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]]'' → [[Sengic family #Sensigh|Sengic family]]
* ''[[Catakleismoid]]'' → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 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°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0018{{c}}, ~3/2 = 701.9851{{c}}, ~5/4 = 386.2710{{c}}
: [[error map]]: {{val| +0.002 +0.032 -0.039 -0.014 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9855{{c}}, ~5/4 = 386.2724{{c}}
: error map: {{val| 0.000 +0.031 -0.041 -0.017 }}


[[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 }}
 
[[Badness]] (Sintel): 0.0515
 
[[Projection pair]]: <code>7 ~4374/625</code>
 
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 tuning]]s:
* [[WE]]: ~2 = 1200.4709{{c}}, ~3/2 = 701.3876{{c}}, ~5/4 = 385.3287{{c}}
: [[error map]]: {{val| +0.471 -0.097 -0.043 -0.427 -0.833 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2311{{c}}, ~5/4 = 385.3753{{c}}
: error map: {{val| 0.000 -0.724 -0.938 -1.709 -2.579 }}
 
{{Optimal ET sequence|legend=1| 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee }}


Lattice basis: 10/9 length = 0.789, 6/5 length = 0.921
[[Badness]] (Sintel): 0.866


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 tunings:  
* WE: ~2 = 1200.6829{{c}}, ~3/2 = 701.4188{{c}}, ~5/4 = 385.0695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1881{{c}}, ~5/4 = 385.1147{{c}}


{{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=0| 19, 26, 45f, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}


[[Projection pair]]: ~7 = ~4374/625
Badness (Sintel): 0.785


Scales: [[Ragismic19]]
== Ulmo ==
[[Subgroup]]: 2.3.5.7.11


= Thor =
[[Comma list]]: 176/175, 2200/2187
[[Comma list]]: 3025/3024, 4375/4374
 
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1423{{c}}, ~3/2 = 703.8558{{c}}, ~5/4 = 389.2089{{c}}
: [[error map]]: {{val| -0.858 +1.043 +1.180 +1.329 -2.745 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9658{{c}}, ~5/4 = 388.9716{{c}}
: error map: {{val| 0.000 +2.011 +2.658 +3.048 -1.501 }}
 
{{Optimal ET sequence|legend=1| 19e, 27e, 34d, 46, 53, 80, 126, 179, 259cd }}


[[Minimax tuning]]:
[[Badness]] (Sintel): 1.48
* 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}}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping generators: ~99/70, ~3, ~5
Comma list: 169/168, 176/175, 325/324


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


[[Badness]]: 0.0000888
Optimal tunings:  
* WE: ~2 = 1199.5491{{c}}, ~3/2 = 703.6725{{c}}, ~5/4 = 388.5875{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7489{{c}}, ~5/4 = 388.5028{{c}}


Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
{{Optimal ET sequence|legend=0| 19e, 27e, 34d, 46, 53, 80, 133d, 179 }}


== Donar ==
Badness (Sintel): 0.871
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 }}
: mapping generators: ~99/70, ~3, ~5


Map to lattice: [{{val|0 2 2 6 4 1}}, {{val|0 -1 -2 1 1 2}}]
[[Optimal tuning]]s:  
* [[WE]]: ~99/70 = 600.0103{{c}}, ~3/2 = 701.9669{{c}}, ~5/4 = 386.2245{{c}}
: [[error map]]: {{val| +0.021 +0.033 -0.048 +0.045 -0.126 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9706{{c}}, ~5/4 = 386.2375{{c}}
: error map: {{val| 0.000 +0.016 -0.076 +0.018 -0.177 }}


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]] (Sintel): 0.107


Badness: 0.000322
[[Projection pair]]s: <code>2 9801/4900, 7 21434787/3062500, 11 2310905821257/210087500000</code> 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
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°
Optimal tunings:
* WE: ~99/70 = 599.9809{{c}}, ~3/2 = 702.0647{{c}}, ~5/4 = 386.3428{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0675{{c}}, ~5/4 = 386.3276{{c}}


