Trienstonic clan: Difference between revisions

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The '''trienstonic clan''' of [[Rank-2 temperament|rank-2]] [[temperament]]s [[Tempering out|tempers out]] [[28/27]], the septimal third-tone or trienstonic comma. This equates very different intervals with each other, in particular [[9/8]] with [[7/6]] and [[8/7]] with [[32/27]]. Trienstonian is at the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].  
The '''trienstonic clan''' of [[Rank-2 temperament|rank-2]] [[temperament]]s [[Tempering out|tempers out]] [[28/27]], the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, [[9/8]] is equated with [[7/6]], [[8/7]] with [[32/27]], and [[4/3]] with [[9/7]]. Trienstonian is inaccurate enough that it is close to the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].  


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:  
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:  
* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
* ''[[Sharp (temperament)|Sharp]]'' (+25/24) → [[Dicot family #Sharp|Dicot family]]
* ''[[Sharp (temperament)|Sharp]]'' (+25/24) → [[Dicot family #Sharp|Dicot family]]
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
* ''[[Blacksmith]]'' (+49/48) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
* ''[[Blacksmith]]'' (+49/48) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]


== Trienstonian ==
== Trienstonian ==
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{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}


: mapping generators: ~2, ~3
: Mapping generators: ~2, ~3


{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
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{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}
{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}


: mapping generators: ~27/20, ~3
: Mapping generators: ~27/20, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:
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{{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
: Mapping generators: ~2, ~10/7


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


: mapping generators: ~22/21, ~13
: Mapping generators: ~22/21, ~13


Optimal tunings:
Optimal tunings:
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Trienstonic clan| ]] <!-- main article -->
[[Category:Trienstonic clan| ]] <!-- Main article -->
[[Category:Trienstonic| ]] <!-- key article -->
[[Category:Trienstonic| ]] <!-- Key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 18:03, 20 November 2024

The trienstonic clan of rank-2 temperaments tempers out 28/27, the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, 9/8 is equated with 7/6, 8/7 with 32/27, and 4/3 with 9/7. Trienstonian is inaccurate enough that it is close to the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

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

Sval 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

The 5-limit restriction of octokaidecal is supersharp, which tempers out 800/729, the difference between the 27/20 wolf fourth and the 40/27 wolf fifth, splitting the octave into two 27/20~40/27 semioctaves. It generally requires a very sharp fifth, even sharper than 3\5, as a generator. This means that five steps from the Zarlino generator sequence starting with 6/5 are tempered to one and a half octaves. The only reasonable 7-limit extension adds 28/27 and 50/49 to the comma list, taking advantage of the existing semioctave.

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

Parakangaroo

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

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