Ragismic family: Difference between revisions

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→Vidar: note the relation with lux
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''ragismic family''' of temperaments are rank-3 microtemperaments which temper out [[4375/4374]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-04 21:50:30 UTC</tt>.<br>
: The original revision id was <tt>167796719</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Comma: 4375/4374


7-limit minimax
Temperaments discussed elsewhere include:
[|1 0 0 0&gt;, |-1/7 0 4/7 1/7&gt;,
* ''[[Sensigh]]'' → [[Sengic family #Sensigh|Sengic family]]
|0 0 1 0&gt;, |0 0 0 1&gt;]
* ''[[Catakleismoid]]'' → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]
Eigenmonzos: 2, 5, 7
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]


9-limit minimax
Considered below are beyla, ulmo, thor, heimdall, bragi, and vidar.
||1 0 0 0&gt;, |0 1 0 0&gt;,  
|1/5 7/5 1/5 -1/5&gt;, |1/5 7/5 -4/5 4/5&gt;]
Eigenmonzos: 2, 3, 7/5


Map: [&lt;1 0 0 1|, &lt;0 1 0 7|, &lt;0 0 1 -4|]
== Ragismic ==
Generators: 2, 3, 5
[[Subgroup]]: 2.3.5.7
Edos: 7, 19, 26, 27, 45, 46, 53, 65, 72, 80, 91, 99, 171, 270, 441, 612, 935, 1106, 1277, 1376, 1547, 1718, 1889, 2923, 3166, 3436, 3877, 3535, 4390, 5496, 5937, 6450, 9058


==11 limit children==
[[Comma list]]: [[4375/4374]]


===Thor===
{{Mapping|legend=1| 1 0 0 1 | 0 1 0 7 | 0 0 1 -4 }}
Commas: 4375/4374, 3025/3024
: mapping generators: ~2, ~3, ~5


Minimax tuning
[[Mapping to lattice]]: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]
[|1 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2&gt;,  
|25/16 0 7/8 5/8 -7/8&gt;, |0 0 0 1 0&gt;,
|25/16 0 -1/8 5/8 1/8&gt;]
Eigenmonzos: 2, 11/10, 8/7


Map: [&lt;2 0 0 2 5|, &lt;0 1 0 7 5|, &lt;0 0 1 -4 -3|]
Lattice basis:  
Generators: 99/70, 3, 5
: 10/9 length = 0.789, 6/5 length = 0.921
: Angle (10/9, 6/5) = 105.299°


13-limit
[[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 }}


====Donar====
[[Minimax tuning]]:
Commas: 4375/4374, 3025/3024, 4225/4224
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


13-limit
{{Optimal ET sequence|legend=1| 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c }}
||1 0 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2 0&gt;,  
|25/16 0 7/8 5/8 -7/8 0&gt;, |0 0 0 1 0 0&gt;,  
|25/16 0 -1/8 5/8 1/8 0&gt;,  
|99/32 0 -11/16 -1/16 11/16 0&gt;]
Eigenmonzos: 2, 11/10, 8/7


15-limit minimax
[[Badness]] (Sintel): 0.0515
[|1 0 0 0 0 0&gt;, |1/2 1/3 1/3 1/3 -1/3 0&gt;,
|9/8 7/12 7/12 1/3 -7/12 0&gt;, |0 0 0 1 0 0&gt;,
|13/8 -1/12 -1/12 2/3 1/12 0&gt;,
|55/16 -11/24 -11/24 1/6 11/24 0&gt;]
Eigenmonzos: 2, 8/7, 15/11


Map: [&lt;2 0 1 -2 2 7|, &lt;0 1 0 7 5 3|, &lt;0 0 2 -8 -6 -5|]
[[Projection pair]]: <code>7 ~4374/625</code>
Generators: 99/70, 3, 220/117
Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390


====Thunor====
Scales: [[Ragismic19]], [[Ragismic chromatic]]
Commas: 4375/4374, 3025/3024, 1716/1715


