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Sengic family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''sengic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the senga a.k.a. sengic comma, [[686/675]].  
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-23 17:33:48 UTC</tt>.<br>
: The original revision id was <tt>238465813</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">These temper out the [[senga]], [[686_675|686/675]].


Comma: 686/675
== Sengic ==
Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It was discovered and named in 2005.


Map: [&lt;1 0 2 1|, &lt;0 1 0 1|, &lt;0 0 3 2|]
[[Subgroup]]: 2.3.5.7
EDOs: [[9edo|9]], [[10edo|10]], [[18edo|18]], [[19edo|19]], [[27edo|27]], [[46edo|46]]
[[Badness]]: 0.000320


==Krypton==
[[Comma list]]: [[686/675]]
Commas: 56/55, 540/539


Map: [&lt;1 0 2 1 2|, &lt;0 1 0 1 1| &lt;0 0 3 2 -1|]
{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}
EDOs: 9, 10, 19, [[36edo|36]]
: mapping generators: ~2, ~3, ~15/14
Badness: 0.000856


===13-limit===
[[Optimal tuning]]s:
Commas: 56/55, 78/77, 91/90
* [[WE]]: ~2 = 1199.7533{{c}}, ~3/2 = 704.0092{{c}}, ~15/14 = 129.7976{{c}}
: [[error map]]: {{val| -0.247 +1.808 +2.586 -5.715 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9671{{c}}, ~15/14 = 129.7330{{c}}
: error map: {{val| 0.000 +2.012 +2.885 -5.393 }}


Map: [&lt;1 0 2 1 2 2|, &lt;0 1 0 1 1 1| &lt;0 0 3 2 -1 1|]
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }}
EDOs: 9, 10, 19, 36
 
Badness: 0.000727</pre></div>
[[Badness]] (Sintel): 1.41
<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;Sengic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These temper out the &lt;a class="wiki_link" href="/senga"&gt;senga&lt;/a&gt;, &lt;a class="wiki_link" href="/686_675"&gt;686/675&lt;/a&gt;.&lt;br /&gt;
[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
&lt;br /&gt;
 
Comma: 686/675&lt;br /&gt;
=== Overview to extensions ===
&lt;br /&gt;
First noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group | ''It's the "thirds", stupid!'']</ref>, sengic is naturally a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 = ([[169/168]])⋅([[196/195]])<sup>2</sup>, as we can see from its [[S-expression]], S13⋅S14<sup>2</sup>. This identifies the last generator as [[13/12]]~[[14/13]]~15/14. This extension is considered immediately below.
Map: [&amp;lt;1 0 2 1|, &amp;lt;0 1 0 1|, &amp;lt;0 0 3 2|]&lt;br /&gt;
 
EDOs: &lt;a class="wiki_link" href="/9edo"&gt;9&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/18edo"&gt;18&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;&lt;br /&gt;
11-limit temperaments considered below are demeter, krypton, and sensigh.
&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.000320&lt;br /&gt;
 
&lt;br /&gt;
=== 2.3.5.7.13 subgroup ===
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Krypton"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Krypton&lt;/h2&gt;
Subgroup: 2.3.5.7.13
Commas: 56/55, 540/539&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 169/168
Map: [&amp;lt;1 0 2 1 2|, &amp;lt;0 1 0 1 1| &amp;lt;0 0 3 2 -1|]&lt;br /&gt;
 
EDOs: 9, 10, 19, &lt;a class="wiki_link" href="/36edo"&gt;36&lt;/a&gt;&lt;br /&gt;
Subgroup-val mapping: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }}
Badness: 0.000856&lt;br /&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-Krypton-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h3&gt;
* WE: ~2 = 1200.3448{{c}}, ~3/2 = 704.1998{{c}}, ~14/13 = 129.5253{{c}}
Commas: 56/55, 78/77, 91/90&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2688{{c}}, ~14/13 = 129.6115{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 2 1 2 2|, &amp;lt;0 1 0 1 1 1| &amp;lt;0 0 3 2 -1 1|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
EDOs: 9, 10, 19, 36&lt;br /&gt;
 
