Rastmic rank-3 clan: Difference between revisions

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m FloraC moved page Rastmic rank three clan to Rastmic rank-3 clan: For obvious reasons
m Units. Misc. cleanup
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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 three clan #Betarabian|Alphaxenic rank-3 clan]]
* ''[[Rabic]]'' (+131769/131072) → [[Alphaxenic rank-3 clan #Betarabian|Alphaxenic rank-3 clan]]


== 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 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.
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.


[[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, ~11/9 = 350.377, ~5/4 = 384.370
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.377{{c}}, ~5/4 = 384.370{{c}}
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.176, ~5/4 = 384.095
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.176{{c}}, ~5/4 = 384.095{{c}}


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 28: Line 28:
{{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 }}


[[Badness]]: 0.499 × 10<sup>-3</sup>
[[Badness]] (Smith): 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
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 350.239, ~5/4 = 384.998
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.239{{c}}, ~5/4 = 384.998{{c}}
* CWE: ~2 = 1200.000, ~11/9 = 350.039, ~5/4 = 384.587
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.039{{c}}, ~5/4 = 384.587{{c}}


{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} <nowiki>*</nowiki>
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<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
<nowiki>*</nowiki> optimal patent val: [[240edo|240]]


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


== Cuckoo ==
== Cuckoo ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.355, ~5/4 = 389.555
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.355{{c}}, ~5/4 = 389.555{{c}}
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.421, ~5/4 = 389.731
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.421{{c}}, ~5/4 = 389.731{{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]]: 0.933 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.933 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 350.585, ~5/4 = 389.302
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.585{{c}}, ~5/4 = 389.302{{c}}
* CWE: ~2 = 1200.000, ~11/9 = 350.568, ~5/4 = 389.610
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.568{{c}}, ~5/4 = 389.610{{c}}


{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
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: mapping generators: ~2, ~128/105, ~5
: mapping generators: ~2, ~128/105, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/105 = 350.174, ~5/4 = 384.014
[[Optimal tuning]] ([[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 }}


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


=== 11-limit ===
=== 11-limit ===
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.217, ~5/4 = 383.962
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.217{{c}}, ~5/4 = 383.962{{c}}
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.003, ~5/4 = 384.078
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.003{{c}}, ~5/4 = 384.078{{c}}


{{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 }}


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


== Mirage ==
== Mirage ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~15/14 = 116.711, ~13/8 = 840.528
* [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.711{{c}}, ~13/8 = 840.528{{c}}
* [[CWE]]: ~2 = 1200.000, ~15/14 = 116.647, ~13/8 = 838.212
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.647{{c}}, ~13/8 = 838.212{{c}}


{{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 (Smith): 0.738 × 10<sup>-3</sup>


=== 17-limit===
=== 17-limit===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~15/14 = 116.704, ~13/8 = 839.452
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.704{{c}}, ~13/8 = 839.452{{c}}
* CWE: ~2 = 1200.000, ~15/14 = 116.640, ~13/8 = 837.142
* CWE: ~2 = 1200.000{{c}}, ~15/14 = 116.640{{c}}, ~13/8 = 837.142{{c}}


{{Optimal ET sequence|legend=0| 10, 31, 41, 62, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 10, 31, 41, 62, 72, 103, 175f }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~11/9 = 350.145, ~5/4 = 389.708
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.145{{c}}, ~5/4 = 389.708{{c}}
* [[CWE]]: ~2 = 1200.000, ~11/9 = 350.555, ~5/4 = 390.269
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.555{{c}}, ~5/4 = 390.269{{c}}


{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}


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


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 350.293, ~5/4 = 389.979
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.293{{c}}, ~5/4 = 389.979{{c}}
* CWE: ~2 = 1200.000, ~11/9 = 351.155, ~5/4 = 391.123
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 351.155{{c}}, ~5/4 = 391.123{{c}}


{{Optimal ET sequence|legend=0| 7, 24, 31, 58 }}
{{Optimal ET sequence|legend=0| 7, 24, 31, 58 }}


