Ragismic family: Difference between revisions

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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>2011-08-22 02:19:52 UTC</tt>.<br>
: The original revision id was <tt>247577137</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
Temperaments discussed elsewhere include:
[|1 0 0 0&gt;, |-1/7 0 4/7 1/7&gt;, |0 0 1 0&gt;, |0 0 0 1&gt;]
* ''[[Sensigh]]'' → [[Sensamagic family #Sensigh|Sensamagic family]]
Eigenmonzos: 2, 8/7, 5/4
* ''[[Catakleismoid]]'' → [[Marvel family #Catakleismoid|Marvel family]]
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]


9-limit
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, 4/3, 7/5


Lattice basis: 10/9 length=0.789 6/5 length=0.921
== Ragismic ==
Angle(10/9, 6/5) = 105.299
[[Subgroup]]: 2.3.5.7
Map to lattice: [&lt;0 -1 -1 -3|, &lt;0 -1 -2 1|]


Map: [&lt;1 0 0 1|, &lt;0 1 0 7|, &lt;0 0 1 -4|]
[[Comma list]]: [[4375/4374]]
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
{{Mapping|legend=1| 1 0 0 1 | 0 1 0 7 | 0 0 1 -4 }}


Scales: [[ragismic19]]
: mapping generators: ~2, ~3, ~5


==11 limit children==
[[Mapping to lattice]]: [{{val| 0 -1 -1 -3 }}, {{val| 0 -1 -2 1 }}]


==Thor==
Lattice basis:
Commas: 4375/4374, 3025/3024
: 10/9 length = 0.789, 6/5 length = 0.921
: Angle (10/9, 6/5) = 105.299°


Minimax tuning
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2&gt;,
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
|25/16 0 7/8 5/8 -7/8&gt;, |0 0 0 1 0&gt;,
* [[9-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
|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|]
{{Optimal ET sequence|legend=1| 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c }}
Generators: 99/70, 3, 5
Badness: 0.0000888


Projection pairs: 2 9801/4900 7 21434787/3062500 11 2310905821257/210087500000 to 3.5.14/11
[[Projection pair]]: ~7 = ~4374/625


===Donar===
Scales: [[Ragismic19]], [[Ragismic chromatic]]
Commas: 4375/4374, 3025/3024, 4225/4224


13-limit
== Beyla ==
||1 0 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2 0&gt;,
[[Subgroup]]: 2.3.5.7.11
|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
[[Comma list]]: 385/384, 4375/4374
[|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


Lattice basis: 44/39 length = 0.3480 6/5 length = 0.7828
{{Mapping|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}
Angle(44/39, 6/5) = 85.3977
Map to lattice: [&lt;0 2 2 6 4 1|, &lt;0 -1 -2 1 1 2|]


Map: [&lt;2 0 1 -2 2 7|, &lt;0 1 0 7 5 3|, &lt;0 0 2 -8 -6 -5|]
{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee }}
Generators: 99/70, 3, 220/117
Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390
Badness: 0.000322


Scales: [[donar46]]
[[Badness]]: 0.721 × 10<sup>-3</sup>


===Thunor===
=== 13-limit ===
Commas: 4375/4374, 3025/3024, 1716/1715
Subgroup: 2.3.5.7.11.13


13 and 15 limit minimax
Comma list: 169/168, 325/324, 385/384
[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,
|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


Lattice basis: 10/9 length 0.4234 6/5 length 0.8426
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}
Angle(10/9, 6/5) = 84.126 degrees
Map to lattice: [&lt;0 -1 -2 -3 -2 -7|, &lt;0 -1 -2 1 1 1|] 


Map: [&lt;2 0 0 2 5 -3|, &lt;0 1 0 7 5 15|, &lt;0 0 1 -4 -3 -8|]
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}
Generators: 99/70, 3, 5
Edos: 26, 46, 72, 178, 224, 270, 494
Badness: 0.000341


===Rym===
Badness: 0.840 × 10<sup>-3</sup>
Commas: 4375/4374, 3025/3024, 4096/4095


Map: [&lt;2 0 0 2 5 22|, &lt;0 1 0 7 5 -9|, &lt;0 0 1 -4 -3 3|]
== Ulmo ==
EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870
[[Subgroup]]: 2.3.5.7.11
Badness: 0.000549


