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This is a miscellaneous collection of rank-3 rastmic temperaments, tempering out the rastma, [[243/242]]. These temperaments divide the fifth in half and use it as an [[11/9]] neutral third.  
{{Technical data page}}
The '''rastmic rank-3 clan''' of temperaments tempers out the rastma, [[243/242]]. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an [[11/9]] neutral third.  


Temperaments not listed include:  
Temperaments discussed elsewhere include:  
* ''[[Urania]]'', {81/80, 121/120} → [[Didymus rank three family #Urania]]
* [[Jove]] (+441/440 or 540/539) → [[Breed family #Jove|Breed family]]
* ''[[Mandos]]'', {176/175, 243/242} → [[Valinorismic temperaments #Mandos]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Spectacle]]'', {225/224, 243/242} → [[Marvel family #Spectacle]]
* ''[[Rabic]]'' (+131769/131072) → [[Alphaxenic rank three clan #Betarabian|Alphaxenic rank-3 clan]]
* ''[[Mirage]]'', {225/224, 243/242, 385/384} → [[Marvel family #Mirage]]
 
* [[Jove]], {243/242, 441/440} → [[Breed family #Jove]]
== Spectacle ==
* ''[[Hagrid]]'', {243/242, 9801/9800} → [[Cataharry family #Hagrid]]
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 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 two 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
 
[[Comma list]]: 225/224, 243/242
 
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 4 5 | 0 0 1 2 0 }}
 
: mapping generators: ~2, ~11/9, ~5
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.377, ~5/4 = 384.370
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.176, ~5/4 = 384.095
 
[[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 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
 
{{Optimal ET sequence|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
 
[[Badness]]: 0.499 × 10<sup>-3</sup>
 
[[Projection pair]]s: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9
 
Scales: [[spectacle31]]
 
[[Associated temperament]]: [[marvo]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 351/350
 
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 4 5 -2 | 0 0 1 2 0 4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/9 = 350.239, ~5/4 = 384.998
* CWE: ~2 = 1200.000, ~11/9 = 350.039, ~5/4 = 384.587
 
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} <nowiki>*</nowiki>
 
<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
 
Badness: 1.009 × 10<sup>-3</sup>
 
== 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:
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.355, ~5/4 = 389.555
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.421, ~5/4 = 389.731
 
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
 
[[Badness]]: 0.933 × 10<sup>-3</sup>
 
=== 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:
* CTE: ~2 = 1200.000, ~11/9 = 350.585, ~5/4 = 389.302
* CWE: ~2 = 1200.000, ~11/9 = 350.568, ~5/4 = 389.610
 
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}


== Mirwomo ==
== Mirwomo ==
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=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 33075/32768
[[Comma list]]: 33075/32768


[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 0 -3 }}, {{val| 0 0 1 -1 }}]
{{Mapping|legend=1| 1 1 0 6 | 0 2 0 -3 | 0 0 1 -1 }}
 
: mapping generators: ~2, ~128/105, ~5
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/105 = 350.174, ~5/4 = 384.014


{{Val list|legend=1| 17, 21, 24, 31, 41, 72, 281d }}
{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d }}
 
[[Badness]]: 0.770 × 10<sup>-3</sup>


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


[[Comma list]]: 243/242, 385/384
[[Comma list]]: 243/242, 385/384


[[Mapping]]: [{{val| 1 1 0 6 2 }}, {{val| 0 2 0 -3 5 }}, {{val| 0 0 1 -1 0 }}]
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 -3 5 | 0 0 1 -1 0 }}


{{Val list|legend=1| 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.217, ~5/4 = 383.962
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.003, ~5/4 = 384.078
 
{{Optimal ET sequence|legend=1| 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde }}
 
[[Badness]]: 0.641 × 10<sup>-3</sup>
 
== Mirage ==
[[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:
* [[CTE]]: ~2 = 1200.000, ~15/14 = 116.711, ~13/8 = 840.528
* [[CWE]]: ~2 = 1200.000, ~15/14 = 116.647, ~13/8 = 838.212
 
{{Optimal ET sequence|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
 
Badness: 0.738 × 10<sup>-3</sup>
 
=== 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:
* CTE: ~2 = 1200.000, ~15/14 = 116.704, ~13/8 = 839.452
* CWE: ~2 = 1200.000, ~15/14 = 116.640, ~13/8 = 837.142
 
{{Optimal ET sequence|legend=0| 10, 31, 41, 62, 72, 103, 175f }}
 
== Mandos ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 176/175, 243/242
 
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}
 
: mapping generators: ~2, ~11/9, ~5
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.145, ~5/4 = 389.708
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.555, ~5/4 = 390.269
 
{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
 
[[Badness]]: 0.751 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 243/242
 
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/9 = 350.293, ~5/4 = 389.979
* CWE: ~2 = 1200.000, ~11/9 = 351.155, ~5/4 = 391.123
 
{{Optimal ET sequence|legend=0| 7, 24, 31, 58 }}
 
Badness: 0.923 × 10<sup>-3</sup>


== Parahemif ==
== Parahemif ==
=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 0 13 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 1 1 0 -1 | 0 2 0 13 | 0 0 1 0 }}


{{Val list|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}
: mapping generators: ~2, ~896/729, ~5
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~896/729 = 351.416, ~5/4 = 386.314
 
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}


[[Badness]]: 0.607 × 10<sup>-3</sup>
[[Badness]]: 0.607 × 10<sup>-3</sup>


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


[[Comma list]]: 243/242, 896/891
[[Comma list]]: 243/242, 896/891


[[Mapping]]: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 0 13 5 }}, {{val| 0 0 1 0 0 }}]
{{Mapping|legend=1| 1 1 0 -1 2 | 0 2 0 13 5 | 0 0 1 0 0 }}


{{Val list|legend=1| 17c, 24, 34d, 41, 58, 99e }} *
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~11/9 = 351.320, ~5/4 = 386.314
* [[CWE]]: ~2 = 1200.000, ~11/9 = 351.459, ~5/4 = 387.423
 
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99e }} *


<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]
<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]
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Comma list: 144/143, 243/242, 364/363
Comma list: 144/143, 243/242, 364/363


Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 0 13 5 -1 }}, {{val| 0 0 1 0 0 0 }}]
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 | 0 0 1 0 0 0 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/9 = 351.343, ~5/4 = 386.314
* CWE: ~2 = 1200.000, ~11/9 = 351.604, ~5/4 = 388.472


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


Badness: 1.19366 × 10<sup>-3</sup>
Badness: 1.19366 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
== Urania  ==
[[Category:Temperament collection]]
[[Subgroup]]: 2.3.5.7.11
[[Category:Rastmic temperaments| ]] <!-- main article -->
 
[[Comma list]]: 81/80, 121/120
 
{{Mapping|legend=1| 1 1 0 0 2 | 0 2 8 0 5 | 0 0 0 1 0 }}
 
[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
 
Lattice basis:
: 11/9 length = 0.2536, 8/7 length = 2.807
: Angle (11/9, 8/7) = 90 degrees
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~11/9 = 348.830, ~7/4 = 968.826
* [[CWE]]: ~2 = 1200.000, ~11/9 = 348.379, ~7/4 = 965.630
 
{{Optimal ET sequence|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
 
[[Badness]]: 0.842 × 10<sup>-3</sup>
 
[[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
 
Scales: [[urania24]]
 
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rastmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Rastmic_rank_three_clan"