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 discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Sensigh]]'' → [[Sensamagic family #Sensigh|Sensamagic family]]
* ''[[Sensigh]]'' → [[Sengic family #Sensigh|Sengic family]]
* ''[[Catakleismoid]]'' → [[Marvel family #Catakleismoid|Marvel family]]
* ''[[Catakleismoid]]'' → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]


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


== Ragismic ==
== Ragismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[4375/4374]]
[[Comma list]]: [[4375/4374]]


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


[[Mapping to lattice]]: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]
[[Mapping to lattice]]: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]
Line 22: Line 22:
: 10/9 length = 0.789, 6/5 length = 0.921
: 10/9 length = 0.789, 6/5 length = 0.921
: Angle (10/9, 6/5) = 105.299°
: 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 }}


{{Val list|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]]: ~7 = ~4374/625
[[Projection pair]]: <code>7 ~4374/625</code>


Scales: [[Ragismic19]]
Scales: [[Ragismic19]], [[Ragismic chromatic]]


== Beyla ==
== Beyla ==
Subgroup: 2.3.5.7.11
[[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|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}


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


{{Val list|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee }}
{{Optimal ET sequence|legend=1| 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee }}


[[Badness]]: 0.721 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.866


=== 13-limit ===
=== 13-limit ===
Line 51: Line 63:
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: {{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{{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}}
 
{{Optimal ET sequence|legend=0| 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|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 }}
 
[[Badness]] (Sintel): 1.48
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}


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


Optimal GPV sequence: {{Val list| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}
{{Optimal ET sequence|legend=0| 19e, 27e, 34d, 46, 53, 80, 133d, 179 }}


Badness: 0.840 × 10<sup>-3</sup>
Badness (Sintel): 0.871


== Thor ==
== Thor ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 4375/4374
[[Comma list]]: 3025/3024, 4375/4374


[[Mapping]]: [{{val| 2 0 0 2 5 }}, {{val| 0 1 0 7 5 }}, {{val| 0 0 1 -4 -3 }}]
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}
: mapping generators: ~99/70, ~3, ~5


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


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]] eigenmonzos: 2, 11/10, 8/7
* [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5


{{Val list|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{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>
[[Badness]] (Sintel): 0.107


Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
[[Projection pair]]s: <code>2 9801/4900, 7 21434787/3062500, 11 2310905821257/210087500000</code> to 3.5.14/11


=== Thunor ===
=== Thunor ===
Line 82: Line 133:
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 }}]   
Mapping to lattice: [{{val| 0 -1 -2 -3 -2 -7 }}, {{val| 0 -1 -2 1 1 1 }}]   
Line 91: Line 140:
: 10/9 length = 0.4234, 6/5 length = 0.8426
: 10/9 length = 0.4234, 6/5 length = 0.8426
: Angle (10/9, 6/5) = 84.126°
: 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
 
{{Optimal ET sequence|legend=0| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}


Optimal GPV sequence: {{Val list| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}
Badness (Sintel): 0.319


Badness: 0.341 × 10<sup>-3</sup>
Scales: [[Thunor46]]


=== Rym ===
=== Rym ===
Line 104: Line 159:
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
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 GPV sequence: {{Val list| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
{{Optimal ET sequence|legend=0| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}


Badness: 0.549 × 10<sup>-3</sup>
Badness (Sintel): 0.514


=== Donar ===
=== Donar ===
Line 117: Line 174:
Comma list: 3025/3024, 4225/4224, 4375/4374
Comma list: 3025/3024, 4225/4224, 4375/4374


Mapping: [{{val| 2 0 1 -2 2 7 }}, {{val| 0 1 0 7 5 3 }}, {{val| 0 0 2 -8 -6 -5 }}]
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
Mapping generators: ~99/70, ~3, ~220/117


Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]
Line 126: Line 182:
: 44/39 length = 0.3480, 6/5 length = 0.7828
: 44/39 length = 0.3480, 6/5 length = 0.7828
: Angle (44/39, 6/5) = 85.3977°
: Angle (44/39, 6/5) = 85.3977°
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}}


Minimax tuning:
Minimax tuning:
* 13-odd-limit eigenmonzos: 2, 11/10, 8/7
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
* 15-odd-limit eigenmonzos: 2, 8/7, 15/11
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11


Optimal GPV sequence: {{Val list| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 144, 178, 190, 224, 270, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}


Badness: 0.322 × 10<sup>-3</sup>
Badness (Sintel): 0.301


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


== Heimdall ==
== Heimdall ==
Subgroup: 2.3.5.7.11
[[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|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
: mapping generators: ~2, ~343/198, ~5


Mapping generators: ~2, ~343/198, ~5
[[Optimal tuning]]s:
* [[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 }}


{{Val list|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}
 
[[Badness]] (Sintel): 0.362


== Bragi ==
== Bragi ==
Subgroup: 2.3.5.7.11
[[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|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, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}
{{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 ===
=== 13-limit ===
Line 164: Line 238:
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 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}


Optimal GPV sequence: {{Val list| 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}
Badness (Sintel): 0.865


== Vidar ==
== Vidar ==
Subgroup: 2.3.5.7.11
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]].
 
[[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|legend=1| 1 0 -5 21 9 | 0 1 4 -9 -3 | 0 0 5 -20 -4 }}
: mapping generators: ~2, ~3, ~55/48


Mapping generators: ~2, ~3, ~55/32
[[Optimal tuning]]s:
* [[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 }}


{{Val list|legend=1| 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}
{{Optimal ET sequence|legend=1| 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 ===
=== 13-limit ===
Line 184: Line 273:
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: {{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{{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}}
 
{{Optimal ET sequence|legend=0| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}


Optimal GPV sequence: {{Val list| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}
Badness (Sintel): 0.593


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Ragismic family| ]] <!-- main article -->
[[Category:Ragismic family| ]] <!-- main article -->
[[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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