==Vidar==
[[Comma list]]: 176/175, 2200/2187
Commas: 4375/4374, 100663296/100656875


==Bragi==
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
Commas: 4375/4374, 2097152/2096325</pre></div>
 
<h4>Original HTML content:</h4>
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
<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;
 
&lt;br /&gt;
[[Badness]]: 1.23 × 10<sup>-3</sup>
7-limit&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |-1/7 0 4/7 1/7&amp;gt;, |0 0 1 0&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
=== 13-limit ===
Eigenmonzos: 2, 8/7, 5/4&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
9-limit&lt;br /&gt;
Comma list: 169/168, 176/175, 325/324
[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |1/5 7/5 1/5 -1/5&amp;gt;, &lt;br /&gt;
 
|1/5 7/5 -4/5 4/5&amp;gt;]&lt;br /&gt;
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}
Eigenmonzos: 2, 4/3, 7/5&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
Lattice basis: 10/9 length=0.789 6/5 length=0.921&lt;br /&gt;
 
Angle(10/9, 6/5) = 105.299&lt;br /&gt;
Badness: 0.932 × 10<sup>-3</sup>
Map to lattice: [&amp;lt;0 -1 -1 -3|, &amp;lt;0 -1 -2 1|]&lt;br /&gt;
 
&lt;br /&gt;
== Thor ==
Map: [&amp;lt;1 0 0 1|, &amp;lt;0 1 0 7|, &amp;lt;0 0 1 -4|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Generators: 2, 3, 5&lt;br /&gt;
 
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 &lt;br /&gt;
[[Comma list]]: 3025/3024, 4375/4374
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 4374/625&lt;br /&gt;
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}
&lt;br /&gt;
 
Scales: &lt;a class="wiki_link" href="/ragismic19"&gt;ragismic19&lt;/a&gt;&lt;br /&gt;
: mapping generators: ~99/70, ~3, ~5
&lt;br /&gt;
 
&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;
[[Minimax tuning]]:
&lt;br /&gt;
* [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Thor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Thor&lt;/h2&gt;
 
Commas: 4375/4374, 3025/3024&lt;br /&gt;
{{Optimal ET sequence|legend=1| 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
[[Badness]]: 0.0888 × 10<sup>-3</sup>
[|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;
[[Projection pair]]s: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
|25/16 0 -1/8 5/8 1/8&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
=== Thunor ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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: 1716/1715, 2080/2079, 3025/3024
Badness: 0.0000888&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 2 0 0 2 5 -3 | 0 1 0 7 5 15 | 0 0 1 -4 -3 -8 }}
Projection pairs: 2 9801/4900 7 21434787/3062500 11 2310905821257/210087500000 to 3.5.14/11&lt;br /&gt;
 
&lt;br /&gt;
Mapping to lattice: [{{val| 0 -1 -2 -3 -2 -7 }}, {{val| 0 -1 -2 1 1 1 }}] 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Thor-Donar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Donar&lt;/h3&gt;
 
Commas: 4375/4374, 3025/3024, 4225/4224&lt;br /&gt;
Lattice basis:  
&lt;br /&gt;
: 10/9 length = 0.4234, 6/5 length = 0.8426
13-limit&lt;br /&gt;
: Angle (10/9, 6/5) = 84.126°
||1 0 0 0 0 0&amp;gt;, |3/4 0 1/2 1/2 -1/2 0&amp;gt;, &lt;br /&gt;
 
|25/16 0 7/8 5/8 -7/8 0&amp;gt;, |0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
Minimax tuning:
|25/16 0 -1/8 5/8 1/8 0&amp;gt;, &lt;br /&gt;
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13
|99/32 0 -11/16 -1/16 11/16 0&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
{{Optimal ET sequence|legend=1| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}
&lt;br /&gt;
 
15-limit minimax&lt;br /&gt;
Badness: 0.341 × 10<sup>-3</sup>
[|1 0 0 0 0 0&amp;gt;, |1/2 1/3 1/3 1/3 -1/3 0&amp;gt;, &lt;br /&gt;
 
|9/8 7/12 7/12 1/3 -7/12 0&amp;gt;, |0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
=== Rym ===
|13/8 -1/12 -1/12 2/3 1/12 0&amp;gt;, &lt;br /&gt;
Subgroup: 2.3.5.7.11.13
|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;
Comma list: 3025/3024, 4096/4095, 4375/4374
&lt;br /&gt;
 
