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Ripple family: Difference between revisions

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m Cleanup
+ CTE and CWE tunings; + error maps
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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:
* [[CTE]]: ~2 = 1200.000, ~27/25 = 100.752
: [[error map]]: {{val| 0.000 -5.717 +7.668 }}
* [[CWE]]: ~2 = 1200.000, ~27/25 = 100.798
: [[error map]]: {{val| 0.000 -5.946 +7.300 }}
* [[POTE]]: ~2 = 1200.000, ~27/25 = 100.838
: error map: {{val| 0.000 -6.145 +6.982 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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* 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]]:
 
* Smith 0.139
[[Badness]] (Dirichlet): 3.259
* Dirichlet: 3.26


== 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
Line 40: Line 46:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~15/14 = 101.538
* [[CTE]]: ~2 = 1200.000, ~15/14 = 101.538
: [[error map]]: {{val| 0 -9.643 1.385 9.647 }}
: [[error map]]: {{val| 0 -9.643 +1.385 +9.647 }}
* [[CE]]: ~15/14 = 101.881
* [[CWE]]: ~2 = 1200.000, ~15/14 = 101.777
: [[error map]]: {{val| 0 -11.361 -1.364 4.837 }}
: error map: {{val| 0 -10.841 -0.531 +6.294 }}
* [[CEE]]: ~2 = 1200.000, ~15/14 = 101.881
: error map: {{val| 0 -11.361 -1.364 +4.837 }}


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


[[Badness]] (Dirichlet): 1.521
[[Badness]]:
* Smith: 0.0601
* Dirichlet: 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
* CTE: ~2 = 1200.000, ~15/14 = 101.538
: [[error map]]: {{val| 0 -11.785, -2.041, 3.651, 13.296 }}
: error map: {{val| 0 -11.785 -2.041 +3.651 +13.296 }}
* [[CE]]: ~15/14 = 102.319
* CWE: ~2 = 1200.000, ~15/14 = 102.297
: [[error map]]: {{val| 0 -13.551 -4.868 -1.296 6.935 }}
: error map: {{val| 0 -13.441 -4.691 -0.986 +7.333 }}
* CEE: ~2 = 1200.000, ~15/14 = 102.319
: error map: {{val| 0 -13.551 -4.868 -1.296 +6.935 }}


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


[[Badness]] (Dirichlet): 1.334
Badness:
* Smith: 0.0403
* Dirichlet: 1.33


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 99.483
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~21/20 = 101.089
: [[error map]]: {{val| 0.000 -7.402 +4.970 +28.995 }}
* [[CWE]]: ~2 = 1200.000, ~21/20 = 100.109
: error map: {{val| 0.000 -2.501 +12.812 +30.956 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 99.483
: error map: {{val| 0.000 +0.632 +17.825 +32.209 }}


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


[[Badness]] (Smith): 0.059735
[[Badness]] (Smith): 0.0597


=== 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:
* CTE: ~2 = 1200.000, ~21/20 = 101.923
* CWE: ~2 = 1200.000, ~21/20 = 100.320
* POTE: ~2 = 1200.000, ~21/20 = 99.385


{{Optimal ET sequence|legend=0| 12 }}
{{Optimal ET sequence|legend=0| 11c, 12 }}


Badness (Smith): 0.038811
Badness (Smith): 0.0388


=== 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:
* CTE: ~2 = 1200.000, ~21/20 = 102.376
* CWE: ~2 = 1200.000, ~21/20 = 99.762
* POTE: ~2 = 1200.000, ~21/20 = 98.572


{{Optimal ET sequence|legend=0| 12f }}
{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }}


Badness (Smith): 0.031639
Badness (Smith): 0.0316


== Hemiripple ==
== Hemiripple ==
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{{Multival|legend=1| 10 16 5 2 -20 -33 }}
{{Multival|legend=1| 10 16 5 2 -20 -33 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~36/35 = 50.826
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~36/35 = 50.231
: [[error map]]: {{val| 0.000 -4.264 +9.991 -19.981 }}
* [[CWE]]: ~2 = 1200.000, ~36/35 = 50.593
: error map: {{val| 0.000 -7.883 +4.201 -21.790 }}
* [[POTE]]: ~2 = 1200.000, ~36/35 = 50.826
: error map: {{val| 0.000 -10.214 +0.472 -22.956 }}


{{Optimal ET sequence|legend=1| 23d, 24, 47d, 71bdd }}
{{Optimal ET sequence|legend=1| 23d, 24, 47d }}


[[Badness]] (Smith): 0.175113
[[Badness]] (Smith): 0.175


=== 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:
* CTE: ~2 = 1200.000, ~36/35 = 50.186
* CWE: ~2 = 1200.000, ~36/35 = 50.587
* POTE: ~2 = 1200.000, ~36/35 = 50.826


{{Optimal ET sequence|legend=0| 23de, 24, 47de, 71bdde }}
{{Optimal ET sequence|legend=0| 23de, 24, 47de }}


Badness (Smith): 0.066834
Badness (Smith): 0.0668


=== 13-limit ===
=== 13-limit ===
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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:
* CTE: ~2 = 1200.000, ~36/35 = 50.225
* CWE: ~2 = 1200.000, ~36/35 = 50.505
* POTE: ~2 = 1200.000, ~36/35 = 50.635


{{Optimal ET sequence|legend=0| 23de, 24, 47de, 71bdde }}
{{Optimal ET sequence|legend=0| 23de, 24 }}


Badness (Smith): 0.046588
Badness (Smith): 0.0466


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 549.944
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~7/5 = 550.063
: [[error map]]: {{val| 0.000 -1.328 +14.690 -17.760 }}
* [[CWE]]: ~2 = 1200.000, ~7/5 = 549.998
: error map: {{val| 0.000 -1.976 +13.653 -18.861 }}
* [[POTE]]: ~2 = 1200.000, ~7/5 = 549.944
: error map: {{val| 0.000 -2.515 +12.791 -19.777 }}


{{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}
{{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}


[[Badness]] (Smith): 0.190208
[[Badness]] (Smith): 0.190


=== 11-limit ===
=== 11-limit ===
Line 176: Line 220:
Mapping: {{mapping| 1 7 11 12 17 | 0 -10 -16 -17 -25 }}
Mapping: {{mapping| 1 7 11 12 17 | 0 -10 -16 -17 -25 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 549.945
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/5 = 550.060
* CWE: ~2 = 1200.000, ~7/5 = 549.997
* POTE: ~2 = 1200.000, ~7/5 = 549.945


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


Badness (Smith): 0.082716
Badness (Smith): 0.0827


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 7 11 12 17 14 | 0 -10 -16 -17 -25 -19 }}
Mapping: {{mapping| 1 7 11 12 17 14 | 0 -10 -16 -17 -25 -19 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 549.958
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/5 = 549.987
* CWE: ~2 = 1200.000, ~7/5 = 549.966
* POTE: ~2 = 1200.000, ~7/5 = 549.958


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


Badness (Smith): 0.049933
Badness (Smith): 0.0499


[[Category:Temperament families]]
[[Category:Temperament families]]
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