13 and 15 limit minimax
== Beyla ==
[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,
[[Subgroup]]: 2.3.5.7.11
|9/8 3/2 0 -3/4 0 1/4&gt;, |7/5 -2/5 0 1/5 0 2/5&gt;, |21/8 -1/2 0 1/4 0 1/4&gt;, |0 0 0 0 0 1&gt;]
Eigenmonzos: 2, 16/13, 9/7


Map: [&lt;2 0 0 2 5 -3|, &lt;0 1 0 7 5 15|, &lt;0 0 1 -4 -3 -8|]
[[Comma list]]: 385/384, 4375/4374
Generators: 99/70, 3, 5
 
Edos: 26, 46, 72, 178, 224, 270, 494</pre></div>
{{Mapping|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ragisma family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Comma: 4375/4374&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1200.4709{{c}}, ~3/2 = 701.3876{{c}}, ~5/4 = 385.3287{{c}}
7-limit minimax&lt;br /&gt;
: [[error map]]: {{val| +0.471 -0.097 -0.043 -0.427 -0.833 }}
[|1 0 0 0&amp;gt;, |-1/7 0 4/7 1/7&amp;gt;, &lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2311{{c}}, ~5/4 = 385.3753{{c}}
|0 0 1 0&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
: error map: {{val| 0.000 -0.724 -0.938 -1.709 -2.579 }}
Eigenmonzos: 2, 5, 7&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee }}
9-limit minimax&lt;br /&gt;
 
||1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, &lt;br /&gt;
[[Badness]] (Sintel): 0.866
|1/5 7/5 1/5 -1/5&amp;gt;, |1/5 7/5 -4/5 4/5&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 3, 7/5&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;1 0 0 1|, &amp;lt;0 1 0 7|, &amp;lt;0 0 1 -4|]&lt;br /&gt;
 
Generators: 2, 3, 5&lt;br /&gt;
Comma list: 169/168, 325/324, 385/384
Edos: 7, 19, 26, 27, 45, 46, 53, 65, 72, 80, 91, 99, 171, 270, 441, 612, 935, 1106, 1277, 1376, 1547, 1718, 1889, 2923, 3166, 3436, 3877, 3535, 4390, 5496, 5937, 6450, 9058&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-11 limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;11 limit children&lt;/h2&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-11 limit children-Thor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Thor&lt;/h3&gt;
* WE: ~2 = 1200.6829{{c}}, ~3/2 = 701.4188{{c}}, ~5/4 = 385.0695{{c}}
Commas: 4375/4374, 3025/3024&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1881{{c}}, ~5/4 = 385.1147{{c}}
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 26, 45f, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}
[|1 0 0 0 0&amp;gt;, |3/4 0 1/2 1/2 -1/2&amp;gt;, &lt;br /&gt;
 
|25/16 0 7/8 5/8 -7/8&amp;gt;, |0 0 0 1 0&amp;gt;, &lt;br /&gt;
Badness (Sintel): 0.785
|25/16 0 -1/8 5/8 1/8&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
== Ulmo ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Map: [&amp;lt;2 0 0 2 5|, &amp;lt;0 1 0 7 5|, &amp;lt;0 0 1 -4 -3|]&lt;br /&gt;
 
Generators: 99/70, 3, 5&lt;br /&gt;
[[Comma list]]: 176/175, 2200/2187
&lt;br /&gt;
 
13-limit&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc2"&gt;&lt;a name="x-11 limit children-Thor-Donar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Donar&lt;/h4&gt;
[[Optimal tuning]]s:
Commas: 4375/4374, 3025/3024, 4225/4224&lt;br /&gt;
* [[WE]]: ~2 = 1199.1423{{c}}, ~3/2 = 703.8558{{c}}, ~5/4 = 389.2089{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.858 +1.043 +1.180 +1.329 -2.745 }}
13-limit&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9658{{c}}, ~5/4 = 388.9716{{c}}
||1 0 0 0 0 0&amp;gt;, |3/4 0 1/2 1/2 -1/2 0&amp;gt;, &lt;br /&gt;
: error map: {{val| 0.000 +2.011 +2.658 +3.048 -1.501 }}
|25/16 0 7/8 5/8 -7/8 0&amp;gt;, |0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
 
|25/16 0 -1/8 5/8 1/8 0&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=1| 19e, 27e, 34d, 46, 53, 80, 126, 179, 259cd }}
|99/32 0 -11/16 -1/16 11/16 0&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
[[Badness]] (Sintel): 1.48
&lt;br /&gt;
 
