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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]]'' → [[Sengic family #Sensigh|Sengic family]]
* ''[[Catakleismoid]]'' → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 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>]
: mapping generators: ~2, ~3, ~5


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


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


Angle(10/9, 6/5) = 105.299
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0018{{c}}, ~3/2 = 701.9851{{c}}, ~5/4 = 386.2710{{c}}
: [[error map]]: {{val| +0.002 +0.032 -0.039 -0.014 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9855{{c}}, ~5/4 = 386.2724{{c}}
: error map: {{val| 0.000 +0.031 -0.041 -0.017 }}


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
[[Badness]] (Sintel): 0.0515


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
[[Projection pair]]: <code>7 ~4374/625</code>


[[Projection pair]]: 7 4374/625
Scales: [[Ragismic19]], [[Ragismic chromatic]]


Scales: [[Ragismic19]]
== Beyla ==
[[Subgroup]]: 2.3.5.7.11


= Thor =
[[Comma list]]: 385/384, 4375/4374
Commas: 3025/3024, 4375/4374


Minimax tuning
{{Mapping|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}


[|1 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2&gt;,
[[Optimal tuning]]s:
|25/16 0 7/8 5/8 -7/8&gt;, |0 0 0 1 0&gt;,  
* [[WE]]: ~2 = 1200.4709{{c}}, ~3/2 = 701.3876{{c}}, ~5/4 = 385.3287{{c}}
|25/16 0 -1/8 5/8 1/8&gt;]
: [[error map]]: {{val| +0.471 -0.097 -0.043 -0.427 -0.833 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2311{{c}}, ~5/4 = 385.3753{{c}}
: error map: {{val| 0.000 -0.724 -0.938 -1.709 -2.579 }}


Eigenmonzos: 2, 11/10, 8/7
{{Optimal ET sequence|legend=1| 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee }}


Map: [&lt;2 0 0 2 5|, &lt;0 1 0 7 5|, &lt;0 0 1 -4 -3|]
[[Badness]] (Sintel): 0.866


Generators: 99/70, 3, 5
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0000888
Comma list: 169/168, 325/324, 385/384


Projection pairs: 2 9801/4900 7 21434787/3062500 11 2310905821257/210087500000 to 3.5.14/11
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}


== Donar ==
Optimal tunings:
Commas: 3025/3024, 4225/4224, 4375/4374
* WE: ~2 = 1200.6829{{c}}, ~3/2 = 701.4188{{c}}, ~5/4 = 385.0695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1881{{c}}, ~5/4 = 385.1147{{c}}


13-limit minimax
{{Optimal ET sequence|legend=0| 19, 26, 45f, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}


[|1 0 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2 0&gt;,
Badness (Sintel): 0.785
|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
== Ulmo ==
[[Subgroup]]: 2.3.5.7.11


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


[|1 0 0 0 0 0&gt;, |1/2 1/3 1/3 1/3 -1/3 0&gt;,
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
|9/8 7/12 7/12 1/3 -7/12 0&gt;, |0 0 0 1 0 0&gt;,
|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
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.1423{{c}}, ~3/2 = 703.8558{{c}}, ~5/4 = 389.2089{{c}}
: [[error map]]: {{val| -0.858 +1.043 +1.180 +1.329 -2.745 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9658{{c}}, ~5/4 = 388.9716{{c}}
: error map: {{val| 0.000 +2.011 +2.658 +3.048 -1.501 }}


Lattice basis: 44/39 length = 0.3480 6/5 length = 0.7828
{{Optimal ET sequence|legend=1| 19e, 27e, 34d, 46, 53, 80, 126, 179, 259cd }}


Angle(44/39, 6/5) = 85.3977
[[Badness]] (Sintel): 1.48


Map to lattice: [&lt;0 2 2 6 4 1|, &lt;0 -1 -2 1 1 2|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 0 1 -2 2 7|, &lt;0 1 0 7 5 3|, &lt;0 0 2 -8 -6 -5|]
Comma list: 169/168, 176/175, 325/324


Generators: 99/70, 3, 220/117
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}


Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390
Optimal tunings:  
* WE: ~2 = 1199.5491{{c}}, ~3/2 = 703.6725{{c}}, ~5/4 = 388.5875{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7489{{c}}, ~5/4 = 388.5028{{c}}


Badness: 0.000322
{{Optimal ET sequence|legend=0| 19e, 27e, 34d, 46, 53, 80, 133d, 179 }}
 
Badness (Sintel): 0.871
 
== Thor ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 4375/4374
 
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}
: mapping generators: ~99/70, ~3, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.0103{{c}}, ~3/2 = 701.9669{{c}}, ~5/4 = 386.2245{{c}}
: [[error map]]: {{val| +0.021 +0.033 -0.048 +0.045 -0.126 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9706{{c}}, ~5/4 = 386.2375{{c}}
: error map: {{val| 0.000 +0.016 -0.076 +0.018 -0.177 }}
 
[[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]] (Sintel): 0.107
 
[[Projection pair]]s: <code>2 9801/4900, 7 21434787/3062500, 11 2310905821257/210087500000</code> 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°
 
Optimal tunings:
* WE: ~99/70 = 599.9809{{c}}, ~3/2 = 702.0647{{c}}, ~5/4 = 386.3428{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0675{{c}}, ~5/4 = 386.3276{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13
 
{{Optimal ET sequence|legend=0| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}
 
Badness (Sintel): 0.319
 
Scales: [[Thunor46]]
 
=== 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 tunings:
* WE: ~99/70 = 599.9913{{c}}, ~3/2 = 702.0254{{c}}, ~5/4 = 386.3155{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0354{{c}}, ~5/4 = 386.3214{{c}}
 
