Rastmic rank three clan: Difference between revisions

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{{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.  


Line 17: Line 18:
: 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 }}
Line 28: Line 30:
[[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]]
Line 41: Line 43:
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]]
Line 57: Line 61:


: 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 }}
Line 69: Line 77:
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 ==
Line 83: Line 95:
: 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 }}
Line 96: Line 108:
{{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 }}
Line 103: Line 117:


== 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 ==
Line 125: Line 154:


: 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 }}
Line 137: Line 170:
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>
Line 151: Line 188:
: 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 }}
Line 164: Line 201:
{{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 }} *
Line 179: Line 218:
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>
Line 198: Line 239:
: 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 }}
Line 209: Line 252:


[[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]]

Latest revision as of 00:27, 24 June 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 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: [⟨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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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

  1. ↑ Yahoo! Tuning Group | Spectacle temperament?
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