15-limit minimax&lt;br /&gt;
=== 13-limit ===
[|1 0 0 0 0 0&amp;gt;, |1/2 1/3 1/3 1/3 -1/3 0&amp;gt;, &lt;br /&gt;
Subgroup: 2.3.5.7.11.13
|9/8 7/12 7/12 1/3 -7/12 0&amp;gt;, |0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
 
|13/8 -1/12 -1/12 2/3 1/12 0&amp;gt;, &lt;br /&gt;
Comma list: 169/168, 176/175, 325/324
|55/16 -11/24 -11/24 1/6 11/24 0&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 8/7, 15/11&lt;br /&gt;
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 0 1 -2 2 7|, &amp;lt;0 1 0 7 5 3|, &amp;lt;0 0 2 -8 -6 -5|]&lt;br /&gt;
Optimal tunings:
Generators: 99/70, 3, 220/117&lt;br /&gt;
* WE: ~2 = 1199.5491{{c}}, ~3/2 = 703.6725{{c}}, ~5/4 = 388.5875{{c}}
Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7489{{c}}, ~5/4 = 388.5028{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;a name="x-11 limit children-Thor-Thunor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Thunor&lt;/h4&gt;
{{Optimal ET sequence|legend=0| 19e, 27e, 34d, 46, 53, 80, 133d, 179 }}
Commas: 4375/4374, 3025/3024, 1716/1715&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.871
13 and 15 limit minimax&lt;br /&gt;
 
[|1 0 0 0 0 0&amp;gt;, |7/10 4/5 0 -2/5 0 1/5&amp;gt;, &lt;br /&gt;
== Thor ==
|9/8 3/2 0 -3/4 0 1/4&amp;gt;, |7/5 -2/5 0 1/5 0 2/5&amp;gt;, |21/8 -1/2 0 1/4 0 1/4&amp;gt;, |0 0 0 0 0 1&amp;gt;]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Eigenmonzos: 2, 16/13, 9/7&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 3025/3024, 4375/4374
Map: [&amp;lt;2 0 0 2 5 -3|, &amp;lt;0 1 0 7 5 15|, &amp;lt;0 0 1 -4 -3 -8|]&lt;br /&gt;
 
Generators: 99/70, 3, 5&lt;br /&gt;
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}
Edos: 26, 46, 72, 178, 224, 270, 494&lt;/body&gt;&lt;/html&gt;</pre></div>
: 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]]:
* [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5
 
{{Optimal ET sequence|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
 
[[Badness]] (Sintel): 0.107
 
[[Projection pair]]s: <code>2 9801/4900, 7 21434787/3062500, 11 2310905821257/210087500000</code> to 3.5.14/11
 
=== Thunor ===
Subgroup: 2.3.5.7.11.13
 
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:
* 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 }}
 
Badness (Sintel): 0.319
 
Scales: [[Thunor46]]
 
=== Rym ===
Subgroup: 2.3.5.7.11.13
 
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
 
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 to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 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{{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:
* 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|legend=0| 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|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
: 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 }}
 
{{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 ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 4375/4374, 2097152/2096325
 
{{Mapping|legend=1| 1 0 2 -7 8 | 0 1 0 7 -3 | 0 0 3 -12 2 }}
: 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 }}
 
{{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
 
Comma list: 2080/2079, 4375/4374, 10985/10976
 
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 }}
 
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 interval|prime]] [[7/1|7]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 4375/4374, 100663296/100656875
 
{{Mapping|legend=1| 1 0 -5 21 9 | 0 1 4 -9 -3 | 0 0 5 -20 -4 }}
: mapping generators: ~2, ~3, ~55/48
 
[[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 }}
 
{{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 ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 6656/6655
 
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 }}
 
Badness (Sintel): 0.593
 
[[Category:Temperament families]]
[[Category:Ragismic family| ]] <!-- main article -->
[[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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