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]]'' → [[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.  
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374
== Ragismic ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 0 0 1}}, {{val|0 1 0 7}}, {{val|0 0 1 -4}}]
[[Comma list]]: [[4375/4374]]


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


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


Lattice basis:  
Lattice basis:  
Line 19: Line 25:


[[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


{{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 }}
{{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
[[Projection pair]]: ~7 = ~4374/625


Scales: [[Ragismic19]]
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 }}
 
[[Badness]]: 0.721 × 10<sup>-3</sup>


= Thor =
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 3025/3024, 4375/4374
Comma list: 169/168, 325/324, 385/384
 
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}


[[Mapping]]: [{{val|2 0 0 2 5}}, {{val|0 1 0 7 5}}, {{val|0 0 1 -4 -3}}]
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}


Mapping generators: ~99/70, ~3, ~5
Badness: 0.840 × 10<sup>-3</sup>


[[Minimax tuning]]:
== Ulmo ==
* 11-odd-limit eigenmonzos: 2, 11/10, 8/7
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 176/175, 2200/2187


{{Val list|legend=1| 72, 118, 152, 224, 270, 342, 494, 612, 764, 836 }}
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}


[[Badness]]: 0.0888 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}


Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
[[Badness]]: 1.23 × 10<sup>-3</sup>


== 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 }}
 
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
 
Badness: 0.932 × 10<sup>-3</sup>


Mapping: [{{val|2 0 1 -2 2 7}}, {{val|0 1 0 7 5 3}}, {{val|0 0 2 -8 -6 -5}}]
== Thor ==
[[Subgroup]]: 2.3.5.7.11


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


Map to lattice: [{{val|0 2 2 6 4 1}}, {{val|0 -1 -2 1 1 2}}]
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}


Lattice basis:  
: mapping generators: ~99/70, ~3, ~5
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


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


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


== Thunor ==
=== Thunor ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Lattice basis:  
Lattice basis:  
Line 87: Line 110:


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


Vals: {{Val list| 26, 46, 72, 178, 224, 270, 494 }}
{{Optimal ET sequence|legend=1| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}


Badness: 0.341 × 10<sup>-3</sup>
Badness: 0.341 × 10<sup>-3</sup>


== Rym ==
=== Rym ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 }}
 
Mapping generators: ~99/70, ~3, ~5


Vals: {{Val list| 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
{{Optimal ET sequence|legend=1| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}


Badness: 0.549 × 10<sup>-3</sup>
Badness: 0.549 × 10<sup>-3</sup>


= Vidar =
=== Donar ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 4375/4374, 100663296/100656875
Comma list: 3025/3024, 4225/4224, 4375/4374


[[Mapping]]: [{{val| 1 0 0 1 5 }}, {{val| 0 1 -1 11 1 }}, {{val| 0 0 5 -20 -4 }}]
Mapping: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}


Mapping generators: ~2, ~3, ~55/32
: mapping generators: ~99/70, ~3, ~220/117


{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]


== 13-limit ==
Lattice basis:
Subgroup: 2.3.5.7.11.13
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


Comma list: 4225/4224, 4375/4374, 6656/6655
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


Mapping: [{{val| 1 0 0 1 5 6 }}, {{val| 0 1 -1 11 1 2 }}, {{val| 0 0 5 -20 -4 -7 }}]
{{Optimal ET sequence|legend=1| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}


Mapping generators: ~2, ~3, ~55/32
Badness: 0.322 × 10<sup>-3</sup>


Vals: {{Val list| 46, 224, 270, 494, 764, 1395, 1889 }}
Scales: [[Donar46]]


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


[[Comma list]]: 4375/4374, 2097152/2096325
[[Comma list]]: 4375/4374, 117649/117612


[[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 ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11


Comma list: 2080/2079, 4375/4374, 10985/10976
[[Comma list]]: 4375/4374, 2097152/2096325


Mapping: [{{val| 1 0 2 -7 8 -6 }}, {{val| 0 1 0 7 -3 7 }}, {{val| 0 0 3 -12 2 -13 }}]
{{Mapping|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}


Mapping generators: 2, ~3, ~14/13
: mapping generators: ~2, ~3, ~320/297


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


= Beyla =
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 385/384, 4375/4374
Comma list: 2080/2079, 4375/4374, 10985/10976


[[Mapping]]: [{{val| 1 0 0 1 6 }}, {{val| 0 1 0 7 -6 }}, {{val| 0 0 1 -4 3 }}]
Mapping: {{mapping| 1 0 2 -7 8 -6 | 0 1 0 7 -3 7 | 0 0 3 -12 2 -13 }}


Mapping generators: ~2, ~3, ~5
{{Optimal ET sequence|legend=1| 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
== Vidar ==
[[Subgroup]]: 2.3.5.7.11


[[Badness]]: 0.000721
[[Comma list]]: 4375/4374, 100663296/100656875


== 13-limit ==
{{Mapping|legend=1| 1 0 0 1 5 | 0 1 4 -9 -3 | 0 0 -5 20 4 }}
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 385/384
: mapping generators: ~2, ~3, ~96/55


Mapping: [{{val| 1 0 0 1 6 2 }}, {{val| 0 1 0 7 -6 4 }}, {{val| 0 0 1 -4 3 -2 }}]
{{Optimal ET sequence|legend=1| 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}


Mapping generators: ~2, ~3, ~5
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}
Comma list: 4225/4224, 4375/4374, 6656/6655
 
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 }}]


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