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Sensamagic family: Difference between revisions

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The '''sensamagic family''' of rank three temperaments tempers out [[245/243]].  
The '''sensamagic family''' of rank-3 temperaments tempers out [[245/243]].  


= Sensamagic =
= Sensamagic =
[[Comma]]: 245/243


[[7-limit]] [[minimax]]
Subgroup: 2.3.5.7


[|1 0 0 0>, |0 0 1/5 2/5>, |0 0 1 0>, |0 0 0 1>]
[[Comma list]]: 245/243


[[Eigenmonzo]]s: 2, 8/7, 5/4
[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]


[[9-limit]] minimax
Mapping generators: ~2, ~3, ~9/7


[|1 0 0 0>, |0 1 0 0>, |0 5/3 2/3 -2/3>, |0 5/3 -1/3 1/3>]
Map to lattice: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]


Eigenmonzos: 2, 4/3, 7/5
Lattice basis:  
: 3/2 length = 0.9644, 9/7 length = 1.0807
: Angle (3/2, 9/7) = 86.5288°


Lattice basis: 3/2 0.9644 9/7 1.0807
[[POTE generator]]s: ~3/2 = 703.7424, ~9/7 = 440.9020


Angle(3/2, 9/7) = 86.5288
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 0 1/5 2/5 }}, {{monzo| 0 0 1 0 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s: 2, 8/7, 5/4
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 0 5/3 2/3 -2/3 }}, {{monzo| 0 5/3 -1/3 1/3 }}]
: [[Eigenmonzo]]s: 2, 4/3, 7/5


Map to lattice: [<0 1 1 2|, <0 0 2 -1|]
{{Val list|legend=1| 17, 19, 22, 24, 27, 41, 46, 60, 68, 87, 109, 128, 169, 177, 237 }}


Map: [<1 0 0 0|, <0 1 1 2|, <0 0 2 -1|]
[[Badness]]: 0.129 × 10<sup>-3</sup>


Generators: 2, 3, 9/7
[[Projection pair]]: 5 243/49 to 2.3.7


Edos: [[17edo|17]], [[19edo|19]], [[22edo|22]], [[24edo|24]], [[27edo|27]], [[41edo|41]], [[46edo|46]], [[60edo|60]], [[68edo|68]], [[87edo|87]], [[109edo|109]], [[128edo|128]], [[169edo|169]], [[177edo|177]], [[237edo|237]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">
 
{2, 3, 7} subgroup
* 12: 729/686, 64/63
* 17: 64/63, 19683/19208
* 19: 49/48, 177147/175616
* 22: 64/63, 537824/531441
* 24: 64/63, 15059072/14348907


[[Badness]]: 0.000129
</div></div>


[[Projection pair]]: 5 243/49 to 2.3.7
= Undecimal sensamagic =


=== [[Minkowski blocks]] ===
Subgroup: 2.3.5.7.11
{2, 3, 7} subgroup


12: 729/686, 64/63
[[Comma list]]: 245/243, 385/384


17: 64/63, 19683/19208
[[Minimax tuning]]:  


19: 49/48, 177147/175616
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [[Eigenmonzo]]s: 2, 11/9, 7/5


22: 64/63, 537824/531441
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 1 2 -2 }}, {{val| 0 0 2 -1 -1 }}]


24: 64/63, 15059072/14348907
{{Val list|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}


== Undecimal sensamagic ==
[[Badness]]: 0.722 × 10<sup>-3</sup>
Commas: 245/243, 385/384


[[11-limit]] minimax
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7


[|1 0 0 0 0&gt;, |21/13 6/13 -1/13 1/13 -3/13&gt;,
== 13-limit ==
|35/13 10/13 7/13 -7/13 -5/13&gt;,
|35/13 10/13 -6/13 6/13 -5/13&gt;,
|42/13 -14/13 -2/13 2/13 7/13&gt;]


Eigenmonzos: 2, 11/9, 7/5
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 0 0 7|, &lt;0 1 1 2 -2|, &lt;0 0 2 -1 -1|]
Comma list: 245/243, 352/351, 385/384


Generators: 2, 3, 9/7
Mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 1 2 -2 -5 }}, {{val| 0 0 2 -1 -1 -1 }}]


EDOs: [[5edo|5]], 17, 19, 22, 41, 68, 87, [[196edo|196]], [[283edo|283]], 607bd, 694bd
{{Val list|legend=1| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}


Badness: 0.000722
Badness: 1.20 × 10<sup>-3</sup>


Projection pairs: 5 243/49 11 896/81 to 2.3.7
= Sensawer =


== 13-limit ==
Subgroup: 2.3.5.7.11
Commas: 245/243, 352/351, 385/384


Map: [&lt;1 0 0 0 7 12|, &lt;0 1 1 2 -2 -5|, &lt;0 0 2 -1 -1 -1|]
[[Comma list]]: 245/243, 441/440


