Trienstonic clan: Difference between revisions
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The '''trienstonic clan''' of | The '''trienstonic clan''' of [[Rank-2 temperament|rank-2]] [[temperament]]s [[Tempering out|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: | 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: | ||
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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|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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== Father == | == Father == | ||
{{Main| Father }} | |||
{{See also| Father family #Father }} | {{See also| Father family #Father }} | ||
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] [[eigenmonzo | * [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3 | ||
* [[9-odd-limit]] [[eigenmonzo | * [[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, 41bcd }} | ||
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd- | * [[7-odd-limit7-]] 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 }} | ||
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[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category:Trienstonic clan| ]] <!-- main article --> | [[Category:Trienstonic clan| ]] <!-- main article --> | ||
[[Category:Trienstonic]] | [[Category:Trienstonic| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 08:32, 22 September 2023
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:
- Sharptone (+21/20) → Meantone family
- Sharp (+25/24) → Dicot family
- Inflated (+128/125) → Augmented family
- Opossum (+126/125) → Porcupine family
- Blacksmith (+49/48) → Limmic temperaments
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 tuning (POTE): ~2 = 1\1, ~3/2 = 721.5586
Optimal ET sequence: 2d, 3d, 5, 78bb
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 tuning (POTE): ~2 = 1\1, ~3/2 = 742.002
- 7-odd-limit: ~3/2 = [1/2 0 -1/4 1/4⟩
- 9-odd-limit: ~3/2 = [1 -2 0 1⟩ = 14/9
Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd
Badness: 0.021312
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 tuning (POTE): ~2 = 1\1, ~3/2 = 731.177
- 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5
Optimal ET sequence: 5, 13d, 18, 23bc, 41bcd
Badness: 0.072653
Wallaby
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 tuning (POTE): ~2 = 1\1, ~3/2 = 691.351
Optimal ET sequence: 2d, 5c, 19ccdd
Badness: 0.058468
Octokaidecal
5-limit (supersharp)
Subgroup: 2.3.5
Comma list: 800/729
Mapping: [⟨2 0 -5], ⟨0 1 3]]
Optimal tuning (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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 728.874
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 tuning (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]]
Wedgie: ⟨⟨ 3 10 9 9 6 -7 ]]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 560.328
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 tuning (POTE): ~2 = 1\1, ~7/5 = 560.155
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 tuning (POTE): ~2 = 1\1, ~7/5 = 559.770
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]]
Optimal tuning (POTE): ~22/21 = 1\15, ~13/8 = 852.924
Optimal ET sequence: 15, 30, 45bc
Badness: 0.028944