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|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: 4375/4374, 3025/3024


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: 4375/4374, 3025/3024, 4225/4224
[[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
 
Comma list: 3025/3024, 4225/4224, 4375/4374


==Thunor==
Mapping: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}
Commas: 4375/4374, 3025/3024, 1716/1715


13 and 15 limit minimax
: mapping generators: ~99/70, ~3, ~220/117


[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,  
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]
|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
Lattice basis:  
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


Lattice basis: 10/9 length 0.4234 6/5 length 0.8426
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


Angle(10/9, 6/5) = 84.126 degrees
{{Optimal ET sequence|legend=1| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}


Map to lattice: [&lt;0 -1 -2 -3 -2 -7|, &lt;0 -1 -2 1 1 1|] 
Badness: 0.322 × 10<sup>-3</sup>


Map: [&lt;2 0 0 2 5 -3|, &lt;0 1 0 7 5 15|, &lt;0 0 1 -4 -3 -8|]
Scales: [[Donar46]]


Generators: 99/70, 3, 5
== Heimdall ==
[[Subgroup]]: 2.3.5.7.11


Edos: 26, 46, 72, 178, 224, 270, 494
[[Comma list]]: 4375/4374, 117649/117612


Badness: 0.000341
{{Mapping|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}


==Rym==
: mapping generators: ~2, ~343/198, ~5
Commas: 4375/4374, 3025/3024, 4096/4095


Map: [&lt;2 0 0 2 5 22|, &lt;0 1 0 7 5 -9|, &lt;0 0 1 -4 -3 3|]
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}


EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870
== Bragi ==
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000549
[[Comma list]]: 4375/4374, 2097152/2096325


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


EDOs: 46, 224, 270, 494, 764, 1395, 1889
{{Optimal ET sequence|legend=1| 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}


==13-limit==
=== 13-limit ===
Commas: 4225/4224, 4375/4374, 6656/6655
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 1 5 6|, &lt;0 1 -1 11 1 2|, &lt;0 0 5 -20 -4 -7|]
Comma list: 2080/2079, 4375/4374, 10985/10976


EDOs: 46, 224, 270, 494, 764, 1395, 1889
Mapping: {{mapping| 1 0 2 -7 8 -6 | 0 1 0 7 -3 7 | 0 0 3 -12 2 -13 }}


=Bragi=
{{Optimal ET sequence|legend=1| 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}
Commas: 4375/4374, 2097152/2096325


Map: [&lt;1 0 2 -7 8|, &lt;0 1 0 7 -3|, &lt;0 0 3 -12 2|]
== Vidar ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 46, 224, 270, 494, 764, 1053, 1547
[[Comma list]]: 4375/4374, 100663296/100656875


=Beyla=
{{Mapping|legend=1| 1 0 0 1 5 | 0 1 4 -9 -3 | 0 0 -5 20 4 }}
Commas: 4374/4375, 385/384


Map: [&lt;1 0 0 1 6|, &lt;0 1 0 7 -6|, &lt;0 0 1 -4 3|]
: mapping generators: ~2, ~3, ~96/55


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


Badness: 0.000721
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 4225/4224, 4375/4374, 6656/6655
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|]
Mapping: {{mapping| 1 0 0 1 5 6 | 0 1 4 -9 -3 -5 | 0 0 -5 20 4 7 }}


EDOs: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef
{{Optimal ET sequence|legend=1| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}


Badness: 0.000840
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragismic family| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 3]]

Latest revision as of 00:32, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The ragismic family of temperaments are rank-3 microtemperaments which temper out 4375/4374.

Temperaments discussed elsewhere include:

Considered below are beyla, ulmo, thor, heimdall, bragi, and vidar.

Ragismic

Subgroup: 2.3.5.7

Comma list: 4375/4374

Mapping: [⟨1 0 0 1], ⟨0 1 0 7], ⟨0 0 1 -4]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 -1 -1 -3], ⟨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:

Optimal ET sequence: 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: [⟨1 0 0 1 6], ⟨0 1 0 7 -6], ⟨0 0 1 -4 3]]

Optimal ET sequence: 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee

Badness: 0.721 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 1 6 2], ⟨0 1 0 7 -6 4], ⟨0 0 1 -4 3 -2]]

Optimal ET sequence: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff

Badness: 0.840 × 10-3

Ulmo

Subgroup: 2.3.5.7.11

Comma list: 176/175, 2200/2187

Mapping: [⟨1 0 0 1 -3], ⟨0 1 0 7 7], ⟨0 0 1 -4 -2]]

Optimal ET sequence: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd

Badness: 1.23 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324

Mapping: [⟨1 0 0 1 -3 2], ⟨0 1 0 7 7 4], ⟨0 0 1 -4 -2 -2]]

Optimal ET sequence: 7, 39, 46, 53, 80, 133d, 179, 312cdf

Badness: 0.932 × 10-3

Thor

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374

Mapping: [⟨2 0 0 2 5], ⟨0 1 0 7 5], ⟨0 0 1 -4 -3]]

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

Minimax tuning:

Optimal ET sequence: 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee

Badness: 0.0888 × 10-3

Projection pairs: 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: [⟨2 0 0 2 5 -3], ⟨0 1 0 7 5 15], ⟨0 0 1 -4 -3 -8]]

Mapping to lattice: [⟨0 -1 -2 -3 -2 -7], ⟨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: 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258

Badness: 0.341 × 10-3

Rym

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨2 0 0 2 5 22], ⟨0 1 0 7 5 -9], ⟨0 0 1 -4 -3 3]]

Optimal ET sequence: 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870

Badness: 0.549 × 10-3

Donar

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374

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: [⟨0 2 2 6 4 1], ⟨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°

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: 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff

Badness: 0.322 × 10-3

Scales: Donar46

Heimdall

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612

Mapping: [⟨1 0 0 1 2], ⟨0 2 0 14 37], ⟨0 0 1 -4 -12]]

mapping generators: ~2, ~343/198, ~5

Optimal ET sequence: 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d

Bragi

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 2097152/2096325

Mapping: [⟨1 0 2 -7 8], ⟨0 1 0 7 -3], ⟨0 0 3 -12 2]]

mapping generators: ~2, ~3, ~320/297

Optimal ET sequence: 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 2 -7 8 -6], ⟨0 1 0 7 -3 7], ⟨0 0 3 -12 2 -13]]

Optimal ET sequence: 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f

Vidar

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 100663296/100656875

Mapping: [⟨1 0 0 1 5], ⟨0 1 4 -9 -3], ⟨0 0 -5 20 4]]

mapping generators: ~2, ~3, ~96/55

Optimal ET sequence: 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 1 5 6], ⟨0 1 4 -9 -3 -5], ⟨0 0 -5 20 4 7]]

Optimal ET sequence: 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f

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