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


Lattice basis: 10/9 length = 0.4234, 6/5 length = 0.8426
{{Optimal ET sequence|legend=0| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}


Angle(10/9, 6/5) = 84.126°
Badness (Sintel): 0.319


Map to lattice: [{{val|0 -1 -2 -3 -2 -7}}, {{val|0 -1 -2 1 1 1}}]
Scales: [[Thunor46]]


Mapping: [{{val|2 0 0 2 5 -3}}, {{val|0 1 0 7 5 15}}, {{val|0 0 1 -4 -3 -8}}]
=== Rym ===
Subgroup: 2.3.5.7.11.13


Mapping generators: ~99/70, ~3, ~5
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 tunings:
* WE: ~99/70 = 599.9913{{c}}, ~3/2 = 702.0254{{c}}, ~5/4 = 386.3155{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0354{{c}}, ~5/4 = 386.3214{{c}}
 
{{Optimal ET sequence|legend=0| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
 
Badness (Sintel): 0.514
 
=== Donar ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374


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


Badness: 0.000341
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]


== Rym ==
Lattice basis:
Comma list: 3025/3024, 4096/4095, 4375/4374
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°


Mapping: [{{val|2 0 0 2 5 22}}, {{val|0 1 0 7 5 -9}}, {{val|0 0 1 -4 -3 3}}]
Optimal tunings:  
* WE: ~99/70 = 600.0175{{c}}, ~3/2 = 701.9460{{c}}, ~220/117 = 1093.1197{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9416{{c}}, ~220/117 = 1093.0962{{c}}


Mapping generators: ~99/70, ~3, ~5
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


Vals: {{Val list| 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 144, 178, 190, 224, 270, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}


Badness: 0.000549
Badness (Sintel): 0.301


= Vidar =
Scales: [[Donar46]]
[[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}}]
== Heimdall ==
[[Subgroup]]: 2.3.5.7.11


Mapping generators: 2, ~3, ~55/32
[[Comma list]]: 4375/4374, 117649/117612


{{Val list|legend=1| 46, 224, 270, 494, 764, 1395, 1889 }}
{{Mapping|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
: mapping generators: ~2, ~343/198, ~5


== 13-limit ==
[[Optimal tuning]]s:
Comma list: 4225/4224, 4375/4374, 6656/6655
* [[WE]]: ~2 = 1200.0031{{c}}, ~343/198 = 950.9942{{c}}, ~5/4 = 386.2812{{c}}
: [[error map]]: {{val| +0.021 +0.033 -0.048 +0.045 -0.126 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9930{{c}}, ~5/4 = 386.2838{{c}}
: error map: {{val| 0.000 +0.016 -0.076 +0.018 -0.177 }}


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| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}


Mapping generators: 2, ~3, ~55/32
[[Badness]] (Sintel): 0.362


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


= Bragi =
[[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|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}
: mapping generators: ~2, ~3, ~320/297


Mapping generators: 2, ~3, ~320/297
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9750{{c}}, ~3/2 = 702.0222{{c}}, ~320/297 = 128.7779{{c}}
: [[error map]]: {{val| -0.025 +0.042 -0.030 -0.006 +0.046 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0330{{c}}, ~320/297 = 128.7791{{c}}
: error map: {{val| 0.000 +0.078 +0.024 +0.056 +0.141 }}


{{Val list|legend=1| 46, 224, 270, 494, 764, 1053, 1547 }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}
 
[[Badness]] (Sintel): 1.03
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
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: {{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
Optimal tunings:  
* WE: ~2 = 1199.9874{{c}}, ~3/2 = 702.0172{{c}}, ~14/13 = 128.7600{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0228{{c}}, ~14/13 = 128.7608{{c}}


Vals: {{Val list| 19, 46, 224, 270, 494, 764 }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


= Beyla =
Badness (Sintel): 0.865
[[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}}]
== Vidar ==
Vidar tempers out the [[luxma]] and shares the same generators with [[lux]], from which it only differs in the mapping of [[prime interval|prime]] [[7/1|7]].


