Ragismic family: Difference between revisions

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The '''ragisma 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 not discussed here include [[Breed family #Ennealimmic|ennealimmic]].


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


[[Minimax tuning]]:
[[Comma list]]: 4375/4374
* 7-odd-limit eigenmonzos: 2, 8/7, 5/4
* 9-odd-limit eigenmonzos: 2, 4/3, 7/5


Lattice basis: 10/9 length = 0.789, 6/5 length = 0.921
[[Mapping]]: [{{val|1 0 0 1}}, {{val|0 1 0 7}}, {{val|0 0 1 -4}}]


Angle(10/9, 6/5) = 105.299°
Mapping generators: ~2, ~3, ~5


Map 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}}]


[[Mapping]]: [{{val|1 0 0 1}}, {{val|0 1 0 7}}, {{val|0 0 1 -4}}]
Lattice basis:  
: 10/9 length = 0.789, 6/5 length = 0.921
: Angle (10/9, 6/5) = 105.299°


Mapping generators: 2, ~3, ~5
[[Minimax tuning]]:
* 7-odd-limit eigenmonzos: 2, 8/7, 5/4
* 9-odd-limit eigenmonzos: 2, 4/3, 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 }}
{{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 }}
Line 27: Line 29:


= Thor =
= Thor =
Subgroup: 2.3.5.7.11
[[Comma list]]: 3025/3024, 4375/4374
[[Comma list]]: 3025/3024, 4375/4374
[[Minimax tuning]]:
* 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}}]
[[Mapping]]: [{{val|2 0 0 2 5}}, {{val|0 1 0 7 5}}, {{val|0 0 1 -4 -3}}]


Mapping generators: ~99/70, ~3, ~5
Mapping generators: ~99/70, ~3, ~5
[[Minimax tuning]]:
* 11-odd-limit eigenmonzos: 2, 11/10, 8/7


{{Val list|legend=1| 72, 118, 152, 224, 270, 342, 494, 612, 764, 836 }}
{{Val list|legend=1| 72, 118, 152, 224, 270, 342, 494, 612, 764, 836 }}


[[Badness]]: 0.0000888
[[Badness]]: 0.0888 × 10<sup>-3</sup>


Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11


== Donar ==
== Donar ==
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374
Comma list: 3025/3024, 4225/4224, 4375/4374


Minimax tuning:
Mapping: [{{val|2 0 1 -2 2 7}}, {{val|0 1 0 7 5 3}}, {{val|0 0 2 -8 -6 -5}}]
* 13-odd-limit eigenmonzos: 2, 11/10, 8/7
* 15-odd-limit eigenmonzos: 2, 8/7, 15/11


Lattice basis: 44/39 length = 0.3480, 6/5 length = 0.7828
Mapping generators: ~99/70, ~3, ~220/117
 
Angle(44/39, 6/5) = 85.3977°


Map to lattice: [{{val|0 2 2 6 4 1}}, {{val|0 -1 -2 1 1 2}}]
Map to lattice: [{{val|0 2 2 6 4 1}}, {{val|0 -1 -2 1 1 2}}]


Mapping: [{{val|2 0 1 -2 2 7}}, {{val|0 1 0 7 5 3}}, {{val|0 0 2 -8 -6 -5}}]
Lattice basis:  
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


Mapping generators: ~99/70, ~3, ~220/117
Minimax tuning:
* 13-odd-limit eigenmonzos: 2, 11/10, 8/7
* 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 }}
Vals: {{Val list| 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390 }}


Badness: 0.000322
Badness: 0.322 × 10<sup>-3</sup>


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


== Thunor ==
== Thunor ==
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024
Comma list: 1716/1715, 2080/2079, 3025/3024


Minimax tuning:
Mapping: [{{val| 2 0 0 2 5 -3 }}, {{val| 0 1 0 7 5 15 }}, {{val| 0 0 1 -4 -3 -8 }}]
* 13- and 15-odd-limit eigenmonzos: 2, 16/13, 9/7


Lattice basis: 10/9 length = 0.4234, 6/5 length = 0.8426
Mapping generators: ~99/70, ~3, ~5


Angle(10/9, 6/5) = 84.126°
Map 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:  
: 10/9 length = 0.4234, 6/5 length = 0.8426
: Angle (10/9, 6/5) = 84.126°


Mapping: [{{val|2 0 0 2 5 -3}}, {{val|0 1 0 7 5 15}}, {{val|0 0 1 -4 -3 -8}}]
Minimax tuning:
 
* 13- and 15-odd-limit eigenmonzos: 2, 16/13, 9/7
Mapping generators: ~99/70, ~3, ~5


