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__FORCETOC__
{{Technical data page}}
=Sensamagic=
The '''sensamagic family''' of [[rank-3 temperament]]s tempers out [[245/243]]. For a list of rank-2 temperaments, see [[Sensamagic clan]].
[[Comma|Comma]]: 245/243


[[7-limit|7-limit]] [[minimax|minimax]]
== Sensamagic ==
{{Main| Sensamagic }}


[|1 0 0 0>, |0 0 1/5 2/5>, |0 0 1 0>, |0 0 0 1>]
[[Subgroup]]: 2.3.5.7


[[Eigenmonzo|Eigenmonzo]]s: 2, 8/7, 5/4
[[Comma list]]: 245/243


[[9-limit|9-limit]] minimax
{{Mapping|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}


[|1 0 0 0>, |0 1 0 0>, |0 5/3 2/3 -2/3>, |0 5/3 -1/3 1/3>]
: mapping generators: ~2, ~3, ~9/7


Eigenmonzos: 2, 4/3, 7/5
[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]


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


Angle(3/2, 9/7) = 86.5288
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020


Map to lattice: [<0 1 1 2|, <0 0 2 -1|]
[[Minimax tuning]]:  
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


Map: [<1 0 0 0|, <0 1 1 2|, <0 0 2 -1|]
{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}


Generators: 2, 3, 9/7
[[Badness]]: 0.129 × 10<sup>-3</sup>


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]]
[[Projection pair]]: 5 243/49 to 2.3.7


[[Badness|Badness]]: 0.000129
{{Databox|[[Minkowski block]]s|
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
}}


[[Projection_pair|Projection pair]]: 5 243/49 to 2.3.7
=== Overview to extensions ===
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.  


===[[Minkowski_blocks|Minkowski blocks]]===
== Undecimal sensamagic ==
{2, 3, 7} subgroup
{{Main| Sensamagic }}


12: 729/686, 64/63
[[Subgroup]]: 2.3.5.7.11


17: 64/63, 19683/19208
[[Comma list]]: 245/243, 385/384


19: 49/48, 177147/175616
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}


22: 64/63, 537824/531441
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178


24: 64/63, 15059072/14348907
[[Minimax tuning]]:  
* [[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 basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


==Undecimal sensamagic==
{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}
Commas: 245/243, 385/384


[[11-limit|11-limit]] minimax
[[Badness]]: 0.722 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |21/13 6/13 -1/13 1/13 -3/13&gt;,
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7
|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
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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, 364/363


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


EDOs: [[5edo|5]], 17, 19, 22, 41, 68, 87, [[196edo|196]], [[283edo|283]], 607bd, 694bd
{{Optimal ET sequence|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 ==
[[Subgroup]]: 2.3.5.7.11


==13-limit==
[[Comma list]]: 245/243, 441/440
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|]
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}


Generators: 2, 3, 9/7
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359


EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}


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


=Sensawer=
=== 13-limit ===
Commas: 245/243, 441/440
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 0 -3|, &lt;0 1 1 2 5|, &lt;0 0 2 -1 -4|]
Comma list: 196/195, 245/243, 352/351


EDOs: 41, 87, 302d, 389d, 476bd
Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }}


Badness: 0.000796
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}


==13-limit==
Badness: 0.928 × 10<sup>-3</sup>
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|]
== Octarod ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 41, 46, 87, 522bd
[[Comma list]]: 100/99, 245/243


Badness: 0.000928
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }}


=Octarod=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050
Commas: 100/99, 245/243


Map: [&lt;1 0 0 0 2|, &lt;0 1 1 2 0|, &lt;0 0 2 -1 4|]
{{Optimal ET sequence|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}


EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e
[[Badness]]: 0.581 × 10<sup>-3</sup>


Badness: 0.000581
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]


=Shrusus=
== Shrusus ==
Commas: 176/175, 245/243
[[Subgroup]]: 2.3.5.7.11


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


EDOs: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}


Badness: 0.000877
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147


==Shrusic==
{{Optimal ET sequence|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
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|]
[[Badness]]: 0.877 × 10<sup>-3</sup>


EDOs: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
=== Shrusic ===
Subgroup: 2.3.5.7.11.13


Badness: 0.001125
Comma list: 91/90, 176/175, 245/243


=Sensigh=
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}
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|]
{{Optimal ET sequence|legend=1| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}


EDOs: 19, 27, 46, 111df, 157df
Badness: 1.125 × 10<sup>-3</sup>


Badness: 0.000939
== Bisector ==
[[Subgroup]]: 2.3.5.7.11


=Bisector=
[[Comma list]]: 121/120, 245/243
Commas: 121/120, 245/243


Map: [&lt;2 0 0 0 3|, &lt;0 1 1 2 1|, &lt;0 0 2 -1 1|]
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}


EDOs: 22, 46, 68, 82e, 106de, 114, 128e
: mapping generators: ~77/54, ~3, ~9/7


Badness: 0.001089
[[Optimal tuning]] ([[POTE]]): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060
[[Category:family]]
 
[[Category:overview]]
{{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
[[Category:planar]]
 
[[Category:sensamagic]]
[[Badness]]: 1.089 × 10<sup>-3</sup>
[[Category:theory]]
 
== Sensigh ==
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168
 
Mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 | 0 0 0 0 1 0 }}
 
: mapping generators: ~2, ~9/7, ~11
 
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df, 157df }}
 
Badness: 0.939 × 10<sup>-3</sup>
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 126/125, 154/153, 169/168
 
Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }}
 
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic| ]] <!-- key article -->
[[Category:Rank 3]]
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