Trienstonic clan: Difference between revisions
Cmloegcmluin (talk | contribs) adding parenthetical "(unchanged intervals)" to eigenmonzo lists |
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Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are [[Dicot family #Sharp|sharp]], [[Archytas clan #Blacksmith|blacksmith]], and [[Meantone family #Sharptone|sharptone]]. | Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are [[Dicot family #Sharp|sharp]], [[Archytas clan #Blacksmith|blacksmith]], and [[Meantone family #Sharptone|sharptone]]. | ||
== No- | == No-five trienstonic == | ||
Subgroup: 2.3.7 | Subgroup: 2.3.7 | ||
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{{Val list|legend=1| 2d, 5c, 19ccdd }} | {{Val list|legend=1| 2d, 5c, 19ccdd }} | ||
[[Badness]]: 0. | [[Badness]]: 0.058468 | ||
== Octokaidecal == | == Octokaidecal == | ||
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{{Val list|legend=1| 8d, 10, 18, 28b }} | {{Val list|legend=1| 8d, 10, 18, 28b }} | ||
[[Badness]]: 0. | [[Badness]]: 0.036747 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Vals: {{Val list| 8d, 10, 18e }} | Vals: {{Val list| 8d, 10, 18e }} | ||
Badness: 0. | Badness: 0.030235 | ||
== Opossum == | == Opossum == | ||
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{{Val list|legend=1| 7d, 8d, 15 }} | {{Val list|legend=1| 7d, 8d, 15 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.040650 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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POTE generator: ~10/9 = 159.807 | POTE generator: ~10/9 = 159.807 | ||
Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7 | Minimax tuning: | ||
* 11-odd-limit eigenmonzo subgroup: 2.7 | |||
Vals: {{Val list| 7d, 8d, 15 }} | Vals: {{Val list| 7d, 8d, 15 }} | ||
Badness: 0. | Badness: 0.022325 | ||
=== 13-limit === | === 13-limit === | ||
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Vals: {{Val list| 7d, 8d, 15, 38bcef }} | Vals: {{Val list| 7d, 8d, 15, 38bcef }} | ||
Badness: 0. | Badness: 0.019389 | ||
== Kangaroo == | == Kangaroo == | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
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{{Val list|legend=1| 15, 47b, 62b }} | {{Val list|legend=1| 15, 47b, 62b }} | ||
[[Badness]]: 0. | [[Badness]]: 0.295767 | ||
=== 7-limit === | === 7-limit === | ||
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{{Val list|legend=1| 15 }} | {{Val list|legend=1| 15 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.077857 | ||
=== 11-limit === | === 11-limit === | ||
| Line 198: | Line 196: | ||
Vals: {{Val list| 15 }} | Vals: {{Val list| 15 }} | ||
Badness: 0. | Badness: 0.043195 | ||
=== 13-limit === | === 13-limit === | ||
| Line 211: | Line 209: | ||
Vals: {{Val list| 15 }} | Vals: {{Val list| 15 }} | ||
Badness: 0. | Badness: 0.032653 | ||
== Quindecic == | == Quindecic == | ||
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Vals: {{Val list| 15, 30, 45bc }} | Vals: {{Val list| 15, 30, 45bc }} | ||
Badness: 0. | Badness: 0.028944 | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
Revision as of 22:56, 6 August 2021
The trienstonic clan of temperaments tempers out 28/27, the septimal third-tone or trienstonic comma.
Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharp, blacksmith, and sharptone.
No-five trienstonic
Subgroup: 2.3.7
Comma list: 28/27
Sval mapping: [⟨1 0 -2], ⟨0 1 3]]
Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]
POTE generator: ~3/2 = 721.5586
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 ]]
POTE generator: ~3/2 = 742.002
- 7-odd-limit: ~3/2 = [1/2 0 -1/4 1/4⟩
- Eigenmonzos (unchanged intervals): 2, 7/5
- 9-odd-limit: ~3/2 = [1 -2 0 1⟩ = 14/9
- Eigenmonzos (unchanged intervals): 2, 9/7
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 ]]
POTE generator: ~3/2 = 731.177
- 7-odd-limit eigenmonzo subgroup: 2.5/3
- 9-odd-limit eigenmonzo subgroup: 2.9/5
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 ]]
POTE generator: ~3/2 = 691.351
Badness: 0.058468
Octokaidecal
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 ]]
POTE generator: ~3/2 = 728.874
- 7-odd-limit eigenmonzo subgroup: 2.5
- 9-odd-limit eigenmonzo subgroup: 2.5
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]]
POTE generator: ~3/2 = 732.330
Vals: Template:Val list
Badness: 0.030235
Opossum
Subgroup: 2.3.5.7
Comma list: 28/27, 126/125
Mapping: [⟨1 2 3 4], ⟨0 -3 -5 -9]]
Wedgie: ⟨⟨ 3 5 9 1 6 7 ]]
POTE generator: ~10/9 = 159.691
- 7-odd-limit eigenmonzo subgroup: 2.7
- 9-odd-limit eigenmonzo subgroup: 2.7
Badness: 0.040650
11-limit
Subgroup: 2.3.5.7.11
Comma list: 28/27, 55/54, 77/75
Mapping: [⟨1 2 3 4 4], ⟨0 -3 -5 -9 -4]]
POTE generator: ~10/9 = 159.807
Minimax tuning:
- 11-odd-limit eigenmonzo subgroup: 2.7
Vals: Template:Val list
Badness: 0.022325
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 55/54, 66/65
Mapping: [⟨1 2 3 4 4 4], ⟨0 -3 -5 -9 -4 -2]]
POTE generator: ~10/9 = 159.805
Minimax tuning:
- 13-odd-limit eigenmonzo subgroup: 2.7
- 15-odd-limit eigenmonzo subgroup: 2.7
Vals: Template:Val list
Badness: 0.019389
Kangaroo
Subgroup: 2.3.5
Comma list: 64000/59049
Mapping: [⟨1 0 -3], ⟨0 3 10]]
POTE generator: ~27/20 = 561.195
Badness: 0.295767
7-limit
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 ]]
POTE generator: ~7/5 = 560.328
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]]
POTE generator: ~7/5 = 560.155
Vals: Template:Val list
Badness: 0.043195
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 77/75, 147/143
Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]
POTE generator: ~7/5 = 559.770
Vals: Template:Val list
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]]
POTE generator: ~13/8 = 852.924
Vals: Template:Val list
Badness: 0.028944