Ragismic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Comma basis optimization
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,019 edits
No edit summary
Line 1: Line 1:
= Ragisma =
The '''ragisma family''' of temperaments are rank-3 microtemperaments which temper out 4375/4374.
Comma: 4375/4374


7-limit minimax
= Ragismic =
[[Comma]]: 4375/4374


[|1 0 0 0>, |-1/7 0 4/7 1/7>, |0 0 1 0>, |0 0 0 1>]
[[Minimax tuning]]:
* 7-odd-limit eigenmonzos: 2, 8/7, 5/4
* 9-odd-limit eigenmonzos: 2, 4/3, 7/5


Eigenmonzos: 2, 8/7, 5/4
Lattice basis: 10/9 length = 0.789, 6/5 length = 0.921


9-limit minimax
Angle(10/9, 6/5) = 105.299°


[|1 0 0 0>, |0 1 0 0>, |1/5 7/5 1/5 -1/5>,  
Map to lattice: [{{val|0 -1 -1 -3}}, {{val|0 -1 -2 1}}]
|1/5 7/5 -4/5 4/5>]


Eigenmonzos: 2, 4/3, 7/5
[[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
Mapping generators: 2, ~3, ~5


Angle(10/9, 6/5) = 105.299
{{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 }}


Map to lattice: [<0 -1 -1 -3|, <0 -1 -2 1|]
[[Projection pair]]: ~7 = ~4374/625
 
Map: [<1 0 0 1|, <0 1 0 7|, <0 0 1 -4|]
 
Generators: 2, 3, 5
 
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
 
[[Projection pair]]: 7 4374/625


Scales: [[Ragismic19]]
Scales: [[Ragismic19]]


= Thor =
= Thor =
Commas: 3025/3024, 4375/4374
[[Comma list]]: 3025/3024, 4375/4374


Minimax tuning
[[Minimax tuning]]:
* 11-odd-limit eigenmonzos: 2, 11/10, 8/7


[|1 0 0 0 0>, |3/4 0 1/2 1/2 -1/2>,
[[Mapping]]: [{{val|2 0 0 2 5}}, {{val|0 1 0 7 5}}, {{val|0 0 1 -4 -3}}]
|25/16 0 7/8 5/8 -7/8>, |0 0 0 1 0>,
|25/16 0 -1/8 5/8 1/8>]


Eigenmonzos: 2, 11/10, 8/7
Mapping generators: ~99/70, ~3, ~5


Map: [<2 0 0 2 5|, <0 1 0 7 5|, <0 0 1 -4 -3|]
{{Val list|legend=1| 72, 118, 152, 224, 270, 342, 494, 612, 764, 836 }}


Generators: 99/70, 3, 5
[[Badness]]: 0.0000888


Badness: 0.0000888
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 ==
Commas: 3025/3024, 4225/4224, 4375/4374
Comma list: 3025/3024, 4225/4224, 4375/4374
 
13-limit minimax
 
[|1 0 0 0 0 0>, |3/4 0 1/2 1/2 -1/2 0>,
|25/16 0 7/8 5/8 -7/8 0>, |0 0 0 1 0 0>,
|25/16 0 -1/8 5/8 1/8 0>,
|99/32 0 -11/16 -1/16 11/16 0>]
 
Eigenmonzos: 2, 11/10, 8/7
 
15-limit minimax
 
[|1 0 0 0 0 0>, |1/2 1/3 1/3 1/3 -1/3 0>,
|9/8 7/12 7/12 1/3 -7/12 0>, |0 0 0 1 0 0>,
|13/8 -1/12 -1/12 2/3 1/12 0>,
|55/16 -11/24 -11/24 1/6 11/24 0>]


Eigenmonzos: 2, 8/7, 15/11
Minimax tuning:
* 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
Lattice basis: 44/39 length = 0.3480, 6/5 length = 0.7828


Angle(44/39, 6/5) = 85.3977
Angle(44/39, 6/5) = 85.3977°


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


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


Generators: 99/70, 3, 220/117
Mapping generators: ~99/70, ~3, ~220/117


Edos: 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: 0.000322
Line 88: Line 64:


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


13 and 15 limit minimax
Minimax tuning:
* 13- and 15-odd-limit eigenmonzos: 2, 16/13, 9/7


[|1 0 0 0 0 0>, |7/10 4/5 0 -2/5 0 1/5>,
Lattice basis: 10/9 length = 0.4234, 6/5 length = 0.8426
|9/8 3/2 0 -3/4 0 1/4>, |7/5 -2/5 0 1/5 0 2/5>, |21/8 -1/2 0 1/4 0 1/4>, |0 0 0 0 0 1>]


Eigenmonzos: 2, 16/13, 9/7
Angle(10/9, 6/5) = 84.126°


Lattice basis: 10/9 length 0.4234 6/5 length 0.8426
Map to lattice: [{{val|0 -1 -2 -3 -2 -7}}, {{val|0 -1 -2 1 1 1}}] 