Lattice basis: 44/39 length = 0.3480 6/5 length = 0.7828&lt;br /&gt;
Mapping: {{mapping| 2 0 0 2 5 22 | 0 1 0 7 5 -9 | 0 0 1 -4 -3 3 }}
Angle(44/39, 6/5) = 85.3977&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 2 2 6 4 1|, &amp;lt;0 -1 -2 1 1 2|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
&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;
Badness: 0.549 × 10<sup>-3</sup>
Generators: 99/70, 3, 220/117&lt;br /&gt;
 
Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390&lt;br /&gt;
=== Donar ===
Badness: 0.000322&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Scales: &lt;a class="wiki_link" href="/donar46"&gt;donar46&lt;/a&gt;&lt;br /&gt;
Comma list: 3025/3024, 4225/4224, 4375/4374
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Thor-Thunor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Thunor&lt;/h3&gt;
Mapping: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}
Commas: 4375/4374, 3025/3024, 1716/1715&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~99/70, ~3, ~220/117
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;
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]
|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;
 
Eigenmonzos: 2, 16/13, 9/7&lt;br /&gt;
Lattice basis:
&lt;br /&gt;
: 44/39 length = 0.3480, 6/5 length = 0.7828
Lattice basis: 10/9 length 0.4234 6/5 length 0.8426&lt;br /&gt;
: Angle (44/39, 6/5) = 85.3977°
Angle(10/9, 6/5) = 84.126 degrees&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 -1 -2 -3 -2 -7|, &amp;lt;0 -1 -2 1 1 1|]  &lt;br /&gt;
Minimax tuning:
&lt;br /&gt;
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
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;
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11
Generators: 99/70, 3, 5&lt;br /&gt;
 
Edos: 26, 46, 72, 178, 224, 270, 494&lt;br /&gt;
{{Optimal ET sequence|legend=1| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}
Badness: 0.000341&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.322 × 10<sup>-3</sup>
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Thor-Rym"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Rym&lt;/h3&gt;
 
Commas: 4375/4374, 3025/3024, 4096/4095&lt;br /&gt;
Scales: [[Donar46]]
&lt;br /&gt;
 
Map: [&amp;lt;2 0 0 2 5 22|, &amp;lt;0 1 0 7 5 -9|, &amp;lt;0 0 1 -4 -3 3|]&lt;br /&gt;
== Heimdall ==
EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Badness: 0.000549 &lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 4375/4374, 117649/117612
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Vidar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Vidar&lt;/h2&gt;
 
Commas: 4375/4374, 100663296/100656875&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 1 2 | 0 2 0 14 37 | 0 0 1 -4 -12 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Bragi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Bragi&lt;/h2&gt;
: mapping generators: ~2, ~343/198, ~5
Commas: 4375/4374, 2097152/2096325&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d }}
 
== 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 ET sequence|legend=1| 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee }}
 
=== 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 ET sequence|legend=1| 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f }}
 
== Vidar ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 4375/4374, 100663296/100656875
 
{{Mapping|legend=1| 1 0 0 1 5 | 0 1 4 -9 -3 | 0 0 -5 20 4 }}
 
: mapping generators: ~2, ~3, ~96/55
 
{{Optimal ET sequence|legend=1| 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894 }}
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 6656/6655
 
Mapping: {{mapping| 1 0 0 1 5 6 | 0 1 4 -9 -3 -5 | 0 0 -5 20 4 7 }}
 
{{Optimal ET sequence|legend=1| 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f }}
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragismic family| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 3]]

Latest revision as of 00:32, 24 June 2025

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°

Minimax tuning:

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

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 ET sequence: 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee

Badness: 0.721 × 10-3

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 ET sequence: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff

Badness: 0.840 × 10-3

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 ET sequence: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd

Badness: 1.23 × 10-3

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 ET sequence: 7, 39, 46, 53, 80, 133d, 179, 312cdf

Badness: 0.932 × 10-3

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:

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

Badness: 0.0888 × 10-3

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°

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: 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]]

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

Badness: 0.549 × 10-3

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°

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

Badness: 0.322 × 10-3

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

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

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

Vidar

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 100663296/100656875

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

mapping generators: ~2, ~3, ~96/55

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

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 4 -9 -3 -5], ⟨0 0 -5 20 4 7]]

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

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