Hemimage family: Difference between revisions

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:

7-odd-limit

[[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

9-odd-limit

[[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:

11-odd-limit

[[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:

11-odd-limit

[[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

@@ 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 '''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>2011-01-02 04:26:43 UTC</tt>.<br>
-: The original revision id was <tt>190465040</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
--limit minimax
+== Hemimage  ==
-[|1 0 0 0&gt;, |5/8 1/8 -1/8 3/8&gt;,
+[[Subgroup]]: 2.3.5.7
-|5/8 -7/8 7/8 3/8&gt;, |0 0 0 1&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 6/5
--limit minimax
+[[Comma list]]: 10976/10935
-[|1 0 0 0&gt;, |5/9 2/9 -1/9 1/3&gt;,
-|10/9 -14/9 7/9 2/3&gt;, |0 0 0 1&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 8/7
-Map: [&lt;1 0 2 -1|, &lt;0 1 2 3|, &lt;0 0 -3 -1|]
+{{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }}
-[[Generator|Generators]]: 2, 3, 27/14
+: mapping generators: ~2, ~3, ~27/14
-[[edo|Edos]]: 14, 19, 41, 58, 77, 80, [[99edo|99]], [[118edo|118]], [[140edo|140]], [[217edo|217]], [[316edo|316]], [[338edo|338]], [[357edo|357]], [[415edo|415]], [[456edo|456]], [[654edo|654]], [[695edo|695]]
-Badness: 0.000199
-==Uni==
+[[Minimax tuning]]:
-[[Comma|Commas]]: 540/539, 896/891
+* [[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
-[[Minimax tuning]]
+{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
-[|1 0 0 0 0&gt;, |16/11 6/11 1/11 0 -3/11&gt;,
-|0 0 1 0 0&gt;, |19/11 14/11 6/11 0 -7/11&gt;,
-|32/11 -10/11 2/11 0 5/11&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4
-Map: [&lt;1 0 2 -1 6|, &lt;0 1 2 3 -1|, &lt;0 0 -3 -1 -1|]
+[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>
-[[Generator|Generators]]: 2, 3, 27/14
-[[edo|Edos]]: [[19edo|19]], [[22edo|22]], [[41edo|41]], [[58edo|58]], [[80edo|80]], [[121edo|121]]
-Badness: 0.000702
-===13-limit===
+[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7
-Commas: 352/351, 364/363, 540/539
-Map: [&lt;1 0 2 -1 6 11|, &lt;0 1 2 3 -1 -4|, &lt;0 0 -3 -1 -1 -1|]
+== Uni ==
-EDOs: 22, 41, 58, 80, 121
+[[Subgroup]]: 2.3.5.7.11
-Badness: 0.00106
-===Tin===
+[[Comma list]]: 540/539, 896/891
-Commas: 144/143, 196/195, 729/728
-Map: [&lt;1 0 2 -1 6 -2|, &lt;0 1 2 3 -1 3|, &lt;0 0 -3 -1 -1 1|]
+{{Mapping|legend=1| 1 0 2 -1 6 | 0 1 2 3 -1 | 0 0 -3 -1 -1 }}
-EDOs: 19, 36, 41, 58
-Badness: 0.00107
-==Turan==
+[[Minimax tuning]]:
-[[Comma|Commas]]: 10976/10935, 441/440
+* [[11-odd-limit]]
+: [{{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 }}]
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9
-[[Minimax tuning]]
+{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
-[|1 0 0 0 0&gt;, |19/12 -2/3 1/12 0 1/4&gt;,
-|19/6 -10/3 7/6 0 1/2&gt;,
-|37/12 -8/3 7/12 0 3/4&gt;,
-|19/6 -10/3 1/6 0 3/2&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9
-Map: [&lt;1 0 2 -1 -7|, &lt;0 1 2 3 6|, &lt;0 0 -3 -1 1|]
+[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>
-[[Generator|Generators]]: 2, 3, 27/14
-[[edo|Edos]]: [[41edo|41]], [[58edo|58]], [[118edo|118]], [[159edo|159]], [[217edo|217]], [[335edo|335]]
+=== 13-limit  ===
-Badness: 0.000936</pre></div>
+Subgroup: 2.3.5.7.11.13
-<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;Hemimage family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&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;
+Comma list: 352/351, 364/363, 540/539
- &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;: hemimage 10976/10935&lt;br /&gt;
-&lt;br /&gt;
+Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}
--limit minimax&lt;br /&gt;
-[|1 0 0 0&amp;gt;, |5/8 1/8 -1/8 3/8&amp;gt;, &lt;br /&gt;
+{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
-|5/8 -7/8 7/8 3/8&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 6/5&lt;br /&gt;
+Badness (Smith): 1.06 × 10<sup>-3</sup>
-&lt;br /&gt;
--limit minimax&lt;br /&gt;
+=== Tin ===
-[|1 0 0 0&amp;gt;, |5/9 2/9 -1/9 1/3&amp;gt;, &lt;br /&gt;
+Subgroup: 2.3.5.7.11.13
