Hemimage family: Difference between revisions

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**Imported revision 165077867 - Original comment: **
m Units, misc. cleanup, - redundant category
 
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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 '''hemimage family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] tempers out the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-24 00:04:39 UTC</tt>.<br>
: The original revision id was <tt>165077867</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">
===Vital statistics===
Comma: hemimage 10976/10935


7-limit minimax
== Hemimage  ==
[[Subgroup]]: 2.3.5.7


Eigenmonzos: 2, 8/7, 6/5
[[Comma list]]: 10976/10935


9-limit minimax
{{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }}
: mapping generators: ~2, ~3, ~27/14


Eigenmonz: 2, 10/9, 8/7
[[Minimax tuning]]:  
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/8 1/8 -1/8 3/8 | 5/8 -7/8 7/8 3/8 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/9 2/9 -1/9 1/3 | 10/9 -14/9 7/9 2/3 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5


==11-limit children==
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}


===Indra===
[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>
Commas: 540/539, 1375/1372


Minimax tuning
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7


Eigenmonzos: 2, 11/9, 5/4
== Uni ==
[[Subgroup]]: 2.3.5.7.11


===Brahma===
[[Comma list]]: 540/539, 896/891
Commas: 10976/10935, 441/440


Minimax tuning
{{Mapping|legend=1| 1 0 2 -1 6 | 0 1 2 3 -1 | 0 0 -3 -1 -1 }}


Eigenmonzos: 2, 10/9, 11/9</pre></div>
[[Minimax tuning]]:
<h4>Original HTML content:</h4>
* [[11-odd-limit]]
<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;Hemimage family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9
Comma: hemimage 10976/10935&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
7-limit minimax&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>
Eigenmonzos: 2, 8/7, 6/5&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit  ===
9-limit minimax&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Eigenmonz: 2, 10/9, 8/7&lt;br /&gt;
Comma list: 352/351, 364/363, 540/539
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-11-limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit children&lt;/h2&gt;
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11-limit children-Indra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Indra&lt;/h3&gt;
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
Commas: 540/539, 1375/1372&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 1.06 × 10<sup>-3</sup>
Minimax tuning&lt;br /&gt;
 
&lt;br /&gt;
=== Tin ===
Eigenmonzos: 2, 11/9, 5/4&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-11-limit children-Brahma"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Brahma&lt;/h3&gt;
Comma list: 144/143, 196/195, 729/728
Commas: 10976/10935, 441/440&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}
Minimax tuning&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
Eigenmonzos: 2, 10/9, 11/9&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Badness (Smith): 1.07 × 10<sup>-3</sup>
 
== Turan ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 10976/10935
 
{{Mapping|legend=1| 1 0 2 -1 -7 | 0 1 2 3 6 | 0 0 -3 -1 1 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
 
{{Optimal ET sequence|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}
 
[[Badness]] (Smith): 0.936 × 10<sup>-3</sup>
 
[[Category:Temperament families]]
[[Category:Hemimage family| ]] <!-- main article -->
[[Category:Rank 3]]

Latest revision as of 18:49, 28 April 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 hemimage family of rank-3 temperaments tempers out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935).

Hemimage

Subgroup: 2.3.5.7

Comma list: 10976/10935

Mapping: [⟨1 0 2 -1], ⟨0 1 2 3], ⟨0 0 -3 -1]]

mapping generators: ~2, ~3, ~27/14

Minimax tuning:

[[1 0 0 0⟩, [5/8 1/8 -1/8 3/8⟩, [5/8 -7/8 7/8 3/8⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.7
[[1 0 0 0⟩, [5/9 2/9 -1/9 1/3⟩, [10/9 -14/9 7/9 2/3⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.7.9/5

Optimal ET sequence: 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695

Badness (Smith): 0.199 × 10-3

Projection pair: 5 10976/2187 to 2.3.7

Uni

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891

Mapping: [⟨1 0 2 -1 6], ⟨0 1 2 3 -1], ⟨0 0 -3 -1 -1]]

Minimax tuning:

[[1 0 0 0 0⟩, [16/11 6/11 1/11 0 -3/11⟩, [0 0 1 0 0⟩, [19/11 14/11 6/11 0 -7/11⟩, [32/11 -10/11 2/11 0 5/11⟩]
unchanged-interval (eigenmonzo) basis: 2.5.11/9

Optimal ET sequence: 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee

Badness (Smith): 0.702 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539

Mapping: [⟨1 0 2 -1 6 11], ⟨0 1 2 3 -1 -4], ⟨0 0 -3 -1 -1 -1]]

Optimal ET sequence: 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef

Badness (Smith): 1.06 × 10-3

Tin

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 729/728

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 1 2 3 -1 3], ⟨0 0 -3 -1 -1 1]]

Optimal ET sequence: 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff

Badness (Smith): 1.07 × 10-3

Turan

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935

Mapping: [⟨1 0 2 -1 -7], ⟨0 1 2 3 6], ⟨0 0 -3 -1 1]]

Minimax tuning:

[[1 0 0 0 0⟩, [19/12 -2/3 1/12 0 1/4⟩, [19/6 -10/3 7/6 0 1/2⟩, [37/12 -8/3 7/12 0 3/4⟩, [19/6 -10/3 1/6 0 3/2⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335

Badness (Smith): 0.936 × 10-3

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