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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
- no-19 23-limit mirage (no need). Request completion of data
 
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* [[Jove]] (+441/440 or 540/539) → [[Breed family #Jove|Breed family]]
* [[Jove]] (+441/440 or 540/539) → [[Breed family #Jove|Breed family]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Rabic]]'' (+131769/131072) → [[Alphaxenic rank-3 clan #Betarabian|Alphaxenic rank-3 clan]]
 
Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, rabic, and mirage.


== 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 & 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.
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.
It is [[associated temperament|associated]] with the [[marvo]] temperamment.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 15: Line 18:


{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 4 5 | 0 0 1 2 0 }}
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 4 5 | 0 0 1 2 0 }}
: mapping generators: ~2, ~11/9, ~5
: mapping generators: ~2, ~11/9, ~5


Line 23: Line 25:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.1758{{c}}, ~5/4 = 384.0951{{c}}
* [[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 }}
: 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]]:
Line 29: Line 30:
: [[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, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=1| 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}


[[Badness]] (Sintel): 0.599
[[Badness]] (Sintel): 0.599
Line 36: Line 37:


Scales: [[spectacle31]]
Scales: [[spectacle31]]
[[Associated temperament]]: [[marvo]]


=== 13-limit ===
=== 13-limit ===
Line 49: Line 48:
* WE: ~2 = 1200.6024{{c}}, ~11/9 = 350.1004{{c}}, ~5/4 = 384.5435{{c}}
* WE: ~2 = 1200.6024{{c}}, ~11/9 = 350.1004{{c}}, ~5/4 = 384.5435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.0393{{c}}, ~5/4 = 384.5866{{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, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=0| 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff }} <nowiki>*</nowiki>
Line 56: Line 54:


Badness (Sintel): 0.944
Badness (Sintel): 0.944
== Cuckoo ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 126/125, 243/242
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
: mapping generators: ~2, ~11/9, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8222{{c}}, ~11/9 = 350.4356{{c}}, ~5/4 = 389.8478{{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 }}
[[Badness]] (Sintel): 1.12
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 243/242
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}
Optimal tunings:
* WE: ~2 = 1199.7103{{c}}, ~11/9 = 350.5840{{c}}, ~5/4 = 389.8071{{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 }}
Badness (Sintel): 1.23


== Mirwomo ==
== Mirwomo ==
The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Mirwomo]].''
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 33075/32768
 
{{Mapping|legend=1| 1 1 0 6 | 0 2 0 -3 | 0 0 1 -1 }}
 
: mapping generators: ~2, ~128/105, ~5
 
[[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, 322cd, 353cd, 425bcdd, 497bcdd }}


[[Badness]] (Sintel): 3.40
Mirwomo tempers out 385/384 and may be described as the {{nowrap| 24 & 31 & 41 }} temperament, equating the undecimal quartertone ~33/32 with the septimal quartertone ~36/35.  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 122: Line 65:


{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 -3 5 | 0 0 1 -1 0 }}
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 -3 5 | 0 0 1 -1 0 }}
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 128: Line 72:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.0035{{c}}, ~5/4 = 384.0785{{c}}
* [[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 }}
: 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, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde }}
{{Optimal ET sequence|legend=1| 17, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde }}
Line 134: Line 77:
[[Badness]] (Sintel): 0.770
[[Badness]] (Sintel): 0.770


== Mirage ==
== Mandos ==
[[Subgroup]]: 2.3.5.7.11.13
Mandos tempers out 176/175 and may be described as the {{nowrap| 24 & 27e & 31 }} temperament.
 
