Trienstonic clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Formatting
Revert databoxes
Line 5: Line 5:
= Trienstonic =
= Trienstonic =


Period: 1\1
Subgroup: 2.3.7


Optimal ([[POTE]]) generator: ~3/2 = 721.5586
[[Comma list]]: 28/27


EDO generators: [[5edo|3\5]], [[8edo|5\8]], [[13edo|8\13]], [[18edo|11\18]], [[23edo|14\23]]
[[Sval]] [[mapping]]: [{{val| 1 0 -2 }}, {{val| 0 1 3 }}]


Scales (Scala files):  
[[Gencom]] mapping: [{{val| 1 0 0 -2 }}, {{val| 0 1 0 3 }}]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[POTE generator]]: ~3/2 = 721.5586
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3..7
 
Comma list: 28/27
 
Mapping: [{{val| 1 0 0 -2 }}, {{val| 0 1 0 3 }}]


{{Val list|legend=1| 2d, 3d, 5, 78bb }}
{{Val list|legend=1| 2d, 3d, 5, 78bb }}
</div></div>


= Father =
= Father =
{{see also| Father family #Father }}
{{see also| Father family #Father }}


Period: 1\1
Subgroup: 2.3.5.7


Optimal ([[POTE]]) generator: ~3/2 = 742.002
[[Comma list]]: 16/15, 28/27


EDO generators: [[5edo|3\5]], [[8edo|5\8]], [[13edo|8\13]]
[[Mapping]]: [{{val| 1 0 4 -2 }}, {{val| 0 1 -1 3 }}]


Scales (Scala files):
{{Multival|legend=1| 1 -1 3 -4 2 10 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[POTE generator]]: ~3/2 = 742.002
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Minimax tuning]]:  
 
* [[7-odd-limit]]
Comma list: 16/15, 28/27
 
Mapping: [{{val| 1 0 4 -2 }}, {{val| 0 1 -1 3 }}]
 
Wedgie: {{wedgie| 1 -1 3 -4 2 10 }}
 
Minimax tuning:  
* 7-odd-limit
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 0 -1/4 1/4 }}, {{monzo| 5/2 0 1/4 -1/4 }}, {{monzo| 5/2 0 -3/4 3/4 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 0 -1/4 1/4 }}, {{monzo| 5/2 0 1/4 -1/4 }}, {{monzo| 5/2 0 -3/4 3/4 }}]
: [[Eigenmonzo subgroup]]: 2.7/5
: [[Eigenmonzo subgroup]]: 2.7/5
* 9-odd-limit
* [[9-odd-limit]]
: Eigenmonzo subgroup: 2.9/5
: Eigenmonzo subgroup: 2.9/5


{{Val list|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
{{Val list|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}


Badness: 0.0213
[[Badness]]: 0.0213
 
</div></div>


= Uncle =
= Uncle =


Period: 1\1
Subgroup: 2.3.5.7


Optimal ([[POTE]]) generator: ~3/2 = 731.1774
[[Comma list]]: 28/27, 256/245


EDO generators: [[18edo|11\18]], [[23edo|14\23]]
[[Mapping]]: [{{val| 1 0 12 -2 }}, {{val| 0 1 -6 3 }}]


Scales (Scala files):
{{Multival|legend=1| 1 -6 3 -12 2 24 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[POTE generator]]: ~3/2 = 731.1774
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Minimax tuning]]:  
 
* [[7-odd-limit]] eigenmonzo subgroup: 2.5/3
Comma list: 28/27, 256/245
* [[9-odd-limit]] eigenmonzo subgroup: 2.9/5
 
Mapping: [{{val| 1 0 12 -2 }}, {{val| 0 1 -6 3 }}]
 
Wedgie: {{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


{{Val list|legend=1| 5, 13d, 18, 23bc, 41bcd }}
{{Val list|legend=1| 5, 13d, 18, 23bc, 41bcd }}


Badness: 0.0727
[[Badness]]: 0.0727
 
</div></div>


= Octokaidecal =
= Octokaidecal =


Period: 1\2
Subgroup: 2.3.5.7
 
Optimal ([[POTE]]) generator: ~3/2 = 728.874
 
EDO generators: [[8edo|5\8]], [[10edo|6\10]], [[18edo|11\18]]
 
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: 2.3.5.7
[[Comma list]]: 28/27, 50/49


Comma list: 28/27, 50/49
[[Mapping]]: [{{val| 2 0 -5 -4 }}, {{val| 0 1 3 3 }}]


