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 = | |||
Period: 1\1 | |||
Optimal ([[POTE]]) generator: ~3/2 = 721.5586 | |||
EDO generators: [[5edo|3\5]], [[8edo|5\8]], [[13edo|8\13]], [[18edo|11\18]], [[23edo|14\23]] | |||
Scales (Scala files): | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
Subgroup: 2.3..7 | |||
Comma list: 28/27 | |||
Mapping: [<1 0 0 -2|, <0 1 0 3|] | |||
{{Val list|legend=1| 2d, 3d, 5, 78bb }} | |||
</div></div> | |||
= Father = | = Father = | ||
Revision as of 07:53, 13 February 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
Period: 1\1
Optimal (POTE) generator: ~3/2 = 721.5586
EDO generators: 3\5, 5\8, 8\13, 11\18, 14\23
Scales (Scala files):
Father
Period: 1\1
Optimal (POTE) generator: ~3/2 = 742.002
EDO generators: 3\5, 5\8, 8\13
Scales (Scala files):
Comma list: 16/15, 28/27
Mapping: [⟨1 0 4 -2], ⟨0 1 -1 3]]
Wedgie: ⟨⟨1 -1 3 -4 2 10]]
Minimax tuning:
- 7-odd-limit
- [[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
- 9-odd-limit
- Eigenmonzo subgroup: 2.9/5
Badness: 0.0213
Uncle
Period: 1\1
Optimal (POTE) generator: ~3/2 = 731.1774
Scales (Scala files):
Comma list: 28/27, 256/245
Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]
Wedgie: ⟨⟨1 -6 3 -12 2 24]]
Minimax tuning:
- 7-odd-limit eigenmonzo subgroup: 2.5/3
- 9-odd-limit eigenmonzo subgroup: 2.9/5
Badness: 0.0727
Octokaidecal
Period: 1\2
Optimal (POTE) generator: ~3/2 = 728.874
EDO generators: 5\8, 6\10, 11\18
Scales (Scala files):
Comma list: 28/27, 50/49
Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]
Wedgie: ⟨⟨2 6 6 5 4 -2]]
Minimax tuning:
- 7-odd-limit eigenmonzo subgroup: 2.5
- 9-odd-limit eigenmonzo subgroup: 2.5
Badness: 0.0367
11-limit
Period: 1\2
Optimal (POTE) generator: ~3/2 = 732.330
EDO generators: 5\8, 6\10, 11\18
Scales (Scala files):
Comma list: 28/27, 50/49, 55/54
Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]
Badness: 0.0302
Opossum
Period: 1\1
Optimal (POTE) generator: ~10/9 = 159.691
EDO generators: 1\7, 1\8, 2\15
Scales (Scala files):
Comma list: 28/27, 126/125
Mapping: [⟨1 2 3 4], ⟨0 -3 -5 -9]]
Wedgie: ⟨⟨3 5 9 1 6 7]]
Minimax tuning:
- 7-odd-limit eigenmonzo subgroup: 2.7
- 9-odd-limit eigenmonzo subgroup: 2.7
Badness: 0.0407
11-limit
Period: 1\1
Optimal (POTE) generator: ~10/9 = 159.807
EDO generators: 1\7, 1\8, 2\15
Scales (Scala files):
Comma list: 28/27, 55/54, 77/75
Mapping: [⟨1 2 3 4 4], ⟨0 -3 -5 -9 -4]]
Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7
Badness: 0.0223
13-limit
Period: 1\1
Optimal (POTE) generator: ~10/9 = 159.805
EDO generators: 1\7, 1\8, 2\15
Scales (Scala files):
Comma list: 28/27, 40/39, 55/54, 66/65
Mapping: [⟨1 2 3 4 4 4], ⟨0 -3 -5 -9 -4 -2]]
Minimax tuning:
- 13-odd-limit eigenmonzo subgroup: 2.7
- 15-odd-limit eigenmonzo subgroup: 2.7
Badness: 0.0194
Wallaby
Period: 1\1
Optimal (POTE) generator: ~3/2 = 691.351
EDO generators: 3\5
Scales (Scala files):
Comma list: 28/27, 35/32
Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]
Wedgie: ⟨⟨1 -3 3 -7 2 15]]
Badness: 0.0585
Kangaroo
5-limit
Period: 1\1
Optimal (POTE) generator: ~27/20 = 561.195
EDO generators: 9\15
Scales (Scala files):
7-limit
Period: 1\1
Optimal (POTE) generator: ~7/5 = 560.328
EDO generators: 7\15
Scales (Scala files):
Comma list: 28/27, 1029/1000
Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]
Wedgie: ⟨⟨3 10 9 9 6 -7]]
Badness: 0.0779
11-limit
Period: 1\1
Optimal (POTE) generator: ~7/5 = 560.155
EDO generators: 7\15
Scales (Scala files):
Comma list: 28/27, 77/75, 245/242
Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]
Badness: 0.0432
13-limit
Period: 1\1
Optimal (POTE) generator: ~7/5 = 559.770
EDO generators: 7\15
Scales (Scala files):
Comma list: 28/27, 40/39, 77/75, 147/143
Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]
Badness: 0.0327
Quindecic
Period: 1\15
Optimal (POTE) generator: ~13/8 = 852.924
Scales (Scala files):
Comma list: 28/27, 49/48, 55/54, 77/75
Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]
Badness: 0.0289