Trienstonic clan: Difference between revisions
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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]]. | ||
= Trienstonic = | == Trienstonic == | ||
Subgroup: 2.3.7 | Subgroup: 2.3.7 | ||
| Line 17: | Line 17: | ||
{{Val list|legend=1| 2d, 3d, 5, 78bb }} | {{Val list|legend=1| 2d, 3d, 5, 78bb }} | ||
= Father = | == Father == | ||
{{see also| Father family #Father }} | {{see also| Father family #Father }} | ||
| Line 41: | Line 41: | ||
[[Badness]]: 0.0213 | [[Badness]]: 0.0213 | ||
= Uncle = | == Uncle == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 61: | Line 61: | ||
[[Badness]]: 0.0727 | [[Badness]]: 0.0727 | ||
= Wallaby = | == Wallaby == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 77: | Line 77: | ||
[[Badness]]: 0.0585 | [[Badness]]: 0.0585 | ||
= Octokaidecal = | == Octokaidecal == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 97: | Line 97: | ||
[[Badness]]: 0.0367 | [[Badness]]: 0.0367 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 111: | Line 111: | ||
Badness: 0.0302 | Badness: 0.0302 | ||
= Opossum = | == Opossum == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 131: | Line 131: | ||
[[Badness]]: 0.0407 | [[Badness]]: 0.0407 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 147: | Line 147: | ||
Badness: 0.0223 | Badness: 0.0223 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 165: | Line 165: | ||
Badness: 0.0194 | Badness: 0.0194 | ||
= Kangaroo = | == Kangaroo == | ||
== 5-limit == | === 5-limit === | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 180: | Line 180: | ||
Badness: 0.2958 | Badness: 0.2958 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 196: | Line 196: | ||
Badness: 0.0779 | Badness: 0.0779 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 210: | Line 210: | ||
Badness: 0.0432 | Badness: 0.0432 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 224: | Line 224: | ||
Badness: 0.0327 | Badness: 0.0327 | ||
= Quindecic = | == Quindecic == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 19:03, 3 June 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.
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
- [[1 0 0 0⟩, [3/2 0 -1/4 1/4⟩, [5/2 0 1/4 -1/4⟩, [5/2 0 -3/4 3/4⟩]
- Eigenmonzo subgroup: 2.7/5
- Eigenmonzo subgroup: 2.9/5
Badness: 0.0213
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 = 731.1774
- 7-odd-limit eigenmonzo subgroup: 2.5/3
- 9-odd-limit eigenmonzo subgroup: 2.9/5
Badness: 0.0727
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: 0.0585
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: 0.0367
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
Badness: 0.0302
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: 0.0407
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
Badness: 0.0223
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
Badness: 0.0194
Kangaroo
5-limit
Subgroup: 2.3.5
Comma list: 64000/59049
Mapping: [⟨1 0 -3], ⟨0 3 10]]
POTE generator: ~27/20 = 561.195
Badness: 0.2958
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: 0.0779
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
Badness: 0.0432
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
Badness: 0.0327
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
Badness: 0.0289