{{Optimal ET sequence|legend=0| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
 
Badness (Sintel): 0.514
 
=== Donar ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374
 
Mapping: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}
: mapping generators: ~99/70, ~3, ~220/117
 
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]
 
Lattice basis:
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°
 
Optimal tunings:
* WE: ~99/70 = 600.0175{{c}}, ~3/2 = 701.9460{{c}}, ~220/117 = 1093.1197{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9416{{c}}, ~220/117 = 1093.0962{{c}}
 
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11
 
{{Optimal ET sequence|legend=0| 34d, 46, 80, 144, 178, 190, 224, 270, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}
 
Badness (Sintel): 0.301


Scales: [[Donar46]]
Scales: [[Donar46]]


== Thunor ==
== Heimdall ==
Commas: 1716/1715, 2080/2079, 3025/3024
[[Subgroup]]: 2.3.5.7.11
 
13 and 15 limit minimax


[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,  
[[Comma list]]: 4375/4374, 117649/117612
|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|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
: mapping generators: ~2, ~343/198, ~5


Lattice basis: 10/9 length 0.4234 6/5 length 0.8426
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0031{{c}}, ~343/198 = 950.9942{{c}}, ~5/4 = 386.2812{{c}}
: [[error map]]: {{val| +0.021 +0.033 -0.048 +0.045 -0.126 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9930{{c}}, ~5/4 = 386.2838{{c}}
: error map: {{val| 0.000 +0.016 -0.076 +0.018 -0.177 }}


Angle(10/9, 6/5) = 84.126 degrees
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}


Map to lattice: [&lt;0 -1 -2 -3 -2 -7|, &lt;0 -1 -2 1 1 1|] 
[[Badness]] (Sintel): 0.362


Map: [&lt;2 0 0 2 5 -3|, &lt;0 1 0 7 5 15|, &lt;0 0 1 -4 -3 -8|]
== Bragi ==
[[Subgroup]]: 2.3.5.7.11


Generators: 99/70, 3, 5
[[Comma list]]: 4375/4374, 2097152/2096325


Edos: 26, 46, 72, 178, 224, 270, 494
{{Mapping|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}
: mapping generators: ~2, ~3, ~320/297


Badness: 0.000341
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9750{{c}}, ~3/2 = 702.0222{{c}}, ~320/297 = 128.7779{{c}}
: [[error map]]: {{val| -0.025 +0.042 -0.030 -0.006 +0.046 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0330{{c}}, ~320/297 = 128.7791{{c}}
: error map: {{val| 0.000 +0.078 +0.024 +0.056 +0.141 }}


== Rym ==
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}
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|]
[[Badness]] (Sintel): 1.03


EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000549
Comma list: 2080/2079, 4375/4374, 10985/10976


= Vidar =
Mapping: {{mapping| 1 0 2 -7 8 -6 | 0 1 0 7 -3 7 | 0 0 3 -12 2 -13 }}
Commas: 4375/4374, 100663296/100656875


Map: [&lt;1 0 0 1 5|, &lt;0 1 -1 11 1|, &lt;0 0 5 -20 -4|]
Optimal tunings:  
* WE: ~2 = 1199.9874{{c}}, ~3/2 = 702.0172{{c}}, ~14/13 = 128.7600{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0228{{c}}, ~14/13 = 128.7608{{c}}


EDOs: 46, 224, 270, 494, 764, 1395, 1889
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


== 13-limit ==
Badness (Sintel): 0.865
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|]
== Vidar ==
Vidar tempers out the [[luxma]] and shares the same generators with [[lux]], from which it only differs in the mapping of [[prime interval|prime]] [[7/1|7]].


EDOs: 46, 224, 270, 494, 764, 1395, 1889
[[Subgroup]]: 2.3.5.7.11


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


Map: [&lt;1 0 2 -7 8|, &lt;0 1 0 7 -3|, &lt;0 0 3 -12 2|]
{{Mapping|legend=1| 1 0 -5 21 9 | 0 1 4 -9 -3 | 0 0 5 -20 -4 }}
: mapping generators: ~2, ~3, ~55/48


EDOs: 46, 224, 270, 494, 764, 1053, 1547
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9928{{c}}, ~3/2 = 701.9969{{c}}, ~55/48 = 235.6566{{c}}
: [[error map]]: {{val| -0.007 +0.035 -0.036 -0.016 +0.022 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9988{{c}}, ~55/48 = 235.6593{{c}}
: error map: {{val| 0.000 +0.044 -0.022 -0.002 +0.048 }}


= Beyla =
{{Optimal ET sequence|legend=1| 45, 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}
Commas: 385/384, 4375/4374


Map: [&lt;1 0 0 1 6|, &lt;0 1 0 7 -6|, &lt;0 0 1 -4 3|]
[[Badness]] (Sintel): 1.32


EDOs: 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000721
Comma list: 4225/4224, 4375/4374, 6656/6655


== 13-limit ==
Mapping: {{mapping| 1 0 -5 21 9 13 | 0 1 4 -9 -3 -5 | 0 0 5 -20 -4 -7 }}
Commas: 169/168, 325/324, 385/384


Map: [&lt;1 0 0 1 6 2|, &lt;0 1 0 7 -6 4|, &lt;0 0 1 -4 3 -2|]
Optimal tunings:  
* WE: ~2 = 1200.0042{{c}}, ~3/2 = 701.9794{{c}}, ~55/48 = 235.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9775{{c}}, ~55/48 = 235.6698{{c}}


EDOs: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef
{{Optimal ET sequence|legend=0| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}


Badness: 0.000840
Badness (Sintel): 0.593


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