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>2010-10-05 03:10:34 UTC</tt>.<br>
: The original revision id was <tt>167840857</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
== Ragismic ==
[|1 0 0 0&gt;, |-1/7 0 4/7 1/7&gt;,
[[Subgroup]]: 2.3.5.7
|0 0 1 0&gt;, |0 0 0 1&gt;]
Eigenmonzos: 2, 5, 7


9-limit minimax
[[Comma list]]: [[4375/4374]]
||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|]
{{Mapping|legend=1| 1 0 0 1 | 0 1 0 7 | 0 0 1 -4 }}
Generators: 2, 3, 5
: mapping generators: ~2, ~3, ~5
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==
[[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
[[Optimal tuning]]s:
[|1 0 0 0 0&gt;, |3/4 0 1/2 1/2 -1/2&gt;,  
* [[WE]]: ~2 = 1200.0018{{c}}, ~3/2 = 701.9851{{c}}, ~5/4 = 386.2710{{c}}
|25/16 0 7/8 5/8 -7/8&gt;, |0 0 0 1 0&gt;,
: [[error map]]: {{val| +0.002 +0.032 -0.039 -0.014 }}
|25/16 0 -1/8 5/8 1/8&gt;]
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9855{{c}}, ~5/4 = 386.2724{{c}}
Eigenmonzos: 2, 11/10, 8/7
: error map: {{val| 0.000 +0.031 -0.041 -0.017 }}


Map: [&lt;2 0 0 2 5|, &lt;0 1 0 7 5|, &lt;0 0 1 -4 -3|]
[[Minimax tuning]]:
Generators: 99/70, 3, 5
* [[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 }}


====Donar====
[[Badness]] (Sintel): 0.0515
Commas: 4375/4374, 3025/3024, 4225/4224


13-limit
[[Projection pair]]: <code>7 ~4374/625</code>
||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
Scales: [[Ragismic19]], [[Ragismic chromatic]]
[|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
=== Overview to extensions ===
Angle(44/39, 6/5) = 85.3977
Temperaments discussed elsewhere include:  
Map to lattice: [&lt;0 2 2 6 4 1|, &lt;0 -1 -2 1 1 2|]
* ''[[Sensigh]]'' → [[Sengic family #Sensigh|Sengic family]]
* ''[[Catakleismoid]]'' → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]
* ''[[Ennealimmic]]'' → [[Breed family #Ennealimmic|Breed family]]


Map: [&lt;2 0 1 -2 2 7|, &lt;0 1 0 7 5 3|, &lt;0 0 2 -8 -6 -5|]
Considered below are beyla, ulmo, raglympic, thor, heimdall, bragi, vidar.
Generators: 99/70, 3, 220/117
Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390


====Thunor====
== Beyla ==
Commas: 4375/4374, 3025/3024, 1716/1715
[[Subgroup]]: 2.3.5.7.11


13 and 15 limit minimax
[[Comma list]]: 385/384, 4375/4374
[|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|legend=1| 1 0 0 1 6 | 0 1 0 7 -6 | 0 0 1 -4 3 }}
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 tuning]]s:  
Generators: 99/70, 3, 5
* [[WE]]: ~2 = 1200.4709{{c}}, ~3/2 = 701.3876{{c}}, ~5/4 = 385.3287{{c}}
Edos: 26, 46, 72, 178, 224, 270, 494
: [[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 }}


====Rym====
{{Optimal ET sequence|legend=1| 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 938bcddeee, 1056bcdddeeee }}
Commas: 4375/4374, 3025/3024, 4096/4095


===Tyr===
[[Badness]] (Sintel): 0.866
Commas: 4375/4374, 131072/130977


===Veur===
=== 13-limit ===
Commas: 4375/4374, 100663296/100656875
Subgroup: 2.3.5.7.11.13


===Veod===
Comma list: 169/168, 325/324, 385/384
Commas: 4375/4374, 2097152/2096325</pre></div>
 
<h4>Original HTML content:</h4>
{{Mapping|legend=0| 1 0 0 1 6 2 | 0 1 0 7 -6 4 | 0 0 1 -4 3 -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 tunings:  
7-limit minimax&lt;br /&gt;
* WE: ~2 = 1200.6829{{c}}, ~3/2 = 701.4188{{c}}, ~5/4 = 385.0695{{c}}
[|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 = 701.1881{{c}}, ~5/4 = 385.1147{{c}}
|0 0 1 0&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 5, 7&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 26, 45f, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff }}
&lt;br /&gt;
 