Mapping generators: 2, ~3, ~5
[[Subgroup]]: 2.3.5.7.11


{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cde }}
[[Comma list]]: 4375/4374, 100663296/100656875


[[Badness]]: 0.000721
{{Mapping|legend=1| 1 0 -5 21 9 | 0 1 4 -9 -3 | 0 0 5 -20 -4 }}
: mapping generators: ~2, ~3, ~55/48


== 13-limit ==
[[Optimal tuning]]s:
Comma list: 169/168, 325/324, 385/384
* [[WE]]: ~2 = 1199.9928{{c}}, ~3/2 = 701.9969{{c}}, ~55/48 = 235.6566{{c}}
: [[error map]]: {{val| -0.007 +0.035 -0.036 -0.016 +0.022 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9988{{c}}, ~55/48 = 235.6593{{c}}
: error map: {{val| 0.000 +0.044 -0.022 -0.002 +0.048 }}


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| 45, 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}


Mapping generators: 2, ~3, ~5
[[Badness]] (Sintel): 1.32


Vals: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190f, 197ef, 243ef }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000840
Comma list: 4225/4224, 4375/4374, 6656/6655


= Heimdall =
Mapping: {{mapping| 1 0 -5 21 9 13 | 0 1 4 -9 -3 -5 | 0 0 5 -20 -4 -7 }}
[[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}}]
Optimal tunings:  
* WE: ~2 = 1200.0042{{c}}, ~3/2 = 701.9794{{c}}, ~55/48 = 235.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9775{{c}}, ~55/48 = 235.6698{{c}}


Mapping generators: 2, ~343/198, ~5
{{Optimal ET sequence|legend=0| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}


{{Val list|legend=1| 270, 342, 612, 954, 1547, 1817 }}
Badness (Sintel): 0.593


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Ragismic family| ]] <!-- main article -->
[[Category:Ragisma]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 10:32, 22 April 2026

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°

Optimal tunings:

  • WE: ~2 = 1200.0018 ¢, ~3/2 = 701.9851 ¢, ~5/4 = 386.2710 ¢
error map: ⟨+0.002 +0.032 -0.039 -0.014]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9855 ¢, ~5/4 = 386.2724 ¢
error map: ⟨0.000 +0.031 -0.041 -0.017]

Minimax tuning:

Optimal ET sequence: 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c

Badness (Sintel): 0.0515

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 tunings:

  • WE: ~2 = 1200.4709 ¢, ~3/2 = 701.3876 ¢, ~5/4 = 385.3287 ¢
error map: ⟨+0.471 -0.097 -0.043 -0.427 -0.833]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2311 ¢, ~5/4 = 385.3753 ¢
error map: ⟨0.000 -0.724 -0.938 -1.709 -2.579]

Optimal ET sequence: 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee

Badness (Sintel): 0.866

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 tunings:

  • WE: ~2 = 1200.6829 ¢, ~3/2 = 701.4188 ¢, ~5/4 = 385.0695 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1881 ¢, ~5/4 = 385.1147 ¢

Optimal ET sequence: 19, 26, 45f, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff

Badness (Sintel): 0.785

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 tunings:

  • WE: ~2 = 1199.1423 ¢, ~3/2 = 703.8558 ¢, ~5/4 = 389.2089 ¢
error map: ⟨-0.858 +1.043 +1.180 +1.329 -2.745]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9658 ¢, ~5/4 = 388.9716 ¢
error map: ⟨0.000 +2.011 +2.658 +3.048 -1.501]

Optimal ET sequence: 19e, 27e, 34d, 46, 53, 80, 126, 179, 259cd

Badness (Sintel): 1.48

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 tunings:

  • WE: ~2 = 1199.5491 ¢, ~3/2 = 703.6725 ¢, ~5/4 = 388.5875 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7489 ¢, ~5/4 = 388.5028 ¢

