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>2011-08-21 16:50:25 UTC</tt>.<br>
: The original revision id was <tt>247447425</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


===Donar===
[[Projection pair]]: ~7 = ~4374/625
Commas: 4375/4374, 3025/3024, 4225/4224


13-limit
Scales: [[Ragismic19]], [[Ragismic chromatic]]
||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
== Beyla ==
[|1 0 0 0 0 0&gt;, |1/2 1/3 1/3 1/3 -1/3 0&gt;,
[[Subgroup]]: 2.3.5.7.11
|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
[[Comma list]]: 385/384, 4375/4374
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|]
{{Mapping|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}
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]]
{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee }}


===Thunor===
[[Badness]]: 0.721 × 10<sup>-3</sup>
Commas: 4375/4374, 3025/3024, 1716/1715


13 and 15 limit minimax
=== 13-limit ===
[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,
Subgroup: 2.3.5.7.11.13
|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
Comma list: 169/168, 325/324, 385/384
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|]
Mapping: {{mapping| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -2 }}
Generators: 99/70, 3, 5
Edos: 26, 46, 72, 178, 224, 270, 494
Badness: 0.000341


===Rym===
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}
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|]
Badness: 0.840 × 10<sup>-3</sup>
EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870
Badness: 0.000549


==Vidar==
== Ulmo ==
Commas: 4375/4374, 100663296/100656875
[[Subgroup]]: 2.3.5.7.11


==Bragi==
[[Comma list]]: 176/175, 2200/2187
Commas: 4375/4374, 2097152/2096325</pre></div>
 
<h4>Original HTML content:</h4>
{{Mapping|legend=1| 1 0 0 1 -3 | 0 1 0 7 7 | 0 0 1 -4 -2 }}
<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;
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
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;
[[Badness]]: 1.23 × 10<sup>-3</sup>
Eigenmonzos: 2, 8/7, 5/4&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
9-limit&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
[|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;
Comma list: 169/168, 176/175, 325/324
Eigenmonzos: 2, 4/3, 7/5&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}
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;
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
Map to lattice: [&amp;lt;0 -1 -1 -3|, &amp;lt;0 -1 -2 1|]&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.932 × 10<sup>-3</sup>
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;
== Thor ==
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;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 4374/625&lt;br /&gt;
[[Comma list]]: 3025/3024, 4375/4374
&lt;br /&gt;
 
Scales: &lt;a class="wiki_link" href="/ragismic19"&gt;ragismic19&lt;/a&gt;&lt;br /&gt;
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}
&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;
: mapping generators: ~99/70, ~3, ~5
&lt;br /&gt;
 
&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;
[[Minimax tuning]]:
Commas: 4375/4374, 3025/3024&lt;br /&gt;
* [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/5
&lt;br /&gt;
 
Minimax tuning&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 }}
[|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]]: 0.0888 × 10<sup>-3</sup>
|25/16 0 -1/8 5/8 1/8&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
[[Projection pair]]s: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
&lt;br /&gt;
 
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;
=== Thunor ===
Generators: 99/70, 3, 5&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0000888&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 1716/1715, 2080/2079, 3025/3024
&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;
Mapping: {{mapping| 2 0 0 2 5 -3 | 0 1 0 7 5 15 | 0 0 1 -4 -3 -8 }}
&lt;br /&gt;
 
13-limit&lt;br /&gt;
Mapping to lattice: [{{val| 0 -1 -2 -3 -2 -7 }}, {{val| 0 -1 -2 1 1 1 }}] 
||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;
Lattice basis:
|25/16 0 -1/8 5/8 1/8 0&amp;gt;, &lt;br /&gt;
: 10/9 length = 0.4234, 6/5 length = 0.8426
|99/32 0 -11/16 -1/16 11/16 0&amp;gt;]&lt;br /&gt;
: Angle (10/9, 6/5) = 84.126°
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:
15-limit minimax&lt;br /&gt;
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13
[|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;
{{Optimal ET sequence|legend=1| 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258 }}
|13/8 -1/12 -1/12 2/3 1/12 0&amp;gt;, &lt;br /&gt;
 
|55/16 -11/24 -11/24 1/6 11/24 0&amp;gt;]&lt;br /&gt;
Badness: 0.341 × 10<sup>-3</sup>
Eigenmonzos: 2, 8/7, 15/11&lt;br /&gt;
 
&lt;br /&gt;
=== Rym ===
Lattice basis: 44/39 length = 0.3480 6/5 length = 0.7828&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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;
Comma list: 3025/3024, 4096/4095, 4375/4374
&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;
Mapping: {{mapping| 2 0 0 2 5 22 | 0 1 0 7 5 -9 | 0 0 1 -4 -3 3 }}
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;
{{Optimal ET sequence|legend=1| 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870 }}
Badness: 0.000322&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.549 × 10<sup>-3</sup>
Scales: &lt;a class="wiki_link" href="/donar46"&gt;donar46&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
=== Donar ===
&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;
Subgroup: 2.3.5.7.11.13
Commas: 4375/4374, 3025/3024, 1716/1715&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 3025/3024, 4225/4224, 4375/4374
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: {{mapping| 2 0 1 -2 2 7 | 0 1 0 7 5 3 | 0 0 2 -8 -6 -5 }}
|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;
: mapping generators: ~99/70, ~3, ~220/117
&lt;br /&gt;
 
Lattice basis: 10/9 length 0.4234 6/5 length 0.8426&lt;br /&gt;
Mapping to lattice: [{{val| 0 2 2 6 4 1 }}, {{val| 0 -1 -2 1 1 2 }}]
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;
Lattice basis:
&lt;br /&gt;
: 44/39 length = 0.3480, 6/5 length = 0.7828
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;
: Angle (44/39, 6/5) = 85.3977°
Generators: 99/70, 3, 5&lt;br /&gt;
 
Edos: 26, 46, 72, 178, 224, 270, 494&lt;br /&gt;
Minimax tuning:
Badness: 0.000341&lt;br /&gt;
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5
&lt;br /&gt;
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11
&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;
{{Optimal ET sequence|legend=1| 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff }}
&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;
Badness: 0.322 × 10<sup>-3</sup>
EDOs: 46, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870&lt;br /&gt;
 
Badness: 0.000549 &lt;br /&gt;
Scales: [[Donar46]]
&lt;br /&gt;
 
&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;
== Heimdall ==
Commas: 4375/4374, 100663296/100656875&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&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;
[[Comma list]]: 4375/4374, 117649/117612
Commas: 4375/4374, 2097152/2096325&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{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 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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