Trienstonic clan: Difference between revisions

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+link to main article of father; misc. cleanup
+CTE tunings as this is a clan where they tend to differ a lot from POTE; misc. cleanup
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{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 721.5586
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 717.5172
* [[POTE]]: ~2 = 1\1, ~3/2 = 721.5586


{{Optimal ET sequence|legend=1| 2d, 3d, 5, 78bb }}
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}


== Father ==
== Father ==
Line 35: Line 37:
{{Multival|legend=1| 1 -1 3 -4 2 10 }}
{{Multival|legend=1| 1 -1 3 -4 2 10 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 742.002
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.8550
* [[POTE]]: ~2 = 1\1, ~3/2 = 742.002


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 46: Line 50:


[[Badness]]: 0.021312
[[Badness]]: 0.021312
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 16/15, 22/21, 28/27
Mapping: {{mapping| 1 0 4 -2 -3 | 0 1 -1 3 4 }}
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 732.2094
* POTE: ~2 = 1\1, ~3/2 = 747.156
Optimal ET sequence: {{Optimal ET sequence| 2de, 3de, 5, 8d }}
Badness: 0.020589


== Uncle ==
== Uncle ==
Line 56: Line 75:
{{Multival|legend=1| 1 -6 3 -12 2 24 }}
{{Multival|legend=1| 1 -6 3 -12 2 24 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 731.177
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 731.3937
* [[POTE]]: ~2 = 1\1, ~3/2 = 731.177


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 62: Line 83:
* [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
* [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5


{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bcd }}
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}


[[Badness]]: 0.072653
[[Badness]]: 0.072653
Line 77: Line 98:
{{Multival|legend=1| 1 -3 3 -7 2 15 }}
{{Multival|legend=1| 1 -3 3 -7 2 15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 691.351
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 691.7571
* [[POTE]]: ~2 = 1\1, ~3/2 = 691.351


{{Optimal ET sequence|legend=1| 2d, 5c, 19ccdd }}
{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}


[[Badness]]: 0.058468
[[Badness]]: 0.058468
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{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}
{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}


[[Optimal tuning]] ([[POTE]]): ~27/20 = 1\2, ~3/2 = 729.097
: mapping generators: ~27/20, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~27/20 = 1\2, ~3/2 = 726.5480
* [[POTE]]: ~27/20 = 1\2, ~3/2 = 729.097


{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
Line 106: Line 133:
{{Multival|legend=1| 2 6 6 5 4 -2 }}
{{Multival|legend=1| 2 6 6 5 4 -2 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 728.874
[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 1\2, ~3/2 = 723.3709
* [[POTE]]: ~7/5 = 1\2, ~3/2 = 728.874


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit7-]] and [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
Line 122: Line 151:
Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}
Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 732.330
Optimal tunings:
* CTE: ~7/5 = 1\2, ~3/2 = 723.3709
* POTE: ~7/5 = 1\2, ~3/2 = 732.330


{{Optimal ET sequence|legend=1| 8d, 10, 18e }}
Optimal ET sequence: {{Optimal ET sequence| 8d, 10, 18e }}


Badness: 0.030235
Badness: 0.030235
Line 136: Line 167:


{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 3 10 9 9 6 -7 }}
{{Multival|legend=1| 3 10 9 9 6 -7 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 560.328
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/7 = 638.8628
* [[POTE]]: ~2 = 1\1, ~10/7 = 639.672


{{Optimal ET sequence|legend=1| 2cd, 13cd, 15 }}
{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}


[[Badness]]: 0.077857
[[Badness]]: 0.077857
Line 152: Line 187:
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 560.155
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 639.0363
* POTE: ~2 = 1\1, ~10/7 = 639.845


{{Optimal ET sequence|legend=1| 15 }}
Optimal ET sequence: {{Optimal ET sequence| 15 }}


Badness: 0.043195
Badness: 0.043195
Line 165: Line 202:
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 559.770
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 638.7168
* POTE: ~2 = 1\1, ~10/7 = 640.230


{{Optimal ET sequence|legend=1| 15 }}
Optimal ET sequence: {{Optimal ET sequence| 15 }}


Badness: 0.032653
Badness: 0.032653
Line 178: Line 217:
Mapping: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
Mapping: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}


Optimal tuning (POTE): ~22/21 = 1\15, ~13/8 = 852.924
: mapping generators: ~22/21, ~13
 
Optimal tunings:
* CTE: ~22/21 = 1\15, ~13/8 = 840.5277 (~40/39 = 39.4723)
* POTE: ~22/21 = 1\15, ~13/8 = 852.924 (~40/39 = 27.076)


{{Optimal ET sequence|legend=1| 15, 30, 45bc }}
Optimal ET sequence: {{Optimal ET sequence| 15, 30 }}


Badness: 0.028944
Badness: 0.028944

Revision as of 10:23, 4 May 2024

The trienstonic clan of rank-2 temperaments tempers out 28/27, the septimal third-tone or trienstonic comma.