Optimal ET sequence: 19e, 27e, 34d, 46, 53, 80, 133d, 179

Badness (Sintel): 0.871

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

Optimal tunings:

  • WE: ~99/70 = 600.0103 ¢, ~3/2 = 701.9669 ¢, ~5/4 = 386.2245 ¢
error map: ⟨+0.021 +0.033 -0.048 +0.045 -0.126]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9706 ¢, ~5/4 = 386.2375 ¢
error map: ⟨0.000 +0.016 -0.076 +0.018 -0.177]

Minimax tuning:

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

Badness (Sintel): 0.107

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°

Optimal tunings:

  • WE: ~99/70 = 599.9809 ¢, ~3/2 = 702.0647 ¢, ~5/4 = 386.3428 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0675 ¢, ~5/4 = 386.3276 ¢

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 (Sintel): 0.319

Scales: Thunor46

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 tunings:

  • WE: ~99/70 = 599.9913 ¢, ~3/2 = 702.0254 ¢, ~5/4 = 386.3155 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0354 ¢, ~5/4 = 386.3214 ¢

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

Badness (Sintel): 0.514

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°

Optimal tunings:

  • WE: ~99/70 = 600.0175 ¢, ~3/2 = 701.9460 ¢, ~220/117 = 1093.1197 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9416 ¢, ~220/117 = 1093.0962 ¢

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, 270, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff

Badness (Sintel): 0.301

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 tunings:

  • WE: ~2 = 1200.0031 ¢, ~343/198 = 950.9942 ¢, ~5/4 = 386.2812 ¢
error map: ⟨+0.021 +0.033 -0.048 +0.045 -0.126]
  • CWE: ~2 = 1200.0000 ¢, ~343/198 = 950.9930 ¢, ~5/4 = 386.2838 ¢
error map: ⟨0.000 +0.016 -0.076 +0.018 -0.177]

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

Badness (Sintel): 0.362

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 tunings:

  • WE: ~2 = 1199.9750 ¢, ~3/2 = 702.0222 ¢, ~320/297 = 128.7779 ¢
error map: ⟨-0.025 +0.042 -0.030 -0.006 +0.046]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0330 ¢, ~320/297 = 128.7791 ¢
error map: ⟨0.000 +0.078 +0.024 +0.056 +0.141]

Optimal ET sequence: 19, 27, 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee

Badness (Sintel): 1.03

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 tunings:

  • WE: ~2 = 1199.9874 ¢, ~3/2 = 702.0172 ¢, ~14/13 = 128.7600 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0228 ¢, ~14/13 = 128.7608 ¢

Optimal ET sequence: 19, 27, 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f

Badness (Sintel): 0.865

Vidar

Vidar tempers out the luxma and shares the same generators with lux, from which it only differs in the mapping of prime 7.

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 100663296/100656875

Mapping: [⟨1 0 -5 21 9], ⟨0 1 4 -9 -3], ⟨0 0 5 -20 -4]]

mapping generators: ~2, ~3, ~55/48

Optimal tunings:

  • WE: ~2 = 1199.9928 ¢, ~3/2 = 701.9969 ¢, ~55/48 = 235.6566 ¢
error map: ⟨-0.007 +0.035 -0.036 -0.016 +0.022]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9988 ¢, ~55/48 = 235.6593 ¢
error map: ⟨0.000 +0.044 -0.022 -0.002 +0.048]

Optimal ET sequence: 45, 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894

Badness (Sintel): 1.32

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -5 21 9 13], ⟨0 1 4 -9 -3 -5], ⟨0 0 5 -20 -4 -7]]

Optimal tunings:

  • WE: ~2 = 1200.0042 ¢, ~3/2 = 701.9794 ¢, ~55/48 = 235.6710 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9775 ¢, ~55/48 = 235.6698 ¢

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

Badness (Sintel): 0.593

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