Mapping: [{{val| 2 0 -5 -4 }}, {{val| 0 1 3 3 }}]
{{Multival|legend=1| 2 6 6 5 4 -2 }}


Wedgie: {{wedgie| 2 6 6 5 4 -2 }}
[[POTE generator]]: ~3/2 = 728.874


Minimax tuning:  
[[Minimax tuning]]:  
* 7-odd-limit eigenmonzo subgroup: 2.5
* [[7-odd-limit]] eigenmonzo subgroup: 2.5
* 9-odd-limit eigenmonzo subgroup: 2.5
* [[9-odd-limit]] eigenmonzo subgroup: 2.5


{{Val list|legend=1| 8d, 10, 18, 28b }}
{{Val list|legend=1| 8d, 10, 18, 28b }}


Badness: 0.0367
[[Badness]]: 0.0367
 
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 732.330
EDO generators: [[8edo|5\8]], [[10edo|6\10]], [[18edo|11\18]]
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: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 146: Line 88:


Mapping: [{{val| 2 0 -5 -4 7 }}, {{val| 0 1 3 3 0 }}]
Mapping: [{{val| 2 0 -5 -4 7 }}, {{val| 0 1 3 3 0 }}]
POTE generator: ~3/2 = 732.330


{{Val list|legend=1| 8d, 10, 18e }}
{{Val list|legend=1| 8d, 10, 18e }}


Badness: 0.0302
Badness: 0.0302
</div></div>


= Opossum =
= Opossum =


Period: 1\1
Subgroup: 2.3.5.7


Optimal ([[POTE]]) generator: ~10/9 = 159.691
[[Comma list]]: 28/27, 126/125


EDO generators: [[7edo|1\7]], [[8edo|1\8]], [[15edo|2\15]]
[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -9 }}]


Scales (Scala files):
{{Multival|legend=1| 3 5 9 1 6 7 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[POTE generator]]: ~10/9 = 159.691
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Minimax tuning]]:  
 
Comma list: 28/27, 126/125
 
Mapping: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -9 }}]
 
Wedgie: {{wedgie| 3 5 9 1 6 7 }}
 
Minimax tuning:  
* 7-odd-limit eigenmonzo subgroup: 2.7
* 7-odd-limit eigenmonzo subgroup: 2.7
* 9-odd-limit eigenmonzo subgroup: 2.7
* 9-odd-limit eigenmonzo subgroup: 2.7
Line 181: Line 113:
{{Val list|legend=1| 7d, 8d, 15 }}
{{Val list|legend=1| 7d, 8d, 15 }}


Badness: 0.0407
[[Badness]]: 0.0407
 
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~10/9 = 159.807
EDO generators: [[7edo|1\7]], [[8edo|1\8]], [[15edo|2\15]]
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: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 204: Line 122:


Mapping: [{{val| 1 2 3 4 4 }}, {{val| 0 -3 -5 -9 -4 }}]
Mapping: [{{val| 1 2 3 4 4 }}, {{val| 0 -3 -5 -9 -4 }}]
POTE generator: ~10/9 = 159.807


Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7
Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7
Line 210: Line 130:


Badness: 0.0223
Badness: 0.0223
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~10/9 = 159.805
EDO generators: [[7edo|1\7]], [[8edo|1\8]], [[15edo|2\15]]
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: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 232: Line 138:


Mapping: [{{val| 1 2 3 4 4 4 }}, {{val| 0 -3 -5 -9 -4 -2 }}]
Mapping: [{{val| 1 2 3 4 4 4 }}, {{val| 0 -3 -5 -9 -4 -2 }}]
POTE generator: ~10/9 = 159.805


Minimax tuning:  
Minimax tuning:  
Line 240: Line 148:


Badness: 0.0194
Badness: 0.0194
</div></div>


= Wallaby =
= Wallaby =


Period: 1\1
Subgroup: 2.3.5.7
 
Optimal ([[POTE]]) generator: ~3/2 = 691.351
 
EDO generators: [[5edo|3\5]]
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 28/27, 35/32
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Mapping]]: [{{val| 1 0 7 -2 }}, {{val| 0 1 -3 3 }}]


Comma list: 28/27, 35/32
[[POTE generator]]: ~3/2 = 691.351


Mapping: [{{val| 1 0 7 -2 }}, {{val| 0 1 -3 3 }}]
{{Multival|legend=1| 1 -3 3 -7 2 15 }}
 
