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Ragismic family: Difference between revisions

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= Ragisma =
{{Technical data page}}
Comma: 4375/4374
The '''ragismic family''' of temperaments are rank-3 microtemperaments which temper out [[4375/4374]].


7-limit minimax
Temperaments discussed elsewhere include:
* ''[[Sensigh]]'' → [[Sensamagic family #Sensigh|Sensamagic family]]
* ''[[Catakleismoid]]'' → [[Marvel family #Catakleismoid|Marvel family]]
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]


[|1 0 0 0>, |-1/7 0 4/7 1/7>, |0 0 1 0>, |0 0 0 1>]
Considered below are beyla, ulmo, thor, heimdall, bragi, and vidar.


Eigenmonzos: 2, 8/7, 5/4
== Ragismic ==
[[Subgroup]]: 2.3.5.7


9-limit minimax
[[Comma list]]: [[4375/4374]]


[|1 0 0 0>, |0 1 0 0>, |1/5 7/5 1/5 -1/5>,
{{Mapping|legend=1| 1 0 0 1 | 0 1 0 7 | 0 0 1 -4 }}
|1/5 7/5 -4/5 4/5>]


Eigenmonzos: 2, 4/3, 7/5
: mapping generators: ~2, ~3, ~5


Lattice basis: 10/9 length=0.789 6/5 length=0.921
[[Mapping to lattice]]: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]


Angle(10/9, 6/5) = 105.299
Lattice basis:
: 10/9 length = 0.789, 6/5 length = 0.921
: Angle (10/9, 6/5) = 105.299°


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


Map: [<1 0 0 1|, <0 1 0 7|, <0 0 1 -4|]
{{Optimal ET sequence|legend=1| 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c }}


Generators: 2, 3, 5
[[Projection pair]]: ~7 = ~4374/625


EDOs: 9, 19, 26, 27, 45, 46, 53, 72, 80, 91, 99, 171, 270, 441, 494, 612, 665, 935, 1106, 1277, 1376, 1718, 1889, 6279
Scales: [[Ragismic19]], [[Ragismic chromatic]]


[[Projection pair]]: 7 4374/625
== Beyla ==
[[Subgroup]]: 2.3.5.7.11


Scales: [[Ragismic19]]
[[Comma list]]: 385/384, 4375/4374


= Thor =
{{Mapping|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}
Commas: 3025/3024, 4375/4374


Minimax tuning
{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee }}


[|1 0 0 0 0>, |3/4 0 1/2 1/2 -1/2>,
[[Badness]]: 0.721 × 10<sup>-3</sup>
|25/16 0 7/8 5/8 -7/8&gt;, |0 0 0 1 0&gt;,
|25/16 0 -1/8 5/8 1/8&gt;]


Eigenmonzos: 2, 11/10, 8/7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 0 0 2 5|, &lt;0 1 0 7 5|, &lt;0 0 1 -4 -3|]
Comma list: 169/168, 325/324, 385/384


Generators: 99/70, 3, 5
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}


Badness: 0.0000888
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}


Projection pairs: 2 9801/4900 7 21434787/3062500 11 2310905821257/210087500000 to 3.5.14/11
Badness: 0.840 × 10<sup>-3</sup>


== Donar ==
== Ulmo ==
Commas: 3025/3024, 4225/4224, 4375/4374
[[Subgroup]]: 2.3.5.7.11


13-limit minimax
[[Comma list]]: 176/175, 2200/2187


[|1 0 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2 0&gt;,
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
|25/16 0 7/8 5/8 -7/8 0&gt;, |0 0 0 1 0 0&gt;,
|25/16 0 -1/8 5/8 1/8 0&gt;,
|99/32 0 -11/16 -1/16 11/16 0&gt;]


Eigenmonzos: 2, 11/10, 8/7
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}


15-limit minimax
[[Badness]]: 1.23 × 10<sup>-3</sup>


[|1 0 0 0 0 0&gt;, |1/2 1/3 1/3 1/3 -1/3 0&gt;,
=== 13-limit ===
|9/8 7/12 7/12 1/3 -7/12 0&gt;, |0 0 0 1 0 0&gt;,
Subgroup: 2.3.5.7.11.13
|13/8 -1/12 -1/12 2/3 1/12 0&gt;,
|55/16 -11/24 -11/24 1/6 11/24 0&gt;]


Eigenmonzos: 2, 8/7, 15/11
Comma list: 169/168, 176/175, 325/324


Lattice basis: 44/39 length = 0.3480 6/5 length = 0.7828
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}


Angle(44/39, 6/5) = 85.3977
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}


Map to lattice: [&lt;0 2 2 6 4 1|, &lt;0 -1 -2 1 1 2|]
Badness: 0.932 × 10<sup>-3</sup>


Map: [&lt;2 0 1 -2 2 7|, &lt;0 1 0 7 5 3|, &lt;0 0 2 -8 -6 -5|]
== Thor ==
[[Subgroup]]: 2.3.5.7.11


Generators: 99/70, 3, 220/117
[[Comma list]]: 3025/3024, 4375/4374


Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}


Badness: 0.000322
: mapping generators: ~99/70, ~3, ~5


Scales: [[Donar46]]
[[Minimax tuning]]:
* [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5
 