Adding 16/15 to 28/27 leads to father, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 35/32 gives wallaby. Other members of the clan discussed elsewhere are:

Trienstonian

Subgroup: 2.3.7

Comma list: 28/27

Subgroup-val mapping: [⟨1 0 -2], ⟨0 1 3]]

mapping generators: ~2, ~3

Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 717.5172
  • POTE: ~2 = 1\1, ~3/2 = 721.5586

Optimal ET sequence: 2d, 3d, 5

Father

Main article: Father
See also: Father family § Father

Subgroup: 2.3.5.7

Comma list: 16/15, 28/27

Mapping: [⟨1 0 4 -2], ⟨0 1 -1 3]]

Wedgie: ⟨⟨ 1 -1 3 -4 2 10 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 727.8550
  • POTE: ~2 = 1\1, ~3/2 = 742.002

Minimax tuning:

eigenmonzo (unchanged-interval) basis: 2.7/5
eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd

Badness: 0.021312

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 28/27

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

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 732.2094
  • POTE: ~2 = 1\1, ~3/2 = 747.156

Optimal ET sequence: 2de, 3de, 5, 8d

Badness: 0.020589

Uncle

Subgroup: 2.3.5.7

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Wedgie: ⟨⟨ 1 -6 3 -12 2 24 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 731.3937
  • POTE: ~2 = 1\1, ~3/2 = 731.177

Minimax tuning:

Optimal ET sequence: 5, 13d, 18, 23bc, 41bbcd

Badness: 0.072653

Wallaby

See also: Pelogic family

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Wedgie: ⟨⟨ 1 -3 3 -7 2 15 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 691.7571
  • POTE: ~2 = 1\1, ~3/2 = 691.351

Optimal ET sequence: 2d, 5c, 7d, 19ccdd

Badness: 0.058468

Octokaidecal

5-limit (supersharp)

Subgroup: 2.3.5

Comma list: 800/729

Mapping: [⟨2 0 -5], ⟨0 1 3]]

mapping generators: ~27/20, ~3

Optimal tunings:

  • CTE: ~27/20 = 1\2, ~3/2 = 726.5480
  • POTE: ~27/20 = 1\2, ~3/2 = 729.097

Optimal ET sequence: 8, 10, 18, 28b

Badness: 0.122848

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Wedgie: ⟨⟨ 2 6 6 5 4 -2 ]]

Optimal tunings:

  • CTE: ~7/5 = 1\2, ~3/2 = 723.3709
  • POTE: ~7/5 = 1\2, ~3/2 = 728.874

Minimax tuning:

Optimal ET sequence: 8d, 10, 18, 28b

Badness: 0.036747

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 50/49, 55/54

Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]

Optimal tunings:

  • CTE: ~7/5 = 1\2, ~3/2 = 723.3709
  • POTE: ~7/5 = 1\2, ~3/2 = 732.330

Optimal ET sequence: 8d, 10, 18e

Badness: 0.030235

Kangaroo

For the 5-limit version of this temperament, see High badness temperaments #Kangaroo.

Subgroup: 2.3.5.7

Comma list: 28/27, 1029/1000

Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]

mapping generators: ~2, ~10/7

Wedgie: ⟨⟨ 3 10 9 9 6 -7 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~10/7 = 638.8628
  • POTE: ~2 = 1\1, ~10/7 = 639.672

Optimal ET sequence: 2cd, …, 13cd, 15

Badness: 0.077857

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 77/75, 245/242

Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~10/7 = 639.0363
  • POTE: ~2 = 1\1, ~10/7 = 639.845

Optimal ET sequence: 15

Badness: 0.043195

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 66/65, 147/143

Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~10/7 = 638.7168
  • POTE: ~2 = 1\1, ~10/7 = 640.230

Optimal ET sequence: 15

Badness: 0.032653

Quindecic

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 49/48, 55/54, 77/75

Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]

mapping generators: ~22/21, ~13

Optimal tunings:

  • CTE: ~22/21 = 1\15, ~13/8 = 840.5277 (~40/39 = 39.4723)
  • POTE: ~22/21 = 1\15, ~13/8 = 852.924 (~40/39 = 27.076)

Optimal ET sequence: 15, 30

Badness: 0.028944

Retrieved from "https://en.xen.wiki/index.php?title=Trienstonic_clan&oldid=142349"