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Sensamagic family: Difference between revisions

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


= Sensamagic =
== Sensamagic ==
[[Comma]]: 245/243
{{Main| Sensamagic }}


[[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|legend=1| 1 0 0 0 | 0 1 1 2 | 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>]
[[Mapping 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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020


Angle(3/2, 9/7) = 86.5288
[[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 to lattice: [<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 }}


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
 
{{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
}}
 
=== Overview to extensions ===
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
 
== Undecimal sensamagic ==
{{Main| Sensamagic }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 245/243, 385/384
 
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178
 
[[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


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]]
{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}


[[Badness]]: 0.000129
[[Badness]]: 0.722 × 10<sup>-3</sup>


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


=== [[Minkowski blocks]] ===
=== 13-limit ===
{2, 3, 7} subgroup
Subgroup: 2.3.5.7.11.13


12: 729/686, 64/63
Comma list: 245/243, 352/351, 364/363


17: 64/63, 19683/19208
Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }}


19: 49/48, 177147/175616
{{Optimal ET sequence|legend=1| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}


22: 64/63, 537824/531441
Badness: 1.20 × 10<sup>-3</sup>


24: 64/63, 15059072/14348907
== Sensawer ==
[[Subgroup]]: 2.3.5.7.11


== Undecimal sensamagic ==
[[Comma list]]: 245/243, 441/440
Commas: 245/243, 385/384


[[11-limit]] minimax
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}


[|1 0 0 0 0&gt;, |21/13 6/13 -1/13 1/13 -3/13&gt;,
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359
|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
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}


Map: [&lt;1 0 0 0 7|, &lt;0 1 1 2 -2|, &lt;0 0 2 -1 -1|]
[[Badness]]: 0.796 × 10<sup>-3</sup>


Generators: 2, 3, 9/7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: [[5edo|5]], 17, 19, 22, 41, 68, 87, [[196edo|196]], [[283edo|283]], 607bd, 694bd
Comma list: 196/195, 245/243, 352/351


Badness: 0.000722
Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }}


Projection pairs: 5 243/49 11 896/81 to 2.3.7
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}


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


Generators: 2, 3, 9/7
[[Comma list]]: 100/99, 245/243


EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }}


Badness: 0.00120
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050


= Sensawer =
{{Optimal ET sequence|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
Commas: 245/243, 441/440


Map: [&lt;1 0 0 0 -3|, &lt;0 1 1 2 5|, &lt;0 0 2 -1 -4|]
[[Badness]]: 0.581 × 10<sup>-3</sup>


EDOs: 41, 87, 302d, 389d, 476bd
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]


Badness: 0.000796
== Shrusus ==
[[Subgroup]]: 2.3.5.7.11


== 13-limit ==
[[Comma list]]: 176/175, 245/243
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|]
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}


EDOs: 41, 46, 87, 522bd
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147


Badness: 0.000928
{{Optimal ET sequence|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}


= Octarod =
[[Badness]]: 0.877 × 10<sup>-3</sup>
Commas: 100/99, 245/243


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


EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e
Comma list: 91/90, 176/175, 245/243


Badness: 0.000581
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}


= Shrusus =
{{Optimal ET sequence|legend=1| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
Commas: 176/175, 245/243


Map: [&lt;1 0 0 0 -4|, &lt;0 1 1 2 4|, &lt;0 0 2 -1 3|]
Badness: 1.125 × 10<sup>-3</sup>


EDOs: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
== Bisector ==
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000877
[[Comma list]]: 121/120, 245/243


== Shrusic ==
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}
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|]
: mapping generators: ~77/54, ~3, ~9/7


EDOs: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
[[Optimal tuning]] ([[POTE]]): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060


Badness: 0.001125
{{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}


= Sensigh =
[[Badness]]: 1.089 × 10<sup>-3</sup>
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|]
== Sensigh ==
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: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 | 0 0 0 0 1 0 }}


== 17-limit ==
: mapping generators: ~2, ~9/7, ~11
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|]
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df, 157df }}


EDOs: 19, 27, 46
Badness: 0.939 × 10<sup>-3</sup>


= Bisector =
=== 17-limit ===
Commas: 121/120, 245/243
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 0 0 3|, &lt;0 1 1 2 1|, &lt;0 0 2 -1 1|]
Comma list: 91/90, 126/125, 154/153, 169/168


EDOs: 22, 46, 68, 82e, 106de, 114, 128e
Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }}


Badness: 0.001089
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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