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The '''ripple family''' tempers out the ripple, 6561/6250 = {{monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]]. As one might expect, [[12edo]] is about as accurate as it can be.  
{{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 =
== 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]].


Period: 1\1
Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].


Optimal ([[POTE]]) generators: ~27/25 = 100.838
[[Subgroup]]: 2.3.5


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
[[Comma list]]: 6561/6250


Scales:
{{Mapping|legend=1| 1 2 3 | 0 -5 -8 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
: mapping generators: ~2, ~27/25
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 6561/6250
[[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 }}


Mapping: [{{val| 1 2 3 }}, {{val| 0 -5 -8 }}]
[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: [92.308, 109.091] (1\13 to 1\11)
* 5-odd-limit [[diamond tradeoff]]: [99.609, 105.214]  


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


Badness: 0.1389
[[Badness]] (Sintel): 3.26


</div></div>
=== 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]];


= 7-limit =
Both use the same nominal generator as ripple.
{{see also| Mint temperaments #Ripple }}


Period: 1\1
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.


Optimal ([[POTE]]) generators: ~21/20 = 99.483
== Septimal ripple ==
{{See also| Dual-fifth temperaments }}


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
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.


Scales:
''If you are looking for the former canonical extension, see [[#Rip]].''


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 36/35, 2560/2401
[[Comma list]]: [[126/125]], [[405/392]]


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


{{Multival|legend=1| 5 8 2 1 -11 -18 }}
[[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 }}


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


Badness: 0.0597
[[Badness]] (Sintel): 1.52


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


== 11-limit ==
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: [[45/44]], [[99/98]], [[126/125]]


Optimal ([[POTE]]) generators: ~21/20 = 99.385
Mapping: {{mapping| 1 2 3 4 5 | 0 -5 -8 -14 -18 }}


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
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 }}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.33
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 36/35, 80/77, 126/121
== 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.


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -5 -8 -2 -6 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 12 }}
[[Comma list]]: 36/35, 2560/2401


Badness: 0.0388
{{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }}


</div></div>
[[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 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 11c, 12 }}


Period: 1\1
[[Badness]] (Sintel): 1.51


Optimal ([[POTE]]) generators: ~21/20 = 98.572
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
Comma list: 36/35, 80/77, 126/121
 
Scales:


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 36/35, 40/39, 66/65, 147/143
Optimal tunings:  
* WE: ~2 = 1192.7877{{c}}, ~21/20 = 98.7876{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 100.3202{{c}}


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -5 -8 -2 -6 -3 }}]
{{Optimal ET sequence|legend=0| 11c, 12 }}


{{Val list|legend=1| 12f }}
Badness (Sintel): 1.28


Badness: 0.0316
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


</div></div>
Comma list: 36/35, 40/39, 66/65, 147/143


= Hemiripple =
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }}
{{see also| Slendro clan #Hemiripple }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1189.8521{{c}}, ~21/20 = 97.7384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 99.7618{{c}}


Optimal ([[POTE]]) generator: ~36/35 = 50.826
{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }}


EDO generators: [[23edo|1\23]], [[24edo|1\24]]
Badness (Sintel): 1.31


Scales (Scala files):
== Hemiripple ==
Hemiripple tempers out 49/48 and splits the semitone generator in two for ~36/35. Its ploidacot is omega-decacot.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Comma list]]: 49/48, 6561/6250


Comma list: 49/48, 6561/6250
{{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}


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


Wedgie: {{wedgie| 10 16 5 2 -20 -33 }}
[[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 }}


{{Val list|legend=1| 23d, 24, 47d, 71bd }}
{{Optimal ET sequence|legend=1| 23d, 24, 47d }}


Badness: 0.1751
[[Badness]] (Sintel): 4.43
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~36/35 = 50.826
 
EDO generators: [[23edo|1\23]], [[24edo|1\24]]
 
Scales (Scala files):  
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 121/120, 567/550
Comma list: 49/48, 121/120, 567/550


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -10 -16 -5 -13 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -10 -16 -5 -13 }}


{{Val list|legend=1| 23de, 24, 47de, 71bde }}
Optimal tunings:
* WE: ~2 = 1203.5344{{c}}, ~36/35 = 50.9757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5870{{c}}


Badness: 0.0668
{{Optimal ET sequence|legend=0| 23de, 24, 47de }}


</div></div>
Badness (Sintel): 2.21
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~36/35 = 50.635
 
EDO generators: [[23edo|1\23]], [[24edo|1\24]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 66/65, 121/120, 351/350
Comma list: 49/48, 66/65, 121/120, 351/350


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -10 -16 -5 -13 -7 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -10 -16 -5 -13 -7 }}
 
{{Val list|legend=1| 23de, 24, 47de, 71bde }}
 
Badness: 0.0466
 
</div></div>
 
= Cohemiripple =
{{see also| Sensamagic clan #Cohemiripple }}
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~7/5 = 549.944
 
EDO generators: [[11edo|5\11]], [[13edo|6\13]], [[12edo|11\24]]
 
Scales:


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1202.0936{{c}}, ~36/35 = 50.7232{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5048{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 23de, 24 }}


Comma list: 245/243, 1323/1250
Badness (Sintel): 1.93


Mapping: [{{val| 1 7 11 12 }}, {{val| 0 -10 -16 -17 }}]
== Cohemiripple ==
{{See also| Sensamagic clan }}


{{Multival|legend=1| 10 16 17 2 -1 -5 }}
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.


{{Val list|legend=1| 11cd, 13cd, 24 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.1902
[[Comma list]]: 245/243, 1323/1250


</div></div>
{{Mapping|legend=1| 1 -3 -5 -5 | 0 10 16 17 }}


== 11-limit ==
: mapping generators: ~2, ~7/5


Period: 1\1
[[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 ([[POTE]]) generators: ~7/5 = 549.945
{{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}


EDO generators: [[11edo|5\11]], [[13edo|6\13]], [[12edo|11\24]]
[[Badness]] (Sintel): 4.81
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 7 11 12 17 }}, {{val| 0 -10 -16 -17 -25 }}]
Mapping: {{mapping| 1 -3 -5 -5 -8 | 0 10 16 17 25 }}
 
{{Val list|legend=1| 11cdee, 13cdee, 24 }}
 
Badness: 0.0827
 
</div></div>
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~7/5 = 549.958


EDO generators: [[11edo|5\11]], [[13edo|6\13]], [[12edo|11\24]]
Optimal tunings:  
* WE: ~2 = 1200.6959{{c}}, ~7/5 = 550.2641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 549.9969{{c}}


Scales:
{{Optimal ET sequence|legend=0| 11cdee, 13cdee, 24 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.73
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 7 11 12 17 14 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
Mapping: {{mapping| 1 -3 -5 -5 -8 -5 | 0 10 16 17 25 19 }}


{{Val list|legend=1| 11cdeef, 13cdeef, 24 }}
Optimal tunings:
* WE: ~2 = 1200.1161{{c}}, ~7/5 = 550.0107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 549.9663{{c}}


Badness: 0.0499
{{Optimal ET sequence|legend=0| 11cdeef, 13cdeef, 24 }}


</div></div>
Badness (Sintel): 2.06


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Ripple family| ]] <!-- main article -->
[[Category:Ripple]]
[[Category:Ripple| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Ripple_family"