Vals: {{Val list| 26, 46, 72, 178, 224, 270, 494 }}
Vals: {{Val list| 26, 46, 72, 178, 224, 270, 494 }}


Badness: 0.000341
Badness: 0.341 × 10<sup>-3</sup>


== Rym ==
== Rym ==
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: [{{val| 2 0 0 2 5 22 }}, {{val| 0 1 0 7 5 -9 }}, {{val| 0 0 1 -4 -3 3 }}]


Mapping generators: ~99/70, ~3, ~5
Mapping generators: ~99/70, ~3, ~5
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Vals: {{Val list| 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
Vals: {{Val list| 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}


Badness: 0.000549
Badness: 0.549 × 10<sup>-3</sup>


= Vidar =
= Vidar =
Subgroup: 2.3.5.7.11
[[Comma list]]: 4375/4374, 100663296/100656875
[[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]]: [{{val| 1 0 0 1 5 }}, {{val| 0 1 -1 11 1 }}, {{val| 0 0 5 -20 -4 }}]


Mapping generators: 2, ~3, ~55/32
Mapping generators: ~2, ~3, ~55/32


{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}
{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 4375/4374, 6656/6655
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}}]
Mapping: [{{val| 1 0 0 1 5 6 }}, {{val| 0 1 -1 11 1 2 }}, {{val| 0 0 5 -20 -4 -7 }}]


Mapping generators: 2, ~3, ~55/32
Mapping generators: ~2, ~3, ~55/32


Vals: {{Val list| 46, 224, 270, 494, 764, 1395, 1889 }}
Vals: {{Val list| 46, 224, 270, 494, 764, 1395, 1889 }}


= Bragi =
= Bragi =
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]]: [{{val| 1 0 2 -7 8 }}, {{val| 0 1 0 7 -3 }}, {{val| 0 0 3 -12 2 }}]


Mapping generators: 2, ~3, ~320/297
Mapping generators: 2, ~3, ~320/297
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== 13-limit ==
== 13-limit ==
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: [{{val| 1 0 2 -7 8 -6 }}, {{val| 0 1 0 7 -3 7 }}, {{val| 0 0 3 -12 2 -13 }}]


Mapping generators: 2, ~3, ~14/13
Mapping generators: 2, ~3, ~14/13
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= Beyla =
= Beyla =
Subgroup: 2.3.5.7.11
[[Comma list]]: 385/384, 4375/4374
[[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}}]
[[Mapping]]: [{{val| 1 0 0 1 6 }}, {{val| 0 1 0 7 -6 }}, {{val| 0 0 1 -4 3 }}]


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


{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
Line 144: Line 164:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 385/384
Comma list: 169/168, 325/324, 385/384


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


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


Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}
Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}


Badness: 0.000840
Badness: 0.840 × 10<sup>-3</sup>


= Heimdall =
= Heimdall =
Subgroup: 2.3.5.7.11
[[Comma list]]: 4375/4374, 117649/117612
[[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]]: [{{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 generators: ~2, ~343/198, ~5


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


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Ragisma]]
[[Category:Ragismic]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 07:29, 24 May 2021

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

Temperaments not discussed here include ennealimmic.

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

Map 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:

  • 7-odd-limit eigenmonzos: 2, 8/7, 5/4
  • 9-odd-limit eigenmonzos: 2, 4/3, 7/5

Template:Val list

Projection pair: ~7 = ~4374/625

Scales: Ragismic19

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:

  • 11-odd-limit eigenmonzos: 2, 11/10, 8/7

Template:Val list

Badness: 0.0888 × 10-3

Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11

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

Map 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 eigenmonzos: 2, 11/10, 8/7
  • 15-odd-limit eigenmonzos: 2, 8/7, 15/11

Vals: Template:Val list

Badness: 0.322 × 10-3

Scales: Donar46

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

Map 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 eigenmonzos: 2, 16/13, 9/7

Vals: Template:Val list

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]]

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

Vals: Template:Val list

Badness: 0.549 × 10-3

Vidar

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 100663296/100656875

Mapping: [⟨1 0 0 1 5], ⟨0 1 -1 11 1], ⟨0 0 5 -20 -4]]

Mapping generators: ~2, ~3, ~55/32

Template:Val list

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 -1 11 1 2], ⟨0 0 5 -20 -4 -7]]

Mapping generators: ~2, ~3, ~55/32

Vals: Template:Val list

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

Template:Val list

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]]

Mapping generators: 2, ~3, ~14/13

Vals: Template:Val list

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]]

Mapping generators: ~2, ~3, ~5

Template:Val list

Badness: 0.000721

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]]

Mapping generators: ~2, ~3, ~5

Vals: Template:Val list

Badness: 0.840 × 10-3

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

Template:Val list

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