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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. 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]].
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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: mapping generators: ~2, ~27/25
: 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]]:  
[[Tuning ranges]]:  
Line 21: Line 25:
* 5-odd-limit [[diamond tradeoff]]: [99.609, 105.214]  
* 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


[[Badness]] (Dirichlet): 3.259
=== 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 ==
{{ See also | Dual-fifth temperaments }}
{{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.
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]].''
''If you are looking for the former canonical extension, see [[#Rip]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 2 3 4 | 0 -5 -8 -14 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -5 -8 -14 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 101.538
* [[WE]]: ~2 = 1201.7546{{c}}, ~15/14 = 102.1309{{c}}
: [[error map]]: {{val| 0 -9.643 1.385 9.647 }}
: error map: {{val| +1.755 -9.100 +1.903 +8.360 }}
* [[CE]]: ~15/14 = 101.881
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 101.7772{{c}}
: [[error map]]: {{val| 0 -11.361 -1.364 4.837 }}
: error map: {{val| 0.000 -10.841 -0.531 +6.294 }}


{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}
{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}


[[Badness]] (Dirichlet): 1.521
[[Badness]] (Sintel): 1.52


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


[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


{{Mapping|legend=1| 1 2 3 4 5 | 0 -5 -8 -14 -18 }}
Mapping: {{mapping| 1 2 3 4 5 | 0 -5 -8 -14 -18 }}


[[Optimal tuning]]s:
Optimal tunings:
* [[CTE]]: ~15/14 = 101.538
* WE: ~2 = 1202.5973{{c}}, ~15/14 = 102.7900{{c}}
: [[error map]]: {{val| 0 -11.785, -2.041, 3.651, 13.296 }}
: error map: {{val| +2.597 -10.710 -0.842 +2.504 +11.449 }}
* [[CE]]: ~15/14 = 102.319
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 102.2972{{c}}
: [[error map]]: {{val| 0 -13.551 -4.868 -1.296 6.935 }}
: error map: {{val| 0.000 -13.441 -4.691 -0.986 +7.333 }}


{{Optimal ET sequence|legend=1| 11cdee, 12, 23de, 35 }}
{{Optimal ET sequence|legend=0| 11cdee, 12, 23de, 35 }}


[[Badness]] (Dirichlet): 1.334
Badness (Sintel): 1.33


== Rip ==
== Rip ==
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{{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }}
{{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}}
: 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 tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 99.483
{{Optimal ET sequence|legend=1| 11c, 12 }}


{{Optimal ET sequence|legend=1| 12 }}
[[Badness]] (Sintel): 1.51
 
[[Badness]] (Smith): 0.059735


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}
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 ===
=== 13-limit ===
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Mapping: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }}
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 ==
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 }}


{{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 ===
=== 11-limit ===
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Mapping: {{mapping| 1 2 3 3 4 | 0 -10 -16 -5 -13 }}
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 ===
=== 13-limit ===
Line 146: Line 172:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -10 -16 -5 -13 -7 }}
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 ==
== Cohemiripple ==
{{See also| Sensamagic clan }}
{{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
[[Subgroup]]: 2.3.5.7
Line 159: Line 189:
[[Comma list]]: 245/243, 1323/1250
[[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 }}
{{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}


[[Badness]] (Smith): 0.190208
[[Badness]] (Sintel): 4.81


=== 11-limit ===
=== 11-limit ===
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Comma list: 77/75, 243/242, 245/242
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 }}
{{Optimal ET sequence|legend=0| 11cdee, 13cdee, 24 }}


Badness (Smith): 0.082716
Badness (Sintel): 2.73


=== 13-limit ===
=== 13-limit ===
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Comma list: 66/65, 77/75, 147/143, 243/242
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 }}
{{Optimal ET sequence|legend=0| 11cdeef, 13cdeef, 24 }}


Badness (Smith): 0.049933
Badness (Sintel): 2.06


[[Category:Temperament families]]
[[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:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Ripple_family"