Wedgie: {{wedgie| 1 -3 3 -7 2 15 }}


{{Val list|legend=1| 2d, 5c, 19ccdd }}
{{Val list|legend=1| 2d, 5c, 19ccdd }}


Badness: 0.0585
[[Badness]]: 0.0585
 
</div></div>


= Kangaroo =
= Kangaroo =
== 5-limit ==
== 5-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~27/20 = 561.195
EDO generators: [[15edo|9\15]]
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: 2.3.5
Subgroup: 2.3.5
Line 291: Line 173:


Mapping: [{{val| 1 0 -3 }}, {{val| 0 3 10 }}]
Mapping: [{{val| 1 0 -3 }}, {{val| 0 3 10 }}]
POTE generator: ~27/20 = 561.195


{{Val list|legend=1| 15, 47b, 62b }}
{{Val list|legend=1| 15, 47b, 62b }}


Badness: 0.2958
Badness: 0.2958
</div></div>


== 7-limit ==
== 7-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~7/5 = 560.328
EDO generators: [[15edo|7\15]]
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: 2.3.5.7
Subgroup: 2.3.5.7
Line 318: Line 188:
Mapping: [{{val| 1 0 -3 -2 }}, {{val| 0 3 10 9 }}]
Mapping: [{{val| 1 0 -3 -2 }}, {{val| 0 3 10 9 }}]


Wedgie: {{wedgie| 3 10 9 9 6 -7 }}
{{Multival|legend=1| 3 10 9 9 6 -7 }}
 
POTE generator: ~7/5 = 560.328


{{Val list|legend=1| 15 }}
{{Val list|legend=1| 15 }}


Badness: 0.0779
Badness: 0.0779
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~7/5 = 560.155
EDO generators: [[15edo|7\15]]
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: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 345: Line 203:


Mapping: [{{val| 1 0 -3 -2 -4 }}, {{val| 0 3 10 9 14 }}]
Mapping: [{{val| 1 0 -3 -2 -4 }}, {{val| 0 3 10 9 14 }}]
POTE generator: ~7/5 = 560.155


{{Val list|legend=1| 15 }}
{{Val list|legend=1| 15 }}


Badness: 0.0432
Badness: 0.0432
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~7/5 = 559.770
EDO generators: [[15edo|7\15]]
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: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 371: Line 217:


Mapping: [{{val| 1 0 -3 -2 -4 0 }}, {{val| 0 3 10 9 14 7 }}]
Mapping: [{{val| 1 0 -3 -2 -4 0 }}, {{val| 0 3 10 9 14 7 }}]
POTE generator: ~7/5 = 559.770


{{Val list|legend=1| 15 }}
{{Val list|legend=1| 15 }}


Badness: 0.0327
Badness: 0.0327
</div></div>


= Quindecic =
= Quindecic =
Period: 1\15
Optimal ([[POTE]]) generator: ~13/8 = 852.924
EDO generators: [[30edo|21\30]], [[45edo|32\45]]
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: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 397: Line 231:


Mapping: [{{val| 15 24 35 42 52 0 }}, {{val| 0 0 0 0 0 1 }}]
Mapping: [{{val| 15 24 35 42 52 0 }}, {{val| 0 0 0 0 0 1 }}]
POTE generator: ~13/8 = 852.924


{{Val list|legend=1| 15, 30, 45bc }}
{{Val list|legend=1| 15, 30, 45bc }}


Badness: 0.0289
Badness: 0.0289
</div></div>


[[Category:Theory]]
[[Category:Theory]]

Revision as of 09:57, 27 March 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

Template:Val list

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 ]]

POTE generator: ~3/2 = 742.002

Minimax tuning:

[[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

Template:Val list

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

Minimax tuning:

Template:Val list

Badness: 0.0727

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

Minimax tuning:

Template:Val list

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

Template:Val list

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

Minimax tuning:

  • 7-odd-limit eigenmonzo subgroup: 2.7
  • 9-odd-limit eigenmonzo subgroup: 2.7

Template:Val list

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

Template:Val list

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

Template:Val list

Badness: 0.0194

Wallaby

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

POTE generator: ~3/2 = 691.351

Wedgie: ⟨⟨ 1 -3 3 -7 2 15 ]]

Template:Val list

Badness: 0.0585

Kangaroo

5-limit

Subgroup: 2.3.5

Comma list: 64000/59049

Mapping: [⟨1 0 -3], ⟨0 3 10]]

POTE generator: ~27/20 = 561.195

Template:Val list

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Template:Val list

Badness: 0.0289

Retrieved from "https://en.xen.wiki/index.php?title=Trienstonic_clan&oldid=64535"