Badness: 0.000727&lt;/body&gt;&lt;/html&gt;</pre></div>
Badness (Sintel): 0.425
 
== Demeter ==
Named by [[Graham Breed]] in 2011, demeter was found to be locally efficient in the 17-limit among all rank-3 temperaments<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 686/675
 
{{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4002{{c}}, ~3/2 = 705.1652{{c}}, ~15/14 = 129.9738{{c}}
: [[error map]]: {{val| -0.600 +2.610 +2.408 -4.913 -1.283 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2789{{c}}, ~15/14 = 129.8330{{c}}
: error map: {{val| 0.000 +3.324 +3.185 -3.881 -0.369 }}
 
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }}
 
[[Badness]] (Sintel): 1.58
 
[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 169/168, 352/351
 
Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }}
 
Optimal tunings:  
* WE: ~2 = 1200.0393{{c}}, ~3/2 = 705.1360{{c}}, ~14/13 = 129.6770{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1277{{c}}, ~14/13 = 129.6854{{c}}
 
{{Optimal ET sequence|legend=0| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}
 
Badness (Sintel): 0.913
 
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168
 
Mapping: {{mapping| 1 0 2 1 -3 2 -1 | 0 1 0 1 4 1 3 | 0 0 3 2 1 1 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.0255{{c}}, ~3/2 = 705.1616{{c}}, ~14/13 = 129.7024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1553{{c}}, ~14/13 = 129.7071{{c}}
 
{{Optimal ET sequence|legend=0| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}
 
Badness (Sintel): 0.789
 
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11
 
== Krypton ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 540/539
 
{{Mapping|legend=1| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.0576{{c}}, ~3/2 = 704.2467{{c}}, ~15/14 = 132.2189{{c}}
: [[error map]]: {{val| -2.942 -0.651 +4.458 -6.026 +11.883 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6964{{c}}, ~15/14 = 131.8166{{c}}
: error map: {{val| 0.000 +1.741 +9.136 -1.496 +20.562 }}
 
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 63cee }} *
 
<nowiki/>* [[optimal patent val]]: [[36edo|36]]
 
[[Badness]] (Sintel): 1.03
 
[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 91/90
 
Mapping: {{mapping| 1 0 2 1 2 2 | 0 1 0 1 1 1 | 0 0 3 2 -1 1 }}
 
Optimal tunings:
* WE: ~2 = 1197.6484{{c}}, ~3/2 = 704.6450{{c}}, ~14/13 = 132.1686{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1616{{c}}, ~14/13 = 131.8445{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27e }} *
 
<nowiki/>* optimal patent val: [[36edo|36]]
 
Badness (Sintel): 0.680
 
Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11
 
== Sensigh ==
Sensigh uses the same mapping as 7-limit [[sensi]] with an independent generator for prime 11.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 245/243
 
{{Mapping|legend=1| 1 -1 -1 -2 3 | 0 7 9 13 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~9/7, ~11
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}, ~11/8 = 552.1736{{c}}
: [[error map]]: {{val| -0.2919 +1.2608 +3.4518 -5.6691 -0.0202 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}, ~11/8 = 551.8069{{c}}
: error map: {{val| 0.0000 +1.4899 +3.8297 -5.2854 +0.4890 }}
 
{{Optimal ET sequence|legend=1| 27e, 38df, 46, 111d }}
 
[[Badness]] (Sintel): 1.48
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168
 
Mapping: {{mapping| 1 -1 -1 -2 3 0 | 0 7 9 13 0 10 | 0 0 0 0 1 0 }}
 
Optimal tunings:  
* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}
 
Badness (Sintel): 0.878
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 126/125, 154/153, 169/168
 
Mapping: {{mapping| 1 -1 -1 -2 3 0 4 | 0 7 9 13 0 10 -1 | 0 0 0 0 1 0 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}
 
Badness (Sintel): 0.917
 
== References ==
 
[[Category:Temperament families]]
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Sengic_family"