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Rastmic rank three clan: Difference between revisions

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{{Technical data page}}<br><br>
{{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.  
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.  


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: mapping generators: ~2, ~11/9, ~5
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 350.0570, ~5/4 = 383.9323
[[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]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[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 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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, 72, 247c, 281, 353c, 425bc, 497bc }}
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[[Badness]]: 0.499 × 10<sup>-3</sup>
[[Badness]]: 0.499 × 10<sup>-3</sup>


[[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


Scales: [[spectacle31]]
Scales: [[spectacle31]]
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Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 4 5 -2 | 0 0 1 2 0 4 }}
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 4 5 -2 | 0 0 1 2 0 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9247, ~5/4 = 384.3505
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=1| 31, 72, 103, 175f }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} <nowiki>*</nowiki>


<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
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: mapping generators: ~2, ~11/9, ~5
: 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 }}
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
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Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}


{{Optimal ET sequence|legend=1| 27e, 31, 58, 96d, 154 }}
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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: mapping generators: ~2, ~128/105, ~5
: mapping generators: ~2, ~128/105, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 350.1375, ~5/4 = 383.8665
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/105 = 350.174, ~5/4 = 384.014


{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d }}
{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d }}
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{{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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 349.9554, ~5/4 = 384.1048
[[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 }}
{{Optimal ET sequence|legend=1| 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde }}
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== Mirage ==
== Mirage ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 385/384
[[Comma list]]: 225/224, 243/242, 385/384


Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
{{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
: mapping generators: ~2, ~15/14, ~13


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.6327, ~13/8 = 837.7040
[[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 }}
{{Optimal ET sequence|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}


Badness: 0.738 × 10<sup>-3</sup>
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 ==
== Mandos ==
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: mapping generators: ~2, ~11/9, ~5
: 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 }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
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Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}


{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }}
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>
Badness: 0.923 × 10<sup>-3</sup>
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: mapping generators: ~2, ~896/729, ~5
: mapping generators: ~2, ~896/729, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~896/729 = 351.4846, ~5/4 = 386.9396
[[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 }}
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}
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{{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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 351.5347, ~5/4 = 388.0217
[[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 }} *
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99e }} *
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Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 | 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 tuning (POTE): ~2 = 1\1, ~11/9 = 351.6908, ~5/4 = 389.1892
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


{{Optimal ET sequence|legend=1| 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>
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: Angle (11/9, 8/7) = 90 degrees
: Angle (11/9, 8/7) = 90 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 348.0938, ~7/4 = 963.6042
[[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 }}
{{Optimal ET sequence|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rastmic| ]] <!-- key article -->
[[Category:Rastmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Rastmic_rank_three_clan"