245/243 and splits the octave complement of the semitone generator of ripple in two, each of which is used for ~7/5. Its ploidacot is delta-decacot.

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 245/243, 1323/1250

: mapping generators: ~2, ~7/5

[[Optimal tuning]]s:

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Comma list: 77/75, 243/242, 245/242

Mapping: {{mapping| 1 ~~7 11 12 17~~ | 0 -10 -16 -17 -25 }}

Mapping: {{mapping| 1 -3 -5 -5 -8 | 0 10 16 17 25 }}

Optimal tunings:

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Comma list: 66/65, 77/75, 147/143, 243/242

Mapping: {{mapping| 1 ~~7 11 12 17 14~~ | 0 -10 -16 -17 -25 -19 }}

Mapping: {{mapping| 1 -3 -5 -5 -8 -5 | 0 10 16 17 25 19 }}

Optimal tunings:

@@ Line 3: / Line 3: @@
 == Ripple ==
-The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. The ploidacot of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered 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 sometimes 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 very flat, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].
+The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. The [[ploidacot]] of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered 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 sometimes 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 very flat, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].
 Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].
@@ Line 28: / Line 28: @@
 [[Badness]] (Sintel): 3.26
+=== Overview to extensions ===
+The second comma of the comma list defines which 7-limit family member we are looking at:
+* Septimal ripple adds [[126/125]];
+* Rip adds [[36/35]];
+Both use the same nominal generator as ripple.
+For weak extensions, we have hemiripple and cohemiripple. Hemiripple adds [[49/48]], spliting the semitone generator in two. Cohemiripple adds [[245/243]], spliting the [[octave complement]] of the semitone generator in two.
 == Septimal ripple ==
@@ Line 121: / Line 130: @@
 == Hemiripple ==
+Hemiripple tempers out 49/48 and splits the semitone generator in two for ~36/35. Its ploidacot is omega-decacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 126: / Line 137: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}
+: mapping generators: ~2, ~36/35
 [[Optimal tuning]]s:
@@ Line 169: / Line 182: @@
 == Cohemiripple ==
 {{See also| Sensamagic clan }}
+Cohemiripple tempers out 245/243 and splits the octave complement of the semitone generator of ripple in two, each of which is used for ~7/5. Its ploidacot is delta-decacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 174: / Line 189: @@
 [[Comma list]]: 245/243, 1323/1250
-{{Mapping|legend=1| 1 7 11 12 | 0 -10 -16 -17 }}
+{{Mapping|legend=1| 1 -3 -5 -5 | 0 10 16 17 }}
+: mapping generators: ~2, ~7/5
 [[Optimal tuning]]s:
@@ Line 191: / Line 208: @@
 Comma list: 77/75, 243/242, 245/242
-Mapping: {{mapping| 1 7 11 12 17 | 0 -10 -16 -17 -25 }}
+Mapping: {{mapping| 1 -3 -5 -5 -8 | 0 10 16 17 25 }}
 Optimal tunings:
@@ Line 206: / Line 223: @@
 Comma list: 66/65, 77/75, 147/143, 243/242
-Mapping: {{mapping| 1 7 11 12 17 14 | 0 -10 -16 -17 -25 -19 }}
+Mapping: {{mapping| 1 -3 -5 -5 -8 -5 | 0 10 16 17 25 19 }}
 Optimal tunings: