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Ragismic family: Difference between revisions

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The '''ragisma family''' of temperaments are rank-3 microtemperaments which temper out [[4375/4374]].
{{Technical data page}}
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]]'' → [[Sensamagic family #Sensigh|Sensamagic family]]
* ''[[Catakleismoid]]'' → [[Marvel family #Catakleismoid|Marvel family]]
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]


= Ragismic =
Considered below are beyla, ulmo, thor, heimdall, bragi, and vidar.
[[Comma]]: 4375/4374
 
== Ragismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: [[4375/4374]]
 
{{Mapping|legend=1| 1 0 0 1 | 0 1 0 7 | 0 0 1 -4 }}
 
: mapping generators: ~2, ~3, ~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
: Angle (10/9, 6/5) = 105.299°


[[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 }}
 
[[Projection pair]]: ~7 = ~4374/625
 
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 ET sequence|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee }}


Lattice basis: 10/9 length = 0.789, 6/5 length = 0.921
[[Badness]]: 0.721 × 10<sup>-3</sup>


Angle(10/9, 6/5) = 105.299°
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map to lattice: [{{val|0 -1 -1 -3}}, {{val|0 -1 -2 1}}]
Comma list: 169/168, 325/324, 385/384


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


Mapping generators: 2, ~3, ~5
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}


{{Val list|legend=1| 9, 19, 26, 27, 45, 46, 53, 72, 80, 91, 99, 171, 270, 441, 494, 612, 665, 935, 1106, 1277, 1376, 1718, 1889, 6279 }}
Badness: 0.840 × 10<sup>-3</sup>


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


Scales: [[Ragismic19]]
[[Comma list]]: 176/175, 2200/2187


= Thor =
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
[[Comma list]]: 3025/3024, 4375/4374


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
* 11-odd-limit eigenmonzos: 2, 11/10, 8/7


[[Mapping]]: [{{val|2 0 0 2 5}}, {{val|0 1 0 7 5}}, {{val|0 0 1 -4 -3}}]
[[Badness]]: 1.23 × 10<sup>-3</sup>


Mapping generators: ~99/70, ~3, ~5
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 72, 118, 152, 224, 270, 342, 494, 612, 764, 836 }}
Comma list: 169/168, 176/175, 325/324


[[Badness]]: 0.0000888
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}


Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}


== Donar ==
Badness: 0.932 × 10<sup>-3</sup>
Comma list: 3025/3024, 4225/4224, 4375/4374


Minimax tuning:
== Thor ==
* 13-odd-limit eigenmonzos: 2, 11/10, 8/7
[[Subgroup]]: 2.3.5.7.11
* 15-odd-limit eigenmonzos: 2, 8/7, 15/11


Lattice basis: 44/39 length = 0.3480, 6/5 length = 0.7828
[[Comma list]]: 3025/3024, 4375/4374


Angle(44/39, 6/5) = 85.3977°
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}


Map to lattice: [{{val|0 2 2 6 4 1}}, {{val|0 -1 -2 1 1 2}}]
: mapping generators: ~99/70, ~3, ~5


Mapping: [{{val|2 0 1 -2 2 7}}, {{val|0 1 0 7 5 3}}, {{val|0 0 2 -8 -6 -5}}]
[[Minimax tuning]]:
* [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5


Mapping generators: ~99/70, ~3, ~220/117
{{Optimal ET sequence|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}


Vals: {{Val list| 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390 }}
[[Badness]]: 0.0888 × 10<sup>-3</sup>


Badness: 0.000322
[[Projection pair]]s: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11


Scales: [[Donar46]]
=== Thunor ===
Subgroup: 2.3.5.7.11.13


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


Minimax tuning:
Mapping: {{mapping| 2 0 0 2 5 -3 | 0 1 0 7 5 15 | 0 0 1 -4 -3 -8 }}
* 13- and 15-odd-limit eigenmonzos: 2, 16/13, 9/7
 
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
Lattice basis:
: 10/9 length = 0.4234, 6/5 length = 0.8426
: Angle (10/9, 6/5) = 84.126°


Angle(10/9, 6/5) = 84.126°
Minimax tuning:
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13


Map to lattice: [{{val|0 -1 -2 -3 -2 -7}}, {{val|0 -1 -2 1 1 1}}] 
{{Optimal ET sequence|legend=1| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}


Mapping: [{{val|2 0 0 2 5 -3}}, {{val|0 1 0 7 5 15}}, {{val|0 0 1 -4 -3 -8}}]
Badness: 0.341 × 10<sup>-3</sup>


Mapping generators: ~99/70, ~3, ~5
=== Rym ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 26, 46, 72, 178, 224, 270, 494 }}
Comma list: 3025/3024, 4096/4095, 4375/4374


Badness: 0.000341
Mapping: {{mapping| 2 0 0 2 5 22 | 0 1 0 7 5 -9 | 0 0 1 -4 -3 3 }}


== Rym ==
{{Optimal ET sequence|legend=1| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
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}}]
Badness: 0.549 × 10<sup>-3</sup>


Mapping generators: ~99/70, ~3, ~5
=== Donar ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
Comma list: 3025/3024, 4225/4224, 4375/4374


Badness: 0.000549
Mapping: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}


= Vidar =
: mapping generators: ~99/70, ~3, ~220/117
[[Comma list]]: 4375/4374, 100663296/100656875


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


Mapping generators: 2, ~3, ~55/32
Lattice basis:  
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


== 13-limit ==
{{Optimal ET sequence|legend=1| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}
Comma list: 4225/4224, 4375/4374, 6656/6655


Mapping: [{{val|1 0 0 1 5 6}}, {{val|0 1 -1 11 1 2}}, {{val|0 0 5 -20 -4 -7}}]
Badness: 0.322 × 10<sup>-3</sup>


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


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


= Bragi =
[[Comma list]]: 4375/4374, 117649/117612
[[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 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}


Mapping generators: 2, ~3, ~320/297
: mapping generators: ~2, ~343/198, ~5


{{Val list|legend=1| 46, 224, 270, 494, 764, 1053, 1547 }}
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}


== 13-limit ==
== Bragi ==
Comma list: 2080/2079, 4375/4374, 10985/10976
[[Subgroup]]: 2.3.5.7.11


Mapping: [{{val|1 0 2 -7 8 -6}}, {{val|0 1 0 7 -3 7}}, {{val|0 0 3 -12 2 -13}}]
[[Comma list]]: 4375/4374, 2097152/2096325


Mapping generators: 2, ~3, ~14/13
{{Mapping|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}


Vals: {{Val list| 19, 46, 224, 270, 494, 764 }}
: mapping generators: ~2, ~3, ~320/297


= Beyla =
{{Optimal ET sequence|legend=1| 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}
[[Comma list]]: 385/384, 4375/4374


[[Mapping]]: [{{val|1 0 0 1 6}}, {{val|0 1 0 7 -6}}, {{val|0 0 1 -4 3}}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping generators: 2, ~3, ~5
Comma list: 2080/2079, 4375/4374, 10985/10976


{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
Mapping: {{mapping| 1 0 2 -7 8 -6 | 0 1 0 7 -3 7 | 0 0 3 -12 2 -13 }}


[[Badness]]: 0.000721
{{Optimal ET sequence|legend=1| 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


== 13-limit ==
== Vidar ==
Comma list: 169/168, 325/324, 385/384
[[Subgroup]]: 2.3.5.7.11


Mapping: [{{val|1 0 0 1 6 2}}, {{val|0 1 0 7 -6 4}}, {{val|0 0 1 -4 3 -2}}]
[[Comma list]]: 4375/4374, 100663296/100656875


Mapping generators: 2, ~3, ~5
{{Mapping|legend=1| 1 0 0 1 5 | 0 1 4 -9 -3 | 0 0 -5 20 4 }}


Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}
: mapping generators: ~2, ~3, ~96/55


Badness: 0.000840
{{Optimal ET sequence|legend=1| 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}


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


[[Mapping]]: [{{val|1 0 0 1 2}}, {{val|0 2 0 14 37}}, {{val|0 0 1 -4 -12}}]
Comma list: 4225/4224, 4375/4374, 6656/6655


Mapping generators: 2, ~343/198, ~5
Mapping: {{mapping| 1 0 0 1 5 6 | 0 1 4 -9 -3 -5 | 0 0 -5 20 4 7 }}


{{Val list|legend=1| 270, 342, 612, 954, 1547, 1817 }}
{{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]]
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