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Rastmic rank-3 clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus
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== Spectacle ==
== Spectacle ==
Spectacle, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17700.html Yahoo! Tuning Group | ''Spectacle temperament?'']</ref>, can be described as the {{nowrap| 31 & 34 & 41 }} temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.
Spectacle, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17700.html Yahoo! Tuning Group | ''Spectacle temperament?'']</ref>, can be described as the {{nowrap| 31 & 34d & 41 }} temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 19: Line 19:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.377{{c}}, ~5/4 = 384.370{{c}}
* [[WE]]: ~2 = 1200.5486{{c}}, ~11/9 = 350.2171{{c}}, ~5/4 = 384.1078{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.176{{c}}, ~5/4 = 384.095{{c}}
: [[error map]]: {{val| +0.549 -0.972 -1.109 +0.806 +0.864 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.1758{{c}}, ~5/4 = 384.0951{{c}}
: error map: {{val| 0.000 -1.603 -2.219 +0.068 -0.439 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.377{{c}}, ~5/4 = 384.370{{c}} -->


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~11/9 = {{monzo| -2/5 0 0 0 1/5 }}, ~5 = {{monzo| 2/5 -2 1 0 4/5 }}
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~11/9 = {{monzo| -2/5 0 0 0 1/5 }}, ~5 = {{monzo| 2/5 -2 1 0 4/5 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11


{{Optimal ET sequence|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=1| 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}


[[Badness]] (Smith): 0.499 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.599


[[Projection pair]]s: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9
[[Projection pair]]s: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9
Line 44: Line 47:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.239{{c}}, ~5/4 = 384.998{{c}}
* WE: ~2 = 1200.6024{{c}}, ~11/9 = 350.1004{{c}}, ~5/4 = 384.5435{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.039{{c}}, ~5/4 = 384.587{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.0393{{c}}, ~5/4 = 384.5866{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.239{{c}}, ~5/4 = 384.998{{c}} -->


{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=0| 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff }} <nowiki>*</nowiki>


<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
<nowiki>*</nowiki> optimal patent val: [[240edo|240]]


Badness (Smith): 1.009 × 10<sup>-3</sup>
Badness (Sintel): 0.944


== Cuckoo ==
== Cuckoo ==
Line 63: Line 67:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.355{{c}}, ~5/4 = 389.555{{c}}
* [[WE]]: ~2 = 1199.8222{{c}}, ~11/9 = 350.4356{{c}}, ~5/4 = 389.8478{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.421{{c}}, ~5/4 = 389.731{{c}}
: [[error map]]: {{val| -0.178 -1.262 +3.178 -1.558 +0.504 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.4213{{c}}, ~5/4 = 389.7308{{c}}
: error map: {{val| 0.000 -1.112 +3.417 -1.318 +0.788 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.355{{c}}, ~5/4 = 389.555{{c}} -->


{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}


[[Badness]] (Smith): 0.933 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.12


=== 13-limit ===
=== 13-limit ===
Line 78: Line 85:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.585{{c}}, ~5/4 = 389.302{{c}}
* WE: ~2 = 1199.7103{{c}}, ~11/9 = 350.5840{{c}}, ~5/4 = 389.8071{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.568{{c}}, ~5/4 = 389.610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.5682{{c}}, ~5/4 = 389.6104{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.585{{c}}, ~5/4 = 389.302{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
Badness (Sintel): 1.23


== Mirwomo ==
== Mirwomo ==
Line 95: Line 105:
: mapping generators: ~2, ~128/105, ~5
: mapping generators: ~2, ~128/105, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~128/105 = 350.174{{c}}, ~5/4 = 384.014{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8046{{c}}, ~11/9 = 350.3723{{c}}, ~5/4 = 384.1239{{c}}
: [[error map]]: {{val| +0.805 -0.406 -0.581 -0.848 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.1448{{c}}, ~5/4 = 383.8961{{c}}
: error map: {{val| 0.000 -1.665 -2.418 -3.157 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~128/105 = 350.174{{c}}, ~5/4 = 384.014{{c}} -->


{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d }}
{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d, 322cd, 353cd, 425bcdd, 497bcdd }}


[[Badness]] (Smith): 0.770 × 10<sup>-3</sup>
[[Badness]] (Sintel): 3.40


=== 11-limit ===
=== 11-limit ===
Line 109: Line 124:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.217{{c}}, ~5/4 = 383.962{{c}}
* [[WE]]: ~2 = 1200.7360{{c}}, ~11/9 = 350.1700{{c}}, ~5/4 = 384.3403{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.003{{c}}, ~5/4 = 384.078{{c}}
: [[error map]]: {{val| +0.736 -0.879 -0.501 -0.733 +1.004 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.0035{{c}}, ~5/4 = 384.0785{{c}}
: error map: {{val| 0.000 -1.948 -2.235 -2.915 -1.301 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.217{{c}}, ~5/4 = 383.962{{c}} -->


{{Optimal ET sequence|legend=1| 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde }}
{{Optimal ET sequence|legend=1| 17, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde }}


[[Badness]] (Smith): 0.641 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.770


== Mirage ==
== Mirage ==
Line 126: Line 144:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.711{{c}}, ~13/8 = 840.528{{c}}
* [[WE]]: ~2 = 1200.7626{{c}}, ~15/4 = 116.7069{{c}}, ~13/8 = 838.2364{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.647{{c}}, ~13/8 = 838.212{{c}}
: [[error map]]: {{val| +0.763 -0.951 -0.974 +0.048 +0.810 -0.004 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}, ~13/8 = 838.2123{{c}}
: error map: {{val| 0.000 -2.074 -2.842 -2.120 -1.615 -2.315 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.711{{c}}, ~13/8 = 840.528{{c}} -->


