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Ripple family: Difference between revisions

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== Ripple ==
== 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]].
Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 100.7982{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 100.7982{{c}}
: [[error map]]: {{val| 0.000 -5.946 +7.300 }}
: [[error map]]: {{val| 0.000 -5.946 +7.300 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~27/25 = 100.752{{c}}
: error map: {{val| 0.000 -5.717 +7.668 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 100.838{{c}}
: error map: {{val| 0.000 -6.145 +6.982 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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[[Badness]] (Sintel): 3.26
[[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 ==
== Septimal ripple ==
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 101.7772{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 101.7772{{c}}
: error map: {{val| 0.000 -10.841 -0.531 +6.294 }}
: error map: {{val| 0.000 -10.841 -0.531 +6.294 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 101.538{{c}}
: error map: {{val| 0.000 -9.643 +1.385 +9.647 }}
* [[CEE]]: ~2 = 1200.000{{c}}, ~15/14 = 101.881{{c}}
: error map: {{val| 0.000 -11.361 -1.364 +4.837 }} -->


{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}
{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}
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* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 102.2972{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 102.2972{{c}}
: error map: {{val| 0.000 -13.441 -4.691 -0.986 +7.333 }}
: error map: {{val| 0.000 -13.441 -4.691 -0.986 +7.333 }}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~15/14 = 101.538{{c}}
: error map: {{val| 0.000 -11.785 -2.041 +3.651 +13.296 }}
* CEE: ~2 = 1200.000{{c}}, ~15/14 = 102.319{{c}}
: error map: {{val| 0.000 -13.551 -4.868 -1.296 +6.935 }} -->


{{Optimal ET sequence|legend=0| 11cdee, 12, 23de, 35 }}
{{Optimal ET sequence|legend=0| 11cdee, 12, 23de, 35 }}
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 100.1093{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 100.1093{{c}}
: error map: {{val| 0.000 -2.501 +12.812 +30.956 }}
: error map: {{val| 0.000 -2.501 +12.812 +30.956 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 101.089{{c}}
: error map: {{val| 0.000 -7.402 +4.970 +28.995 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 99.483{{c}}
: error map: {{val| 0.000 +0.632 +17.825 +32.209 }} -->


{{Optimal ET sequence|legend=1| 11c, 12 }}
{{Optimal ET sequence|legend=1| 11c, 12 }}
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* WE: ~2 = 1192.7877{{c}}, ~21/20 = 98.7876{{c}}
* WE: ~2 = 1192.7877{{c}}, ~21/20 = 98.7876{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 100.3202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 100.3202{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 101.923{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 99.385{{c}} -->


{{Optimal ET sequence|legend=0| 11c, 12 }}
{{Optimal ET sequence|legend=0| 11c, 12 }}
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* WE: ~2 = 1189.8521{{c}}, ~21/20 = 97.7384{{c}}
* WE: ~2 = 1189.8521{{c}}, ~21/20 = 97.7384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 99.7618{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 99.7618{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 102.376{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 98.572{{c}} -->


{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }}
{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }}
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== Hemiripple ==
== 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
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}
: mapping generators: ~2, ~36/35


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.5561{{c}}, ~36/35 = 50.9765{{c}}
* [[WE]]: ~2 = 1203.5561{{c}}, ~36/35 = 50.9765{{c}}
: error map: {{val| +3.556 -4.608 +8.730 -13.040 }}
: error map: {{val| +3.556 -4.608 +8.730 -13.040 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~36/35 = 50.5928{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 50.5928{{c}}
: error map: {{val| 0.000 -7.883 +4.201 -21.790 }}
: error map: {{val| 0.000 -7.883 +4.201 -21.790 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~36/35 = 50.231{{c}}
: [[error map]]: {{val| 0.000 -4.264 +9.991 -19.981 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 50.826{{c}}
: error map: {{val| 0.000 -10.214 +0.472 -22.956 }} -->


{{Optimal ET sequence|legend=1| 23d, 24, 47d }}
{{Optimal ET sequence|legend=1| 23d, 24, 47d }}
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* WE: ~2 = 1203.5344{{c}}, ~36/35 = 50.9757{{c}}
* WE: ~2 = 1203.5344{{c}}, ~36/35 = 50.9757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5870{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5870{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 50.186{{c}}
* POTE: ~2 = 1200.000{{c}}, ~36/35 = 50.826{{c}} -->


{{Optimal ET sequence|legend=0| 23de, 24, 47de }}
{{Optimal ET sequence|legend=0| 23de, 24, 47de }}
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* WE: ~2 = 1202.0936{{c}}, ~36/35 = 50.7232{{c}}
* WE: ~2 = 1202.0936{{c}}, ~36/35 = 50.7232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5048{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5048{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 50.225{{c}}
* POTE: ~2 = 1200.000{{c}}, ~36/35 = 50.635{{c}} -->


{{Optimal ET sequence|legend=0| 23de, 24 }}
{{Optimal ET sequence|legend=0| 23de, 24 }}
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Badness (Sintel): 1.93
Badness (Sintel): 1.93


== Cohemiripple ==
{{See also| Sensamagic clan }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 245/243, 1323/1250
{{Mapping|legend=1| 1 7 11 12 | 0 -10 -16 -17 }}
[[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~7/5 = 550.063{{c}}
: error map: {{val| 0.000 -1.328 +14.690 -17.760 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~7/5 = 549.944{{c}}
: error map: {{val| 0.000 -2.515 +12.791 -19.777 }} -->
{{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}
[[Badness]] (Sintel): 4.81
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 77/75, 243/242, 245/242
Mapping: {{mapping| 1 7 11 12 17 | 0 -10 -16 -17 -25 }}
Optimal tunings:
* WE: ~2 = 1200.6959{{c}}, ~7/5 = 550.2641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 549.9969{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/5 = 550.060{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/5 = 549.945{{c}} -->
{{Optimal ET sequence|legend=0| 11cdee, 13cdee, 24 }}
Badness (Sintel): 2.73
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 77/75, 147/143, 243/242
Mapping: {{mapping| 1 7 11 12 17 14 | 0 -10 -16 -17 -25 -19 }}
Optimal tunings:
* WE: ~2 = 1200.1161{{c}}, ~7/5 = 550.0107{{c}}
* CWE: ~2 = 1200.000{{c}}, ~7/5 = 549.9663{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/5 = 549.987{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/5 = 549.958{{c}} -->
{{Optimal ET sequence|legend=0| 11cdeef, 13cdeef, 24 }}
Badness (Sintel): 2.06
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Ripple family| ]] <!-- main article -->
[[Category:Ripple family| ]] <!-- main article -->
[[Category:Ripple| ]] <!-- key article -->
[[Category:Ripple| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
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