Angle(10/9, 6/5) = 84.126 degrees
Mapping: [{{val|2 0 0 2 5 -3}}, {{val|0 1 0 7 5 15}}, {{val|0 0 1 -4 -3 -8}}]


Map to lattice: [<0 -1 -2 -3 -2 -7|, <0 -1 -2 1 1 1|] 
Mapping generators: ~99/70, ~3, ~5


Map: [<2 0 0 2 5 -3|, <0 1 0 7 5 15|, <0 0 1 -4 -3 -8|]
Vals: {{Val list| 26, 46, 72, 178, 224, 270, 494 }}
 
Generators: 99/70, 3, 5
 
Edos: 26, 46, 72, 178, 224, 270, 494


Badness: 0.000341
Badness: 0.000341


== Rym ==
== Rym ==
Commas: 3025/3024, 4096/4095, 4375/4374
Comma list: 3025/3024, 4096/4095, 4375/4374


Map: [<2 0 0 2 5 22|, <0 1 0 7 5 -9|, <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}}]


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


Badness: 0.000549  
Badness: 0.000549  


= Vidar =
= Vidar =
Commas: 4375/4374, 100663296/100656875
[[Comma list]]: 4375/4374, 100663296/100656875


Map: [<1 0 0 1 5|, <0 1 -1 11 1|, <0 0 5 -20 -4|]
[[Mapping]]: [{{val|1 0 0 1 5}}, {{val|0 1 -1 11 1}}, {{val|0 0 5 -20 -4}}]


EDOs: 46, 224, 270, 494, 764, 1395, 1889
Mapping generators: 2, ~3, ~55/32
 
{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}


== 13-limit ==
== 13-limit ==
Commas: 4225/4224, 4375/4374, 6656/6655
Comma list: 4225/4224, 4375/4374, 6656/6655


Map: [<1 0 0 1 5 6|, <0 1 -1 11 1 2|, <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}}]


EDOs: 46, 224, 270, 494, 764, 1395, 1889
Mapping generators: 2, ~3, ~55/32
 
Vals: {{Val list| 46, 224, 270, 494, 764, 1395, 1889 }}


= Bragi =
= Bragi =
Commas: 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 generators: 2, ~3, ~320/297


Map: [<1 0 2 -7 8|, <0 1 0 7 -3|, <0 0 3 -12 2|]
{{Val list|legend=1| 46, 224, 270, 494, 764, 1053, 1547 }}


EDOs: 46, 224, 270, 494, 764, 1053, 1547
== 13-limit ==
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 generators: 2, ~3, ~14/13
 
Vals: {{Val list| 19, 46, 224, 270, 494, 764 }}


= Beyla =
= Beyla =
Commas: 385/384, 4375/4374
[[Comma list]]: 385/384, 4375/4374


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


EDOs: 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde
Mapping generators: 2, ~3, ~5


Badness: 0.000721
{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
 
[[Badness]]: 0.000721


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


Map: [<1 0 0 1 6 2|, <0 1 0 7 -6 4|, <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}}]


EDOs: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef
Mapping generators: 2, ~3, ~5
 
Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}


Badness: 0.000840
Badness: 0.000840
= Heimdall =
[[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
{{Val list|legend=1| 270, 342, 612, 954, 1547, 1817 }}


[[Category:Theory]]
[[Category:Theory]]

Revision as of 23:35, 14 May 2021

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

Ragismic

Comma: 4375/4374

Minimax tuning:

  • 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

Angle(10/9, 6/5) = 105.299°

Map to lattice: [⟨0 -1 -1 -3], ⟨0 -1 -2 1]]

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

Mapping generators: 2, ~3, ~5

Template:Val list

Projection pair: ~7 = ~4374/625

Scales: Ragismic19

Thor

Comma list: 3025/3024, 4375/4374

Minimax tuning:

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

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

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

Template:Val list

Badness: 0.0000888

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

Donar

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

Minimax tuning:

  • 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

Angle(44/39, 6/5) = 85.3977°

Map to lattice: [⟨0 2 2 6 4 1], ⟨0 -1 -2 1 1 2]]

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

Vals: Template:Val list

Badness: 0.000322

Scales: Donar46

Thunor

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

Minimax tuning:

  • 13- and 15-odd-limit eigenmonzos: 2, 16/13, 9/7

Lattice basis: 10/9 length = 0.4234, 6/5 length = 0.8426

Angle(10/9, 6/5) = 84.126°

Map to lattice: [⟨0 -1 -2 -3 -2 -7], ⟨0 -1 -2 1 1 1]]

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

Vals: Template:Val list

Badness: 0.000341

Rym

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: 0.000549

Vidar

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

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

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

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

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

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: 0.000840

Heimdall

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

Retrieved from "https://en.xen.wiki/index.php?title=Ragismic_family&oldid=69855"