[[Comma list]]: 225/224, 243/242, 385/384
 
{{Mapping|legend=1| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
 
: mapping generators: ~2, ~15/14, ~13


[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7626{{c}}, ~15/4 = 116.7069{{c}}, ~13/8 = 838.2364{{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| 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
Badness (Sintel): 0.691
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 243/242, 273/272, 385/384
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 0 4 | 0 0 0 0 0 1 1 }}
Optimal tunings:
* WE: ~2 = 1200.7628{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{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| 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff }}
Badness (Sintel): 0.715
== Mandos ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 176: Line 85:


{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}
: mapping generators: ~2, ~11/9, ~5
: mapping generators: ~2, ~11/9, ~5


Line 184: Line 92:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5548{{c}}, ~5/4 = 390.2690{{c}}
* [[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 }}
: 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| 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde }}
{{Optimal ET sequence|legend=1| 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde }}
Line 200: Line 107:
* WE: ~2 = 1198.5555{{c}}, ~11/9 = 351.0300{{c}}, ~5/4 = 391.0458{{c}}
* WE: ~2 = 1198.5555{{c}}, ~11/9 = 351.0300{{c}}, ~5/4 = 391.0458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.1554{{c}}, ~5/4 = 391.1227{{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| 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff }}
{{Optimal ET sequence|legend=0| 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff }}
Line 206: Line 112:
Badness (Sintel): 0.863
Badness (Sintel): 0.863


== Parahemif ==
== Cuckoo ==
=== 7-limit ===
Cuckoo, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref>, tempers out 126/125 and may be described as the {{nowrap| 24d & 27e & 31 }} temperament.  
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1605632/1594323
[[Subgroup]]: 2.3.5.7.11


{{Mapping|legend=1| 1 1 0 -1 | 0 2 0 13 | 0 0 1 0 }}
[[Comma list]]: 126/125, 243/242


: mapping generators: ~2, ~896/729, ~5
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7303{{c}}, ~896/729 = 351.4056{{c}}, ~5/4 = 386.8527{{c}}
* [[WE]]: ~2 = 1199.8222{{c}}, ~11/9 = 350.4356{{c}}, ~5/4 = 389.8478{{c}}
: [[error map]]: {{val| -0.270 +0.586 -0.000 -0.284 }}
: [[error map]]: {{val| -0.178 -1.262 +3.178 -1.558 +0.504 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/729 = 351.4569{{c}}, ~5/4 = 386.6884{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.4213{{c}}, ~5/4 = 389.7308{{c}}
: error map: {{val| 0.000 +0.959 +0.375 +0.114 }}
: error map: {{val| 0.000 -1.112 +3.417 -1.318 +0.788 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~896/729 = 351.416{{c}}, ~5/4 = 386.314{{c}} -->
 
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
 
[[Badness]] (Sintel): 1.12
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}
Comma list: 126/125, 196/195, 243/242


[[Badness]] (Sintel): 8.77
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.7103{{c}}, ~11/9 = 350.5840{{c}}, ~5/4 = 389.8071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.5682{{c}}, ~5/4 = 389.6104{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
 
Badness (Sintel): 1.23
 
== Parahemif ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Parahemif]].''
 
Parahemif tempers out 896/891 and may be described as the {{nowrap| 24 & 34d & 41 }} temperament. It is related to [[hemif]], the no-5 rank-2 temperament that tempers out the same list of commas. As such, it finds the interval class of 7 at +13 generator steps, as a semi-augmented sixth (C–At). In the 13-limit, it tempers out 144/143, 352/351, 364/363 among others, and finds ~16/13 at the same neutral third as ~11/9.


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 233: Line 157:


{{Mapping|legend=1| 1 1 0 -1 2 | 0 2 0 13 5 | 0 0 1 0 0 }}
{{Mapping|legend=1| 1 1 0 -1 2 | 0 2 0 13 5 | 0 0 1 0 0 }}
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 239: Line 164:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 351.4593{{c}}, ~5/4 = 387.4226{{c}}
* [[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 }}
: 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 257: Line 181:
* WE: ~2 = 1198.7603{{c}}, ~11/9 = 351.3275{{c}}, ~5/4 = 388.7872{{c}}
* WE: ~2 = 1198.7603{{c}}, ~11/9 = 351.3275{{c}}, ~5/4 = 388.7872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.6042{{c}}, ~5/4 = 388.4720{{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 }}
Line 264: Line 187:


== Urania ==
== Urania ==
Urania tempers out 81/80, the syntonic comma. It is essentially [[mohaha]] with an independent generator for prime 7.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 269: Line 194:


{{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 generators: ~2, ~11/9, ~7


