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Ragismic family: Difference between revisions

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


Temperaments not discussed here include [[Breed family #Ennealimmic|ennealimmic]].
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]]


== Ragismic ==
Considered below are beyla, ulmo, thor, heimdall, bragi, and vidar.
Subgroup: 2.3.5.7
 
== Ragismic ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[4375/4374]]
[[Comma list]]: [[4375/4374]]


[[Mapping]]: [{{val| 1 0 0 1 }}, {{val| 0 1 0 7 }}, {{val| 0 0 1 -4 }}]
{{Mapping|legend=1| 1 0 0 1 | 0 1 0 7 | 0 0 1 -4 }}
: mapping generators: ~2, ~3, ~5


Mapping generators: ~2, ~3, ~5
[[Mapping to lattice]]: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]
 
Map to lattice: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]


Lattice basis:  
Lattice basis:  
: 10/9 length = 0.789, 6/5 length = 0.921
: 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 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]] eigenmonzos: 2, 8/7, 5/4
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]] eigenmonzos: 2, 4/3, 7/5
* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
 
{{Optimal ET sequence|legend=1| 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c }}
 
[[Badness]] (Sintel): 0.0515
 
[[Projection pair]]: <code>7 ~4374/625</code>
 
Scales: [[Ragismic19]], [[Ragismic chromatic]]
 
== Beyla ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 385/384, 4375/4374
 
{{Mapping|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4709{{c}}, ~3/2 = 701.3876{{c}}, ~5/4 = 385.3287{{c}}
: [[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 }}
 
{{Optimal ET sequence|legend=1| 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee }}
 
[[Badness]] (Sintel): 0.866


{{Val list|legend=1| 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Projection pair]]: ~7 = ~4374/625
Comma list: 169/168, 325/324, 385/384


Scales: [[Ragismic19]]
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}


== Thor ==
Optimal tunings:
Subgroup: 2.3.5.7.11
* 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}}
 
{{Optimal ET sequence|legend=0| 19, 26, 45f, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}


[[Comma list]]: 3025/3024, 4375/4374
Badness (Sintel): 0.785


[[Mapping]]: [{{val|2 0 0 2 5}}, {{val|0 1 0 7 5}}, {{val|0 0 1 -4 -3}}]
== Ulmo ==
[[Subgroup]]: 2.3.5.7.11


Mapping generators: ~99/70, ~3, ~5
[[Comma list]]: 176/175, 2200/2187


[[Minimax tuning]]:
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
* [[11-odd-limit]] eigenmonzos: 2, 11/10, 8/7


{{Val list|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
[[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 }}


[[Badness]]: 0.0888 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 19e, 27e, 34d, 46, 53, 80, 126, 179, 259cd }}


Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
[[Badness]] (Sintel): 1.48


=== Donar ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4225/4224, 4375/4374
Comma list: 169/168, 176/175, 325/324
 
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}


Mapping: [{{val|2 0 1 -2 2 7}}, {{val|0 1 0 7 5 3}}, {{val|0 0 2 -8 -6 -5}}]
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}}


Mapping generators: ~99/70, ~3, ~220/117
{{Optimal ET sequence|legend=0| 19e, 27e, 34d, 46, 53, 80, 133d, 179 }}


Map to lattice: [{{val|0 2 2 6 4 1}}, {{val|0 -1 -2 1 1 2}}]
Badness (Sintel): 0.871


Lattice basis:  
== Thor ==
: 44/39 length = 0.3480, 6/5 length = 0.7828
[[Subgroup]]: 2.3.5.7.11
: Angle (44/39, 6/5) = 85.3977°
 
[[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:
[[Minimax tuning]]:
* 13-odd-limit eigenmonzos: 2, 11/10, 8/7
* [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5
* 15-odd-limit eigenmonzos: 2, 8/7, 15/11


Vals: {{Val list| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}
{{Optimal ET sequence|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}


Badness: 0.322 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.107


Scales: [[Donar46]]
[[Projection pair]]s: <code>2 9801/4900, 7 21434787/3062500, 11 2310905821257/210087500000</code> to 3.5.14/11


=== Thunor ===
=== Thunor ===
Line 76: Line 133:
Comma list: 1716/1715, 2080/2079, 3025/3024
Comma list: 1716/1715, 2080/2079, 3025/3024


Mapping: [{{val| 2 0 0 2 5 -3 }}, {{val| 0 1 0 7 5 15 }}, {{val| 0 0 1 -4 -3 -8 }}]
Mapping: {{mapping| 2 0 0 2 5 -3 | 0 1 0 7 5 15 | 0 0 1 -4 -3 -8 }}
 
Mapping generators: ~99/70, ~3, ~5


Map to lattice: [{{val| 0 -1 -2 -3 -2 -7 }}, {{val| 0 -1 -2 1 1 1 }}]   
Mapping to lattice: [{{val| 0 -1 -2 -3 -2 -7 }}, {{val| 0 -1 -2 1 1 1 }}]   


