@@ Line 1: / Line 1: @@
-The '''ripple family''' of temperaments [[tempering out|tempers out]] the [[ripple comma]], 6561/6250 = {{monzo| -1 8 -5 }}, which equates a stack of five [[27/25]]'s with [[4/3]].
+{{Technical data page}}
+The '''ripple family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ripple comma]] ([[ratio]]: 6561/6250, {{monzo|legend=1| -1 8 -5 }}), which equates a stack of five [[27/25]]'s with [[4/3]].
 == Ripple ==
-The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. 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. 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]].
-A good patent val tuning not appearing in the optimal ET sequence is [[35edo]].
+Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].
 [[Subgroup]]: 2.3.5
@@ Line 14: / Line 15: @@
 : mapping generators: ~2, ~27/25
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 100.838
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2636{{c}}, ~27/25 = 100.8602{{c}}
+: [[error map]]: {{val| +0.264 -5.729 +7.596 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 100.7982{{c}}
+: [[error map]]: {{val| 0.000 -5.946 +7.300 }}
 [[Tuning ranges]]:
@@ Line 20: / Line 25: @@
 * 5-odd-limit [[diamond tradeoff]]: [99.609, 105.214]
-{{Optimal ET sequence|legend=1| 11c, 12, 71b, 83b, 95b, 107bc, 119bc }}
+{{Optimal ET sequence|legend=1| 11c, 12, 71b, 83b }}
-[[Badness]] (Smith): 0.138948
+[[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 ==
+{{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.
+''If you are looking for the former canonical extension, see [[#Rip]].''
+[[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:
+* [[WE]]: ~2 = 1201.7546{{c}}, ~15/14 = 102.1309{{c}}
+: error map: {{val| +1.755 -9.100 +1.903 +8.360 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 101.7772{{c}}
+: error map: {{val| 0.000 -10.841 -0.531 +6.294 }}
+{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}
+[[Badness]] (Sintel): 1.52
+=== 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: [[45/44]], [[99/98]], [[126/125]]
+Mapping: {{mapping| 1 2 3 4 5 | 0 -5 -8 -14 -18 }}
+Optimal tunings:
+* WE: ~2 = 1202.5973{{c}}, ~15/14 = 102.7900{{c}}
+: error map: {{val| +2.597 -10.710 -0.842 +2.504 +11.449 }}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 102.2972{{c}}
+: error map: {{val| 0.000 -13.441 -4.691 -0.986 +7.333 }}
+{{Optimal ET sequence|legend=0| 11cdee, 12, 23de, 35 }}
+Badness (Sintel): 1.33
+== Rip ==
+Formerly known as ''septimal 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 31: / Line 89: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }}
-{{Multival|legend=1| 5 8 2 1 -11 -18 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1195.0347{{c}}, ~21/20 = 99.0710{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 99.483
+: error map: {{val| -4.965 -7.240 +6.223 +18.136 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 100.1093{{c}}
+: error map: {{val| 0.000 -2.501 +12.812 +30.956 }}
-{{Optimal ET sequence|legend=1| 12 }}
+{{Optimal ET sequence|legend=1| 11c, 12 }}
-[[Badness]] (Smith): 0.059735
+[[Badness]] (Sintel): 1.51
 === 11-limit ===
@@ Line 46: / Line 106: @@
 Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 99.385
+Optimal tunings:
+* WE: ~2 = 1192.7877{{c}}, ~21/20 = 98.7876{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 100.3202{{c}}
-{{Optimal ET sequence|legend=0| 12 }}
+{{Optimal ET sequence|legend=0| 11c, 12 }}
-Badness (Smith): 0.038811
+Badness (Sintel): 1.28
 === 13-limit ===
@@ Line 59: / Line 121: @@
 Mapping: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 98.572
+Optimal tunings:
+* WE: ~2 = 1189.8521{{c}}, ~21/20 = 97.7384{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 99.7618{{c}}
-{{Optimal ET sequence|legend=0| 12f }}
+{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }}
-Badness (Smith): 0.031639
+Badness (Sintel): 1.31
 == 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 72: / Line 138: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}
-{{Multival|legend=1| 10 16 5 2 -20 -33 }}
+: mapping generators: ~2, ~36/35
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~36/35 = 50.826
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.5561{{c}}, ~36/35 = 50.9765{{c}}
+: error map: {{val| +3.556 -4.608 +8.730 -13.040 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 50.5928{{c}}
+: error map: {{val| 0.000 -7.883 +4.201 -21.790 }}
-{{Optimal ET sequence|legend=1| 23d, 24, 47d, 71bdd }}
+{{Optimal ET sequence|legend=1| 23d, 24, 47d }}
-[[Badness]] (Smith): 0.175113
+[[Badness]] (Sintel): 4.43
 === 11-limit ===
@@ Line 87: / Line 157: @@
 Mapping: {{mapping| 1 2 3 3 4 | 0 -10 -16 -5 -13 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~36/35 = 50.826
+Optimal tunings:
+* WE: ~2 = 1203.5344{{c}}, ~36/35 = 50.9757{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5870{{c}}
-{{Optimal ET sequence|legend=0| 23de, 24, 47de, 71bdde }}
+{{Optimal ET sequence|legend=0| 23de, 24, 47de }}
-Badness (Smith): 0.066834
+Badness (Sintel): 2.21
 === 13-limit ===
@@ Line 100: / Line 172: @@
 Mapping: {{mapping| 1 2 3 3 4 4 | 0 -10 -16 -5 -13 -7 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~36/35 = 50.635
+Optimal tunings:
+* WE: ~2 = 1202.0936{{c}}, ~36/35 = 50.7232{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5048{{c}}
-{{Optimal ET sequence|legend=0| 23de, 24, 47de, 71bdde }}
+{{Optimal ET sequence|legend=0| 23de, 24 }}
-Badness (Smith): 0.046588
+Badness (Sintel): 1.93
 == 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 113: / 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 }}
-{{Multival|legend=1| 10 16 17 2 -1 -5 }}
+: mapping generators: ~2, ~7/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 549.944
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.6977{{c}}, ~7/5 = 550.2638{{c}}
+: error map: {{val| +0.698 -1.410 +14.418 -17.830 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/5 = 549.9979{{c}}
+: error map: {{val| 0.000 -1.976 +13.653 -18.861 }}
 {{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}
-[[Badness]] (Smith): 0.190208
+[[Badness]] (Sintel): 4.81
 === 11-limit ===
@@ Line 128: / 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 tuning (POTE): ~2 = 1200.000, ~7/5 = 549.945
+Optimal tunings:
+* WE: ~2 = 1200.6959{{c}}, ~7/5 = 550.2641{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 549.9969{{c}}
 {{Optimal ET sequence|legend=0| 11cdee, 13cdee, 24 }}
-Badness (Smith): 0.082716
+Badness (Sintel): 2.73
 === 13-limit ===
@@ Line 141: / 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 tuning (POTE): ~2 = 1200.000, ~7/5 = 549.958
+Optimal tunings:
+* WE: ~2 = 1200.1161{{c}}, ~7/5 = 550.0107{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 549.9663{{c}}
 {{Optimal ET sequence|legend=0| 11cdeef, 13cdeef, 24 }}
-Badness (Smith): 0.049933
+Badness (Sintel): 2.06
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Ripple family| ]] <!-- main article -->
 [[Category:Ripple| ]] <!-- key article -->
 [[Category:Rank 2]]

Ripple family: Difference between revisions