9-limit minimax&lt;br /&gt;
Badness (Sintel): 0.785
||1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, &lt;br /&gt;
 
|1/5 7/5 1/5 -1/5&amp;gt;, |1/5 7/5 -4/5 4/5&amp;gt;]&lt;br /&gt;
== Ulmo ==
Eigenmonzos: 2, 3, 7/5&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 1|, &amp;lt;0 1 0 7|, &amp;lt;0 0 1 -4|]&lt;br /&gt;
[[Comma list]]: 176/175, 2200/2187
Generators: 2, 3, 5&lt;br /&gt;
 
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;
{{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: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;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1199.1423{{c}}, ~3/2 = 703.8558{{c}}, ~5/4 = 389.2089{{c}}
&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;
: [[error map]]: {{val| -0.858 +1.043 +1.180 +1.329 -2.745 }}
Commas: 4375/4374, 3025/3024&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9658{{c}}, ~5/4 = 388.9716{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +2.011 +2.658 +3.048 -1.501 }}
Minimax tuning&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |3/4 0 1/2 1/2 -1/2&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=1| 19e, 27e, 34d, 46, 53, 80, 126, 179, 259cd }}
|25/16 0 7/8 5/8 -7/8&amp;gt;, |0 0 0 1 0&amp;gt;, &lt;br /&gt;
 
|25/16 0 -1/8 5/8 1/8&amp;gt;]&lt;br /&gt;
[[Badness]] (Sintel): 1.48
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
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;
Subgroup: 2.3.5.7.11.13
Generators: 99/70, 3, 5&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 169/168, 176/175, 325/324
13-limit&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 0 0 1 -3 2 | 0 1 0 7 7 4 | 0 0 1 -4 -2 -2 }}
&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;
 
Commas: 4375/4374, 3025/3024, 4225/4224&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.5491{{c}}, ~3/2 = 703.6725{{c}}, ~5/4 = 388.5875{{c}}
13-limit&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7489{{c}}, ~5/4 = 388.5028{{c}}
||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;
{{Optimal ET sequence|legend=0| 19e, 27e, 34d, 46, 53, 80, 133d, 179 }}
|25/16 0 -1/8 5/8 1/8 0&amp;gt;, &lt;br /&gt;
 
|99/32 0 -11/16 -1/16 11/16 0&amp;gt;]&lt;br /&gt;
Badness (Sintel): 0.871
Eigenmonzos: 2, 11/10, 8/7&lt;br /&gt;
 
&lt;br /&gt;
== Raglympic ==
15-limit minimax&lt;br /&gt;
Named by [[Eufalesio]] in 2026, raglympic is the intersection of [[olympic]] and ragismic, as its name implies. It is also a detemperament of [[ponta]], achieved by observing the schisma. As such, the 5 generator can be taken to be a schisma.
[|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;
[[Subgroup]]: 2.3.5.7.11
|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;
[[Comma list]]: 4375/4374, 131072/130977
Eigenmonzos: 2, 8/7, 15/11&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 1 15 | 0 1 0 7 −19 | 0 0 1 −4 8 }}
Lattice basis: 44/39 length = 0.3480 6/5 length = 0.7828&lt;br /&gt;
 
Angle(44/39, 6/5) = 85.3977&lt;br /&gt;
[[Optimal tuning]]s:
Map to lattice: [&amp;lt;0 2 2 6 4 1|, &amp;lt;0 -1 -2 1 1 2|]&lt;br /&gt;
* [[WE]]: ~2 = 1199.9495{{c}}, ~3/2 = 702.0619{{c}}, ~5/4 = 386.3920{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.0505 +0.0565 -0.0226 +0.0395 +0.0362 }}
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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0892{{c}}, ~5/4 = 386.4019{{c}}
Generators: 99/70, 3, 220/117&lt;br /&gt;
: error map: {{val| 0.0000 +0.1342 +0.0882 +0.1909 +0.2024 }}
Edos: 46, 80, 92, 126, 144, 178, 190, 224, 494, 684, 954, 1178, 1942, 3120, 3390&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99, 171, 217, 224, 270, 494, 764, 1205, 1475, 1969, 2239, 4208de, 9891bbcddeee }}
&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;
 