Generators: 2, 3, 9/7
[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 1 2 5 }}, {{val| 0 0 2 -1 -4 }}]


EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
{{Val list|legend=1| 41, 87, 302d, 389d, 476bd }}


Badness: 0.00120
[[Badness]]: 0.796 × 10<sup>-3</sup>


= Sensawer =
== 13-limit ==
Commas: 245/243, 441/440


Map: [&lt;1 0 0 0 -3|, &lt;0 1 1 2 5|, &lt;0 0 2 -1 -4|]
Subgroup: 2.3.5.7.11.13


EDOs: 41, 87, 302d, 389d, 476bd
Comma list: 196/195, 245/243, 352/351


Badness: 0.000796
Mapping: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 1 2 5 2 }}, {{val| 0 0 2 -1 -4 -4 }}]


== 13-limit ==
{{Val list|legend=1| 41, 46, 87, 522bd }}
Commas: 196/195, 245/243, 352/351


Map: [&lt;1 0 0 0 -3 2|, &lt;0 1 1 2 5 2|, &lt;0 0 2 -1 -4 -4|]
Badness: 0.928 × 10<sup>-3</sup>


EDOs: 41, 46, 87, 522bd
= Octarod =


Badness: 0.000928
Subgroup: 2.3.5.7.11


= Octarod =
[[Comma list]]: 100/99, 245/243
Commas: 100/99, 245/243


Map: [&lt;1 0 0 0 2|, &lt;0 1 1 2 0|, &lt;0 0 2 -1 4|]
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 1 2 0 }}, {{val| 0 0 2 -1 4 }}]


EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e  
{{Val list|legend=1| 19, 22, 41, 63, 104, 109e, 131e, 172e }}


Badness: 0.000581
[[Badness]]: 0.581 × 10<sup>-3</sup>


= Shrusus =
= Shrusus =
Commas: 176/175, 245/243


Map: [&lt;1 0 0 0 -4|, &lt;0 1 1 2 4|, &lt;0 0 2 -1 3|]
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 176/175, 245/243
 
[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 1 2 4 }}, {{val| 0 0 2 -1 3 }}]


EDOs: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
{{Val list|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}


Badness: 0.000877
[[Badness]]: 0.877 × 10<sup>-3</sup>


== Shrusic ==
== Shrusic ==
Commas: 91/90, 176/175, 245/243


Map: [&lt;1 0 0 0 -4 1|, &lt;0 1 1 2 4 1|, &lt;0 0 2 -1 3 3|]
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 176/175, 245/243


EDOs: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
Mapping: [{{val| 1 0 0 0 -4 1 }}, {{val| 0 1 1 2 4 1 }}, {{val| 0 0 2 -1 3 3 }}]


Badness: 0.001125
{{Val list|legend=1| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
 
Badness: 1.125 × 10<sup>-3</sup>


= Sensigh =
= Sensigh =
Commas: 91/90, 126/125, 169/168


Map: [&lt;1 6 8 11 0 10|, &lt;0 -7 -9 -13 0 -10|, &lt;0 0 0 0 1 0|]
Subgroup: 2.3.5.7.11.13


EDOs: 19, 27, 46, 111df, 157df
Comma list: 91/90, 126/125, 169/168


Badness: 0.000939
Mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}, {{val| 0 0 0 0 1 0 }}]
 
{{Val list|legend=1| 19, 27, 46, 111df, 157df }}
 
Badness: 0.939 × 10<sup>-3</sup>


== 17-limit ==
== 17-limit ==
Commas: 91/90, 126/125, 154/153, 169/168


Map: [&lt;1 6 8 11 0 10 0|, &lt;0 -7 -9 -13 0 -10 1|, &lt;0 0 0 0 1 0 1|]
Subgroup: 2.3.5.7.11.13.17


EDOs: 19, 27, 46
Comma list: 91/90, 126/125, 154/153, 169/168
 
Mapping: [{{val| 1 6 8 11 0 10 0 }}, {{val| 0 -7 -9 -13 0 -10 1 }}, {{val| 0 0 0 0 1 0 1 }}]
 
{{Val list|legend=1| 19, 27, 46 }}


= Bisector =
= Bisector =
Commas: 121/120, 245/243


Map: [&lt;2 0 0 0 3|, &lt;0 1 1 2 1|, &lt;0 0 2 -1 1|]
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 121/120, 245/243
 
[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 1 2 1 }}, {{val| 0 0 2 -1 1 }}]


EDOs: 22, 46, 68, 82e, 106de, 114, 128e
{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}


Badness: 0.001089
[[Badness]]: 1.089 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Sensamagic]]
[[Category:Sensamagic]]
[[Category:Rank 3]]
[[Category:Rank 3]]
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