{{Optimal ET sequence|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
{{Optimal ET sequence|legend=1| 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}


Badness (Smith): 0.738 × 10<sup>-3</sup>
Badness (Sintel): 0.691


=== 17-limit===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 141: Line 162:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.704{{c}}, ~13/8 = 839.452{{c}}
* WE: ~2 = 1200.7628{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{c}}
* CWE: ~2 = 1200.000{{c}}, ~15/14 = 116.640{{c}}, ~13/8 = 837.142{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6395{{c}}, ~13/8 = 837.1424{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.704{{c}}, ~13/8 = 839.452{{c}} -->


{{Optimal ET sequence|legend=0| 10, 31, 41, 62, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff }}
 
Badness (Sintel): 0.715


== Mandos ==
== Mandos ==
Line 156: Line 180:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.145{{c}}, ~5/4 = 389.708{{c}}
* [[WE]]: ~2 = 1199.1949{{c}}, ~11/9 = 350.6135{{c}}, ~5/4 = 390.4090{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.555{{c}}, ~5/4 = 390.269{{c}}
: [[error map]]: {{val| -0.805 -1.533 +2.485 +1.814 +0.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5548{{c}}, ~5/4 = 390.2690{{c}}
: error map: {{val| 0.000 -0.845 +3.955 +3.410 +1.456 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.145{{c}}, ~5/4 = 389.708{{c}} -->


{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde }}


[[Badness]] (Smith): 0.751 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.902


=== 13-limit ===
=== 13-limit ===
Line 171: Line 198:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.293{{c}}, ~5/4 = 389.979{{c}}
* WE: ~2 = 1198.5555{{c}}, ~11/9 = 351.0300{{c}}, ~5/4 = 391.0458{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 351.155{{c}}, ~5/4 = 391.123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.1554{{c}}, ~5/4 = 391.1227{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.293{{c}}, ~5/4 = 389.979{{c}} -->


{{Optimal ET sequence|legend=0| 7, 24, 31, 58 }}
{{Optimal ET sequence|legend=0| 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff }}


Badness (Smith): 0.923 × 10<sup>-3</sup>
Badness (Sintel): 0.863


== Parahemif ==
== Parahemif ==
Line 188: Line 216:
: mapping generators: ~2, ~896/729, ~5
: mapping generators: ~2, ~896/729, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~896/729 = 351.416{{c}}, ~5/4 = 386.314{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7303{{c}}, ~896/729 = 351.4056{{c}}, ~5/4 = 386.8527{{c}}
: [[error map]]: {{val| -0.270 +0.586 -0.000 -0.284 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/729 = 351.4569{{c}}, ~5/4 = 386.6884{{c}}
: error map: {{val| 0.000 +0.959 +0.375 +0.114 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~896/729 = 351.416{{c}}, ~5/4 = 386.314{{c}} -->


{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}


[[Badness]] (Smith): 0.607 × 10<sup>-3</sup>
[[Badness]] (Sintel): 8.77


=== 11-limit ===
=== 11-limit ===
Line 202: Line 235:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 351.320{{c}}, ~5/4 = 386.314{{c}}
* [[WE]]: ~2 = 1199.2633{{c}}, ~11/9 = 351.3189{{c}}, ~5/4 = 387.7835{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 351.459{{c}}, ~5/4 = 387.423{{c}}
: [[error map]]: {{val| -0.737 -0.054 -0.004 -0.944 +3.803 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 351.4593{{c}}, ~5/4 = 387.4226{{c}}
: error map: {{val| 0.000 +0.964 +1.109 +0.145 +5.979 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 351.320{{c}}, ~5/4 = 386.314{{c}} -->


{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99e }} *
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99e }} *
Line 209: Line 245:
<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]
<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]


[[Badness]] (Smith): 1.34547 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.62


=== 13-limit ===
=== 13-limit ===
Line 219: Line 255:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 351.343{{c}}, ~5/4 = 386.314{{c}}
* WE: ~2 = 1198.7603{{c}}, ~11/9 = 351.3275{{c}}, ~5/4 = 388.7872{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 351.604{{c}}, ~5/4 = 388.472{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.6042{{c}}, ~5/4 = 388.4720{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 351.343{{c}}, ~5/4 = 386.314{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff }}
{{Optimal ET sequence|legend=0| 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff }}


Badness (Smith): 1.19366 × 10<sup>-3</sup>
Badness (Sintel): 1.12


== Urania ==
== Urania ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 232: Line 269:


{{Mapping|legend=1| 1 1 0 0 2 | 0 2 8 0 5 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 1 0 0 2 | 0 2 8 0 5 | 0 0 0 1 0 }}
: mapping generators: ~2, ~11/9, ~7


[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
Line 240: Line 279:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 348.830{{c}}, ~7/4 = 968.826{{c}}
* [[WE]]: ~2 = 1201.8548{{c}}, ~11/9 = 348.6318{{c}}, ~5/4 = 965.0936{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 348.379{{c}}, ~7/4 = 965.630{{c}}
: [[error map]]: {{val| +1.855 -2.836 +2.741 -0.023 -4.449 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.3793{{c}}, ~7/4 = 965.6304{{c}}
: error map: {{val| 0.000 -5.196 +0.721 -3.196 -9.421 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 348.830{{c}}, ~7/4 = 968.826{{c}} -->


{{Optimal ET sequence|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
{{Optimal ET sequence|legend=1| 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee }}


[[Badness]] (Smith): 0.842 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.01


[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
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