Line 283: Line 207:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.3793{{c}}, ~7/4 = 965.6304{{c}}
* [[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 }}
: 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| 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee }}
{{Optimal ET sequence|legend=1| 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee }}
Line 292: Line 215:


Scales: [[urania24]]
Scales: [[urania24]]
== Rabic ==
If the rastma is added to the list of commas along with the [[Alpharabian comma]], you end up with rabic, which splits the octave into 24 equal parts. This temperament is named as such by [[Aura]] in 2022 because tempering out both the Alpharabian comma and the rastma automatically tempers out the [[Betarabian comma]].
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 243/242, 131769/131072
{{Mapping|legend=1| 24 38 0 0 83 | 0 0 1 0 0 | 0 0 0 1 0 }}
: mapping generators: ~33/32, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~33/32 = 50.0220{{c}}, ~5/4 = 385.0647{{c}}, ~7/4 = 967.3158{{c}}
: [[error map]]: {{val| +0.538 -1.104 -0.002 -0.002 +0.541 }}
* [[CWE]]: ~33/32 = 50.0000{{c}}, ~5/4 = 385.0647{{c}}, ~7/4 = 967.3158{{c}}
: error map: {{val| 0.000 -1.955 -0.937 -1.133 -1.318 }}
{{Optimal ET sequence|legend=1| 24, 48(d), 72, 264, 336b, 408b, 480bcde }}
[[Badness]] (Sintel): 7.30
=== Gunn ===
{{Todo|inline=1|complete section|unify precision|comment= Sort the comma basis. Normalize the mapping. Unify the cent values in optimal tunings to 4 decimal places. Add the optimal ET sequence. <br>Add the data for the 13- and 17-limit versions. }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 729/728, 273/272, 153/152, 364/363, 1729/1728
Mapping: {{mapping| 24 38 48 48 83 108 98 102 | 0 0 1 0 0 0 0 0 | 0 0 0 1 0 -1 0 0 }}
Optimal tunings:
* WE: ~33/32 = 50.030{{c}}, ~5/4 = 384.859{{c}}, ~7/4 = 965.679{{c}}
: error map: {{val| +0.724 -0.809 -0.007 -1.700 +1.185 -2.949 -2.000 +5.563 }}
* CWE: ~33/32 = 50.000{{c}}, ~5/4 = 384.837{{c}}, ~7/4 = 965.136{{c}}
: error map: {{val| -0.000 -1.955 -1.476 -3.690 -1.318 -5.663 -4.955 2.487 }}
Supporting ETs: {{Edos| 24p, 72, 48f, 24c, 24df, 48cf, 48d, 96d, 48cd, 24cdf }}
Badness (Sintel): 3.285
== Mirage ==
Mirage is [[miracle]] with an independent generator for prime 13.
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 225/224, 243/242, 385/384
{{Mapping|legend=1| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
: mapping generators: ~2, ~15/14, ~13
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}, ~13/8 = 838.2364{{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 }}
{{Optimal ET sequence|legend=1| 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
Badness (Sintel): 0.691
=== 17-limit ===
Mirage is very naturally a 17-limit temperament, relating 13 and 17 by tempering out [[273/272]], [[715/714]], [[833/832]], and [[936/935]]. Instead of 13/8, the second generator can also be the small comma tempered out by [[miraculous]].
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 243/242, 273/272, 385/384
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 0 4 | 0 0 0 0 0 1 1 }}
Optimal tunings:
* WE: ~2 = 1200.7628{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6395{{c}}, ~13/8 = 837.1424{{c}}
{{Optimal ET sequence|legend=0| 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff }}
Badness (Sintel): 0.715
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 210/209, 225/224, 243/242, 273/272, 343/342
{{Todo|complete temperament data|inline=1}}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384
{{Todo|complete temperament data|inline=1}}
=== Prism ===
{{Redirect|Prism|the scale|Prism (scale)}}
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 225/224, 243/242, 273/272, 324/323, 385/384
{{Todo|complete temperament data|inline=1}}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 225/224, 243/242, 273/272, 300/299, 324/323, 385/384
{{Todo|complete temperament data|inline=1}}


== References ==
== References ==


[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rastmic| ]] <!-- key article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Rastmic_rank-3_clan"