Keemic family: Difference between revisions

Latest revision as of 12:27, 27 November 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 keemic family is a family of rank-3 temperaments which temper out the keema (monzo: [-5 -3 3 1⟩, ratio: 875/864). For rank-2 temperaments, see keemic temperaments.

Keemic

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.

Subgroup: 2.3.5.7

Comma list: 875/864

Mapping: [⟨1 0 0 5], ⟨0 1 0 3], ⟨0 0 1 -3]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 0 -1 3], ⟨0 1 1 0]]

Lattice basis:

6/5 length = 0.8879, 3/2 length = 1.3391

Angle (6/5, 3/2) = 77.834

Optimal tunings:

WE: ~2 = 1201.0490 ¢, ~3/2 = 702.4868 ¢, ~5/4 = 380.8010 ¢

error map: ⟨+1.049 +1.581 -3.415 -1.670]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8060 ¢, ~5/4 = 381.0536 ¢

error map: ⟨0.000 +0.851 -5.260 -3.569]

Minimax tuning:

7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [5/4 3/4 1/4 -1/4⟩, [5/4 3/4 -3/4 3/4⟩]

unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 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: [⟨1 0 0 5 2], ⟨0 1 0 3 -2], ⟨0 0 1 -3 2]]

Optimal tunings:

WE: ~2 = 1199.9868 ¢, ~3/2 = 704.1908 ¢, ~5/4 = 381.5352 ¢

error map: ⟨-0.013 +2.223 -4.805 -0.885 +3.318]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1922 ¢, ~5/4 = 381.5339 ¢

error map: ⟨0.000 +2.237 -4.780 -0.851 +3.365]

Optimal ET sequence: 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: [⟨1 0 2 -1 0], ⟨0 5 1 12 0], ⟨0 0 0 0 1]]

mapping generators: ~2, ~5, ~11

Optimal tunings:

WE: ~2 = 1201.0786 ¢, ~5/4 = 380.6939 ¢, ~11/8 = 548.0690 ¢

error map: ⟨+1.079 +1.514 -3.463 -1.578 -0.013]

CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4576 ¢, ~11/8 = 548.8962 ¢

error map: ⟨0.000 +0.333 -5.856 -3.335 -2.422]

Optimal ET sequence: 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde *

* optimal patent val: 104

Badness (Sintel): 1.07

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243

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 ¢, ~5/4 = 380.4641 ¢, ~11/8 = 547.2772 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1532 ¢, ~11/8 = 548.2678 ¢

Optimal ET sequence: 19, 22f, 38df, 41, 60e, 79d, 101cd

Badness (Sintel): 1.07

@@ Line 1: / Line 1: @@
-<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>2012-02-29 00:31:25 UTC</tt>.<br>
-: The original revision id was <tt>306237162</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">[[toc|flat]]
-These are the [[planar temperament]]s tempering out the keema, 875/864.
+== Keemic ==
+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.
-=Supermagic=
+[[Subgroup]]: 2.3.5.7
-Comma: 875/864
-and 9 limit minimax
+[[Comma list]]: 875/864
-[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/4 3/4 1/4 -1/4&gt;,
-|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
+{{Mapping|legend=1| 1 0 0 5 | 0 1 0 3 | 0 0 1 -3 }}
-Angle(6/5, 3/2) = 77.834
+: mapping generators: ~2, ~3, ~5
-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|]
+[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]
-Generators: 2, 3, 5
-Edos: 7, 8, 15, 18, 19, 22, 26, 34, 37, 41, 56, 60
+Lattice basis:
+: 6/5 length = 0.8879, 3/2 length = 1.3391
+: Angle (6/5, 3/2) = 77.834
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.0490{{c}}, ~3/2 = 702.4868{{c}}, ~5/4 = 380.8010{{c}}
+: [[error map]]: {{val| +1.049 +1.581 -3.415 -1.670 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}
+: error map: {{val| 0.000 +0.851 -5.260 -3.569 }}
+[[Minimax tuning]]:
+* [[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: [[keemic15]]
+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 }}
-==11-limit==
+Optimal tunings:
-Commas: 100/99, 385/384
+* 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}}
-Map: [&lt;1 0 0 5 2|, &lt;0 1 0 3 -2|, &lt;0 0 1 -3 2|]
+{{Optimal ET sequence|legend=0| 19, 22f, 38df, 41, 60e, 79d, 101cd }}
-EDOs: 7, 15, 19, 22, 37, 41, 63, 104, 145c, 201ce, 242ce, 264bce, 305bce
-Badness: 0.000641
-=Supernatural=
+Badness (Sintel): 1.07
-Commas: 225/224, 245/243
-Map:[&lt;1 0 2 -1 0|, &lt;0 5 1 12 0|, &lt;0 0 0 0 1|]
+[[Category:Temperament families]]
-EDOs: 19, 22, 41, 82e, 101cd, 104, 123ce, 142cd, 224cde, 265cde
+[[Category:Keemic family| ]] <!-- main article -->
-Badness: 0.000888
+[[Category:Rank 3]]
-</pre></div>
-<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;Keemic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Supermagic"&gt;Supermagic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#Supernatural"&gt;Supernatural&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
-These are the &lt;a class="wiki_link" href="/planar%20temperament"&gt;planar temperament&lt;/a&gt;s tempering out the keema, 875/864.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Supermagic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Supermagic&lt;/h1&gt;
-Comma: 875/864&lt;br /&gt;
-&lt;br /&gt;
-and 9 limit minimax&lt;br /&gt;
-[|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;
-|5/4 3/4 -3/4 3/4&amp;gt;]&lt;br /&gt;
-Eigenmonzos: 2, 4/3, 7/5&lt;br /&gt;
-&lt;br /&gt;
-Lattice basis: 6/5 length=0.8879 3/2 length=1.3391&lt;br /&gt;
-Angle(6/5, 3/2) = 77.834&lt;br /&gt;
-Map to lattice: [&amp;lt;0 0 -1 3|, &amp;lt;0 1 1 0|]&lt;br /&gt;
-&lt;br /&gt;
-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;
-Edos: 7, 8, 15, 18, 19, 22, 26, 34, 37, 41, 56, 60&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 864/125&lt;br /&gt;
-&lt;br /&gt;
-Scales: &lt;a class="wiki_link" href="/keemic15"&gt;keemic15&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Supermagic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
-Commas: 100/99, 385/384&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 5 2|, &amp;lt;0 1 0 3 -2|, &amp;lt;0 0 1 -3 2|]&lt;br /&gt;
-EDOs: 7, 15, 19, 22, 37, 41, 63, 104, 145c, 201ce, 242ce, 264bce, 305bce&lt;br /&gt;
-Badness: 0.000641&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Supernatural"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Supernatural&lt;/h1&gt;
-Commas: 225/224, 245/243&lt;br /&gt;
-&lt;br /&gt;
-Map:[&amp;lt;1 0 2 -1 0|, &amp;lt;0 5 1 12 0|, &amp;lt;0 0 0 0 1|]&lt;br /&gt;
-EDOs: 19, 22, 41, 82e, 101cd, 104, 123ce, 142cd, 224cde, 265cde&lt;br /&gt;
-Badness: 0.000888&lt;/body&gt;&lt;/html&gt;</pre></div>

Keemic family: Difference between revisions

Latest revision as of 12:27, 27 November 2025

Keemic

Undecimal keemic

Supernatural

13-limit

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