{{Optimal ET sequence|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
 
[[Badness]]: 0.0888 × 10<sup>-3</sup>
 
[[Projection pair]]s: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
 
=== Thunor ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024
 
Mapping: {{mapping| 2 0 0 2 5 -3 | 0 1 0 7 5 15 | 0 0 1 -4 -3 -8 }}
 
Mapping to lattice: [{{val| 0 -1 -2 -3 -2 -7 }}, {{val| 0 -1 -2 1 1 1 }}] 
 
Lattice basis:
: 10/9 length = 0.4234, 6/5 length = 0.8426
: Angle (10/9, 6/5) = 84.126°
 
Minimax tuning:
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13
 
{{Optimal ET sequence|legend=1| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}
 
Badness: 0.341 × 10<sup>-3</sup>
 
=== Rym ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4096/4095, 4375/4374
 
Mapping: {{mapping| 2 0 0 2 5 22 | 0 1 0 7 5 -9 | 0 0 1 -4 -3 3 }}
 
{{Optimal ET sequence|legend=1| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
 
Badness: 0.549 × 10<sup>-3</sup>
 
=== Donar ===
Subgroup: 2.3.5.7.11.13


== Thunor ==
Comma list: 3025/3024, 4225/4224, 4375/4374
Commas: 1716/1715, 2080/2079, 3025/3024


13 and 15 limit minimax
Mapping: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}


[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,  
: mapping generators: ~99/70, ~3, ~220/117
|9/8 3/2 0 -3/4 0 1/4&gt;, |7/5 -2/5 0 1/5 0 2/5&gt;, |21/8 -1/2 0 1/4 0 1/4&gt;, |0 0 0 0 0 1&gt;]


Eigenmonzos: 2, 16/13, 9/7
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]


Lattice basis: 10/9 length 0.4234 6/5 length 0.8426
Lattice basis:  
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


Angle(10/9, 6/5) = 84.126 degrees
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


Map to lattice: [&lt;0 -1 -2 -3 -2 -7|, &lt;0 -1 -2 1 1 1|] 
{{Optimal ET sequence|legend=1| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}


Map: [&lt;2 0 0 2 5 -3|, &lt;0 1 0 7 5 15|, &lt;0 0 1 -4 -3 -8|]
Badness: 0.322 × 10<sup>-3</sup>


Generators: 99/70, 3, 5
Scales: [[Donar46]]


Edos: 26, 46, 72, 178, 224, 270, 494
== Heimdall ==
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000341
[[Comma list]]: 4375/4374, 117649/117612


== Rym ==
{{Mapping|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
Commas: 3025/3024, 4096/4095, 4375/4374


Map: [&lt;2 0 0 2 5 22|, &lt;0 1 0 7 5 -9|, &lt;0 0 1 -4 -3 3|]
: mapping generators: ~2, ~343/198, ~5


EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}


Badness: 0.000549
== Bragi ==
[[Subgroup]]: 2.3.5.7.11


= Vidar =
[[Comma list]]: 4375/4374, 2097152/2096325
Commas: 4375/4374, 100663296/100656875


Map: [&lt;1 0 0 1 5|, &lt;0 1 -1 11 1|, &lt;0 0 5 -20 -4|]
{{Mapping|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}


EDOs: 46, 224, 270, 494, 764, 1395, 1889
: mapping generators: ~2, ~3, ~320/297


== 13-limit ==
{{Optimal ET sequence|legend=1| 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}
Commas: 4225/4224, 4375/4374, 6656/6655


Map: [&lt;1 0 0 1 5 6|, &lt;0 1 -1 11 1 2|, &lt;0 0 5 -20 -4 -7|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 46, 224, 270, 494, 764, 1395, 1889
Comma list: 2080/2079, 4375/4374, 10985/10976


= Bragi =
Mapping: {{mapping| 1 0 2 -7 8 -6 | 0 1 0 7 -3 7 | 0 0 3 -12 2 -13 }}
Commas: 4375/4374, 2097152/2096325


Map: [&lt;1 0 2 -7 8|, &lt;0 1 0 7 -3|, &lt;0 0 3 -12 2|]
{{Optimal ET sequence|legend=1| 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


EDOs: 46, 224, 270, 494, 764, 1053, 1547
== Vidar ==
[[Subgroup]]: 2.3.5.7.11


= Beyla =
[[Comma list]]: 4375/4374, 100663296/100656875
Commas: 385/384, 4375/4374


Map: [&lt;1 0 0 1 6|, &lt;0 1 0 7 -6|, &lt;0 0 1 -4 3|]
{{Mapping|legend=1| 1 0 0 1 5 | 0 1 4 -9 -3 | 0 0 -5 20 4 }}


EDOs: 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde
: mapping generators: ~2, ~3, ~96/55


Badness: 0.000721
{{Optimal ET sequence|legend=1| 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}


== 13-limit ==
=== 13-limit ===
Commas: 169/168, 325/324, 385/384
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 1 6 2|, &lt;0 1 0 7 -6 4|, &lt;0 0 1 -4 3 -2|]
Comma list: 4225/4224, 4375/4374, 6656/6655


EDOs: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef
Mapping: {{mapping| 1 0 0 1 5 6 | 0 1 4 -9 -3 -5 | 0 0 -5 20 4 7 }}


Badness: 0.000840
{{Optimal ET sequence|legend=1| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragisma]]
[[Category:Ragismic family| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Ragismic_family"