Commas: 4375/4374, 3025/3024, 1716/1715&lt;br /&gt;
[[Badness]] (Sintel): 0.957
&lt;br /&gt;
 
13 and 15 limit minimax&lt;br /&gt;
=== 13-limit ===
[|1 0 0 0 0 0&amp;gt;, |7/10 4/5 0 -2/5 0 1/5&amp;gt;, &lt;br /&gt;
Subgroup: 2.3.5.7.11.13
|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;
Comma list: 2080/2079, 4096/4095, 4375/4374
&lt;br /&gt;
 
Lattice basis: 10/9 length 0.4234 6/5 length 0.8426&lt;br /&gt;
{{Mapping|legend=0| 1 0 0 1 15 11 | 0 1 0 7 −19 -9 | 0 0 1 −4 8 3 }}
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;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1199.9615{{c}}, ~3/2 = 702.0459{{c}}, ~5/4 = 386.3501{{c}}
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;
: error map: {{val| -0.0385 +0.0523 -0.0407 +0.0949 +0.1485 -0.1987 }}
Generators: 99/70, 3, 5&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0686{{c}}, ~5/4 = 386.3614{{c}}
Edos: 26, 46, 72, 178, 224, 270, 494&lt;br /&gt;
: error map: {{val| 0.0000 +0.1136 +0.0477 +0.2089 +0.2692 -0.0612 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-11 limit children-Thor-Rym"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Rym&lt;/h4&gt;
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99, 171, 217, 224, 270, 494, 764, 935, 1205, 1699 }}
Commas: 4375/4374, 3025/3024, 4096/4095&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.431
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-11 limit children-Tyr"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Tyr&lt;/h3&gt;
 
Commas: 4375/4374, 131072/130977&lt;br /&gt;
== Thor ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-11 limit children-Veur"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Veur&lt;/h3&gt;
 
Commas: 4375/4374, 100663296/100656875&lt;br /&gt;
[[Comma list]]: 3025/3024, 4375/4374
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-11 limit children-Veod"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Veod&lt;/h3&gt;
{{Mapping|legend=1| 2 0 0 2 5 | 0 1 0 7 5 | 0 0 1 -4 -3 }}
Commas: 4375/4374, 2097152/2096325&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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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:Ragismic family| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]

Latest revision as of 13:53, 24 October 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.

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

Overview to extensions

Temperaments discussed elsewhere include:

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

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

Raglympic

Named by Eufalesio in 2026, raglympic is the intersection of olympic and ragismic, as its name implies. It is also a detemperament of ponta, achieved by observing the schisma. As such, the 5 generator can be taken to be a schisma.

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 131072/130977

Mapping: [⟨1 0 0 1 15], ⟨0 1 0 7 −19], ⟨0 0 1 −4 8]]

Optimal tunings:

  • WE: ~2 = 1199.9495 ¢, ~3/2 = 702.0619 ¢, ~5/4 = 386.3920 ¢
error map: ⟨-0.0505 +0.0565 -0.0226 +0.0395 +0.0362]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0892 ¢, ~5/4 = 386.4019 ¢
error map: ⟨0.0000 +0.1342 +0.0882 +0.1909 +0.2024]

Optimal ET sequence: 7, 39, 46, 53, 99, 171, 217, 224, 270, 494, 764, 1205, 1475, 1969, 2239, 4208de, 9891bbcddeee

Badness (Sintel): 0.957

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 4375/4374

Mapping: [⟨1 0 0 1 15 11], ⟨0 1 0 7 −19 -9], ⟨0 0 1 −4 8 3]]

Optimal tunings:

  • WE: ~2 = 1199.9615 ¢, ~3/2 = 702.0459 ¢, ~5/4 = 386.3501 ¢
error map: ⟨-0.0385 +0.0523 -0.0407 +0.0949 +0.1485 -0.1987]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0686 ¢, ~5/4 = 386.3614 ¢
error map: ⟨0.0000 +0.1136 +0.0477 +0.2089 +0.2692 -0.0612]

Optimal ET sequence: 7, 39, 46, 53, 99, 171, 217, 224, 270, 494, 764, 935, 1205, 1699

Badness (Sintel): 0.431

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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