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


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~896/729 = 351.416, ~5/4 = 386.314
[[Optimal tuning]] ([[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]]: 0.607 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.607 × 10<sup>-3</sup>


=== 11-limit ===
=== 11-limit ===
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~11/9 = 351.320, ~5/4 = 386.314
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 351.320{{c}}, ~5/4 = 386.314{{c}}
* [[CWE]]: ~2 = 1200.000, ~11/9 = 351.459, ~5/4 = 387.423
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 351.459{{c}}, ~5/4 = 387.423{{c}}


{{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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<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]
<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]


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


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 351.343, ~5/4 = 386.314  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 351.343{{c}}, ~5/4 = 386.314{{c}}
* CWE: ~2 = 1200.000, ~11/9 = 351.604, ~5/4 = 388.472
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 351.604{{c}}, ~5/4 = 388.472{{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: 1.19366 × 10<sup>-3</sup>
Badness (Smith): 1.19366 × 10<sup>-3</sup>


== Urania  ==
== Urania  ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~11/9 = 348.830, ~7/4 = 968.826
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 348.830{{c}}, ~7/4 = 968.826{{c}}
* [[CWE]]: ~2 = 1200.000, ~11/9 = 348.379, ~7/4 = 965.630
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 348.379{{c}}, ~7/4 = 965.630{{c}}


{{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 }}


[[Badness]]: 0.842 × 10<sup>-3</sup>
[[Badness]] (Smith): 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
[[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]]
Scales: [[urania24]]
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 16:07, 8 September 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 discussed elsewhere include:

Spectacle

Spectacle, named by Gene Ward Smith in 2010[1], 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 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: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]

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

Optimal tunings:

  • 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 = [1 0 0 0 0⟩, ~11/9 = [-2/5 0 0 0 1/5⟩, ~5 = [2/5 -2 1 0 4/5⟩
unchanged-interval (eigenmonzo) basis: 2.9/5.11

Optimal ET sequence: 31, 41, 72, 247c, 281, 353c, 425bc, 497bc

Badness (Smith): 0.499 × 10-3

Projection pairs: 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: [⟨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: 31, 72, 103, 175f *

* optimal patent val: 240

Badness (Smith): 1.009 × 10-3

Cuckoo

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 -4 5], ⟨0 0 1 3 0]]

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

Optimal tunings:

  • 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: 24d, 27e, 31, 58, 89, 154, 185

Badness (Smith): 0.933 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

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: 27e, 31, 58, 96d, 154

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.

7-limit

Subgroup: 2.3.5.7

Comma list: 33075/32768

Mapping: [⟨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 ¢

Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d

Badness (Smith): 0.770 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]

Optimal tunings:

  • 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: 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde

Badness (Smith): 0.641 × 10-3

Mirage

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨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 tunings:

  • 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: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness (Smith): 0.738 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384

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: 10, 31, 41, 62, 72, 103, 175f

Mandos

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]

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

Optimal tunings:

  • 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: 7, 24, 31, 58, 89

Badness (Smith): 0.751 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

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: 7, 24, 31, 58

Badness (Smith): 0.923 × 10-3

Parahemif

7-limit

Subgroup: 2.3.5.7

Comma list: 1605632/1594323

Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]

mapping generators: ~2, ~896/729, ~5

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~896/729 = 351.416 ¢, ~5/4 = 386.314 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338

Badness (Smith): 0.607 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891

Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]

Optimal tunings:

  • 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: 17c, 24, 34d, 41, 58, 99e *

* optimal patent val: 123

Badness (Smith): 1.34547 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 243/242, 364/363

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 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff

Badness (Smith): 1.19366 × 10-3

Urania

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

Mapping to lattice: [⟨0 2 8 0 5], ⟨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 tunings:

  • 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: 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde

Badness (Smith): 0.842 × 10-3

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

References

  1. ↑ Yahoo! Tuning Group | Spectacle temperament?
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