Lattice basis:  
Lattice basis:  
: 10/9 length = 0.4234, 6/5 length = 0.8426
: 10/9 length = 0.4234, 6/5 length = 0.8426
: Angle (10/9, 6/5) = 84.126°
: 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:
Minimax tuning:
* 13- and 15-odd-limit eigenmonzos: 2, 16/13, 9/7
* 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 }}


Vals: {{Val list| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}
Badness (Sintel): 0.319


Badness: 0.341 × 10<sup>-3</sup>
Scales: [[Thunor46]]


=== Rym ===
=== Rym ===
Line 98: Line 159:
Comma list: 3025/3024, 4096/4095, 4375/4374
Comma list: 3025/3024, 4096/4095, 4375/4374


Mapping: [{{val| 2 0 0 2 5 22 }}, {{val| 0 1 0 7 5 -9 }}, {{val| 0 0 1 -4 -3 3 }}]
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 generators: ~99/70, ~3, ~5
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]


Vals: {{Val list| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
Lattice basis:  
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


Badness: 0.549 × 10<sup>-3</sup>
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}}


== Vidar ==
Minimax tuning:
Subgroup: 2.3.5.7.11
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


[[Comma list]]: 4375/4374, 100663296/100656875
{{Optimal ET sequence|legend=0| 34d, 46, 80, 144, 178, 190, 224, 270, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}


[[Mapping]]: [{{val| 1 0 0 1 5 }}, {{val| 0 1 -1 11 1 }}, {{val| 0 0 5 -20 -4 }}]
Badness (Sintel): 0.301


Mapping generators: ~2, ~3, ~55/32
Scales: [[Donar46]]


{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}
== Heimdall ==
[[Subgroup]]: 2.3.5.7.11


=== 13-limit ===
[[Comma list]]: 4375/4374, 117649/117612
Subgroup: 2.3.5.7.11.13


Comma list: 4225/4224, 4375/4374, 6656/6655
{{Mapping|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
: mapping generators: ~2, ~343/198, ~5


Mapping: [{{val| 1 0 0 1 5 6 }}, {{val| 0 1 -1 11 1 2 }}, {{val| 0 0 5 -20 -4 -7 }}]
[[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 }}


Mapping generators: ~2, ~3, ~55/32
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}


Vals: {{Val list| 46, 224, 270, 494, 764, 1395, 1889 }}
[[Badness]] (Sintel): 0.362


== Bragi ==
== Bragi ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 4375/4374, 2097152/2096325
[[Comma list]]: 4375/4374, 2097152/2096325


[[Mapping]]: [{{val| 1 0 2 -7 8 }}, {{val| 0 1 0 7 -3 }}, {{val| 0 0 3 -12 2 }}]
{{Mapping|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}
: mapping generators: ~2, ~3, ~320/297


Mapping generators: 2, ~3, ~320/297
[[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 }}


{{Val list|legend=1| 46, 224, 270, 494, 764, 1053, 1547 }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}


=== 13-limit ===
[[Badness]] (Sintel): 1.03
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 2 -7 8 -6 }}, {{val| 0 1 0 7 -3 7 }}, {{val| 0 0 3 -12 2 -13 }}]
Mapping: {{mapping| 1 0 2 -7 8 -6 | 0 1 0 7 -3 7 | 0 0 3 -12 2 -13 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


Mapping generators: 2, ~3, ~14/13
Badness (Sintel): 0.865


Vals: {{Val list| 19, 46, 224, 270, 494, 764 }}
== 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]].


== Beyla ==
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 0 1 6 }}, {{val| 0 1 0 7 -6 }}, {{val| 0 0 1 -4 3 }}]
{{Mapping|legend=1| 1 0 -5 21 9 | 0 1 4 -9 -3 | 0 0 5 -20 -4 }}
: mapping generators: ~2, ~3, ~55/48


Mapping generators: ~2, ~3, ~5
[[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 }}


{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
{{Optimal ET sequence|legend=1| 45, 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}


[[Badness]]: 0.721 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.32


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 385/384
Comma list: 4225/4224, 4375/4374, 6656/6655


Mapping: [{{val| 1 0 0 1 6 2 }}, {{val| 0 1 0 7 -6 4 }}, {{val| 0 0 1 -4 3 -2 }}]
Mapping: {{mapping| 1 0 -5 21 9 13 | 0 1 4 -9 -3 -5 | 0 0 5 -20 -4 -7 }}
 
Mapping generators: ~2, ~3, ~5
 
Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}
 
Badness: 0.840 × 10<sup>-3</sup>
 
== Heimdall ==
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 4375/4374, 117649/117612


[[Mapping]]: [{{val| 1 0 0 1 2 }}, {{val| 0 2 0 14 37 }}, {{val| 0 0 1 -4 -12 }}]
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}}


Mapping generators: ~2, ~343/198, ~5
{{Optimal ET sequence|legend=0| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}


{{Val list|legend=1| 270, 342, 612, 954, 1547, 1817 }}
Badness (Sintel): 0.593


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