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Keemic 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 '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2 temperament]]s, see [[keemic temperaments]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-24 00:48:23 UTC</tt>.<br>
: The original revision id was <tt>173068611</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: 875/864


7 and 9 limit minimax
== Keemic ==
[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/4 3/4 1/4 -1/4&gt;,  
Keemic, also known as ''supermagic'' in earlier materials, has the same lattice structure as [[5-limit]] JI, and [[7/4]] is found by a stack of three [[~]][[6/5]]'s.
|5/4 3/4 -3/4 3/4&gt;]
Eigenmonzos: 2, 4/3, 7/5


Lattice basis: 6/5 length=0.8879 3/2 length=1.3391
[[Subgroup]]: 2.3.5.7
Angle(6/5, 3/2) = 77.834
Map to lattice: [&lt;0 0 -1 3|, &lt;0 1 1 0|]


Map: [&lt;1 0 0 5|, &lt;0 1 0 3|, &lt;0 0 1 -3|]
[[Comma list]]: 875/864
Generators: 2, 3, 5
Edos: 7, 8, 15, 18, 19, 22, 26, 34, 37, 41, 56, 60


</pre></div>
{{Mapping|legend=1| 1 0 0 5 | 0 1 0 3 | 0 0 1 -3 }}
<h4>Original HTML content:</h4>
: mapping generators: ~2, ~3, ~5
<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;Keemic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
 
Comma: 875/864&lt;br /&gt;
[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]
&lt;br /&gt;
 
7 and 9 limit minimax&lt;br /&gt;
Lattice basis:  
[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |5/4 3/4 1/4 -1/4&amp;gt;, &lt;br /&gt;
: 6/5 length = 0.8879, 3/2 length = 1.3391
|5/4 3/4 -3/4 3/4&amp;gt;]&lt;br /&gt;
: Angle (6/5, 3/2) = 77.834
Eigenmonzos: 2, 4/3, 7/5&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
Lattice basis: 6/5 length=0.8879 3/2 length=1.3391&lt;br /&gt;
* [[WE]]: ~2 = 1201.0490{{c}}, ~3/2 = 702.4868{{c}}, ~5/4 = 380.8010{{c}}
Angle(6/5, 3/2) = 77.834&lt;br /&gt;
: [[error map]]: {{val| +1.049 +1.581 -3.415 -1.670 }}
Map to lattice: [&amp;lt;0 0 -1 3|, &amp;lt;0 1 1 0|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +0.851 -5.260 -3.569 }}
Map: [&amp;lt;1 0 0 5|, &amp;lt;0 1 0 3|, &amp;lt;0 0 1 -3|]&lt;br /&gt;
 
Generators: 2, 3, 5&lt;br /&gt;
[[Minimax tuning]]:
Edos: 7, 8, 15, 18, 19, 22, 26, 34, 37, 41, 56, 60&lt;/body&gt;&lt;/html&gt;</pre></div>
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/4 3/4 1/4 -1/4 | 5/4 3/4 -3/4 3/4 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
 
{{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}
 
[[Badness]] (Sintel): 0.937
 
[[Projection pair]]: 7 864/125
 
Scales: [[supermagic15]]
 
== Undecimal keemic ==
{{See also| Ptolemismic clan #Keemic }}
 
Keemic is naturally an [[11-limit]] temperament due to the identity 875/864 = ([[100/99]])⋅([[385/384]]). This identifies a stack of two ~6/5's as ~[[16/11]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 100/99, 385/384
 
{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9868{{c}}, ~3/2 = 704.1908{{c}}, ~5/4 = 381.5352{{c}}
: [[error map]]: {{val| -0.013 +2.223 -4.805 -0.885 +3.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1922{{c}}, ~5/4 = 381.5339{{c}}
: error map: {{val| 0.000 +2.237 -4.780 -0.851 +3.365 }}
 
{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}
 
[[Badness]] (Sintel): 0.770
 
Scales: [[supermagic15]]
 
== Supernatural ==
Supernatural uses the same mapping as [[7-limit]] [[magic]] with an independent generator for prime 11.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 245/243
 
{{Mapping|legend=1| 1 0 2 -1 0 | 0 5 1 12 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~5, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}, ~11/8 = 548.0690{{c}}
: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 -0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}, ~11/8 = 548.8962{{c}}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 -2.422 }}
 
{{Optimal ET sequence|legend=1| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde }} *
 
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]
 
[[Badness]] (Sintel): 1.07
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 245/243
 
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 0 18 | 0 0 0 0 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1201.3389{{c}}, ~5/4 = 380.4641{{c}}, ~11/8 = 547.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}
 
{{Optimal ET sequence|legend=0| 19, 22f, 38df, 41, 60e, 79d, 101cd }}
 
Badness (Sintel): 1.07
 
[[Category:Temperament families]]
[[Category:Keemic family| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Keemic_family"