-|10/9 -14/9 7/9 2/3&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 8/7&lt;br /&gt;
+Comma list: 144/143, 196/195, 729/728
-&lt;br /&gt;
-Map: [&amp;lt;1 0 2 -1|, &amp;lt;0 1 2 3|, &amp;lt;0 0 -3 -1|]&lt;br /&gt;
+Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 27/14&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: 14, 19, 41, 58, 77, 80, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/217edo"&gt;217&lt;/a&gt;, &lt;a class="wiki_link" href="/316edo"&gt;316&lt;/a&gt;, &lt;a class="wiki_link" href="/338edo"&gt;338&lt;/a&gt;, &lt;a class="wiki_link" href="/357edo"&gt;357&lt;/a&gt;, &lt;a class="wiki_link" href="/415edo"&gt;415&lt;/a&gt;, &lt;a class="wiki_link" href="/456edo"&gt;456&lt;/a&gt;, &lt;a class="wiki_link" href="/654edo"&gt;654&lt;/a&gt;, &lt;a class="wiki_link" href="/695edo"&gt;695&lt;/a&gt;&lt;br /&gt;
+{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
-Badness: 0.000199&lt;br /&gt;
-&lt;br /&gt;
+Badness (Smith): 1.07 × 10<sup>-3</sup>
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Uni"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Uni&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 540/539, 896/891&lt;br /&gt;
+== Turan ==
-&lt;br /&gt;
+[[Subgroup]]: 2.3.5.7.11
-&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |16/11 6/11 1/11 0 -3/11&amp;gt;, &lt;br /&gt;
+[[Comma list]]: 441/440, 10976/10935
-|0 0 1 0 0&amp;gt;, |19/11 14/11 6/11 0 -7/11&amp;gt;, &lt;br /&gt;
-|32/11 -10/11 2/11 0 5/11&amp;gt;]&lt;br /&gt;
+{{Mapping|legend=1| 1 0 2 -1 -7 | 0 1 2 3 6 | 0 0 -3 -1 1 }}
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 5/4&lt;br /&gt;
-&lt;br /&gt;
+[[Minimax tuning]]:
-Map: [&amp;lt;1 0 2 -1 6|, &amp;lt;0 1 2 3 -1|, &amp;lt;0 0 -3 -1 -1|]&lt;br /&gt;
+* [[11-odd-limit]]
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 27/14&lt;br /&gt;
+: [{{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 }}]
-&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/80edo"&gt;80&lt;/a&gt;, &lt;a class="wiki_link" href="/121edo"&gt;121&lt;/a&gt;&lt;br /&gt;
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
-Badness: 0.000702&lt;br /&gt;
-&lt;br /&gt;
+{{Optimal ET sequence|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Uni-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h3&gt;
-Commas: 352/351, 364/363, 540/539&lt;br /&gt;
+[[Badness]] (Smith): 0.936 × 10<sup>-3</sup>
-&lt;br /&gt;
-Map: [&amp;lt;1 0 2 -1 6 11|, &amp;lt;0 1 2 3 -1 -4|, &amp;lt;0 0 -3 -1 -1 -1|]&lt;br /&gt;
+[[Category:Temperament families]]
-EDOs: 22, 41, 58, 80, 121&lt;br /&gt;
+[[Category:Hemimage family| ]] <!-- main article -->
-Badness: 0.00106&lt;br /&gt;
+[[Category:Rank 3]]
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Uni-Tin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Tin&lt;/h3&gt;
-Commas: 144/143, 196/195, 729/728&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 2 -1 6 -2|, &amp;lt;0 1 2 3 -1 3|, &amp;lt;0 0 -3 -1 -1 1|]&lt;br /&gt;
-EDOs: 19, 36, 41, 58&lt;br /&gt;
-Badness: 0.00107&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Turan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Turan&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 10976/10935, 441/440&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |19/12 -2/3 1/12 0 1/4&amp;gt;, &lt;br /&gt;
-|19/6 -10/3 7/6 0 1/2&amp;gt;, &lt;br /&gt;
-|37/12 -8/3 7/12 0 3/4&amp;gt;, &lt;br /&gt;
-|19/6 -10/3 1/6 0 3/2&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 11/9&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 2 -1 -7|, &amp;lt;0 1 2 3 6|, &amp;lt;0 0 -3 -1 1|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 27/14&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/217edo"&gt;217&lt;/a&gt;, &lt;a class="wiki_link" href="/335edo"&gt;335&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000936&lt;/body&gt;&lt;/html&gt;</pre></div>

Hemimage family: Difference between revisions

Latest revision as of 18:49, 28 April 2026

Contents

Hemimage

Uni

13-limit

Tin

Turan

Navigation menu

Hemimage family: Difference between revisions

Latest revision as of 18:49, 28 April 2026

Hemimage

Uni

13-limit

Tin

Turan

Navigation menu

Search