Hemimage family: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~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 15:14:37 UTC</tt>.<br>~~

~~: The original revision id was <tt>165246311</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 ==

[~~|1 0 0 0>, |5/8 1/8 -1/8 3/8>,~~

[[Subgroup]]: 2.3.5.7

~~|5/8 -7/8 7/8 3/8>, |0 0 0 1>~~]

~~Eigenmonzos~~: 2~~, 8/~~7~~, 6/5~~

~~9-limit minimax~~

[[Comma list]]: 10976/10935

[~~|1 0 0 0>, |5/9 2/9 -1/9 1/3>,~~

~~|10/9 -14/9 7/9 2/3>, |0 0 0 1>~~]

~~Eigenmonzos~~: ~~2, 10/9, 8~~/7

~~Map: [<~~1 0 2 -1|~~, <~~0 1 2 3|~~, <~~0 0 -3 -1|]

~~Generators~~: 2, 3, 28/27

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

~~Edos: 99, 118, 140, 316, 415, 654~~

~~==11~~-limit ~~children==~~

[[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

=~~==Indra===~~

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

~~Commas: 540/539~~, ~~1375/1372~~

~~Minimax tuning~~

[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>

~~Eigenmonzos~~: ~~2, 11~~/~~9, 5~~/4

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

==~~=Brahma=~~==

== Uni ==

~~Commas~~: ~~10976/10935, 441/440~~

[[Subgroup]]: 2.3.5.7.11

~~Minimax tuning~~

[[Comma list]]: 540/539, 896/891

~~Eigenmonzos:~~ 2~~, 10/9, 11/9</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"><html><head><title>Hemimage family</title></head><body><br />~~

[[Minimax tuning]]:

~~<h3 id="toc0"><a name="x--Vital statistics"></a>Vital statistics</h3>~~

* [[11-odd-limit]]

~~Comma~~: ~~hemimage 10976/10935<br />~~

: [{{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 }}]

~~<br />~~

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9

7-limit ~~minimax<br />~~

[|1 0 0 0~~&gt;~~, |5/~~8 1~~/~~8 -~~1/~~8 3/8&gt;, <br />~~

{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}

~~|5/8~~ -~~7/8 7/8~~ 3/~~8&gt;~~, |0 0 0 ~~1&gt;]<br~~ /~~>~~

~~Eigenmonzos: 2, 8~~/7, 6/~~5<br />~~

[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>

~~<br />~~

9-~~limit minimax<br~~ /~~>~~

=== 13-limit ===

~~[|1 0 0 0&gt;~~, |5/~~9 2/9~~ -1/~~9 1~~/~~3&gt;, <br~~ /~~>~~

Subgroup: 2.3.5.7.11.13

|~~10/9~~ -14/9 ~~7/9 2/3&gt;,~~ |~~0 0 0~~ 1~~&gt;~~]~~<br~~ /~~>~~

~~Eigenmonzos~~: 2, 10/9, 8/~~7<br />~~

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

~~<br />~~

~~Map~~: ~~[&lt;~~1 0 2 -1|~~, &lt;~~0 1 2 3|~~, &lt;~~0 0 -3 -1|~~]<br />~~

Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}

~~Generators: 2~~, 3, ~~28/27<br />~~

~~Edos: 99~~, ~~118~~, ~~140~~, ~~316~~, ~~415~~, ~~654<br />~~

{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}

~~<br />~~

~~~~11~~-limit children<~~/~~h2>~~

Badness (Smith): 1.06 × 10<sup>-3</sup>

~~<br />~~

~~Indra</h3>~~

=== Tin ===

~~Commas: 540/539~~, ~~1375/1372<br />~~

Subgroup: 2.3.5.7.11.13

~~<br~~ /~~>~~

~~Minimax tuning<br />~~

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

~~<br />~~

~~Eigenmonzos~~: 2, 11/9, 5/~~4<br />~~

Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}

~~<br />~~

~~<!~~-- ~~ws:start:WikiTextHeadingRule:~~6~~:&lt;h3&gt;~~ --~~><h3 id="toc3"><a name="x~~-11-limit ~~children~~-~~Brahma"><~~/~~a><!~~-~~- ws:end:WikiTextHeadingRule:~~6 -~~->Brahma<~~/~~h3>~~

{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}

~~Commas: 10976~~/~~10935~~, ~~441~~/~~440<br~~ /~~>~~

~~<br~~ /~~>~~

Badness (Smith): 1.07 × 10<sup>-3</sup>

~~Minimax tuning<br~~ /~~>~~

~~<br />~~

== Turan ==

~~Eigenmonzos~~: 2~~, 10~~/9, 11/9~~</body></html>~~</~~pre~~><~~/div~~>

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 441/440, 10976/10935

[[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| ]]

[[Category:Rank 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>2010-09-24 15:14:37 UTC</tt>.<br>
-: The original revision id was <tt>165246311</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;]
-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;]
-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 }}
-Generators: 2, 3, 28/27
+: mapping generators: ~2, ~3, ~27/14
-Edos: 99, 118, 140, 316, 415, 654
-==11-limit children==
+[[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
-===Indra===
+{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
-Commas: 540/539, 1375/1372
-Minimax tuning
+[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>
-Eigenmonzos: 2, 11/9, 5/4
+[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7
-===Brahma===
+== Uni ==
-Commas: 10976/10935, 441/440
+[[Subgroup]]: 2.3.5.7.11
-Minimax tuning
+[[Comma list]]: 540/539, 896/891
-Eigenmonzos: 2, 10/9, 11/9</pre></div>
+{{Mapping|legend=1| 1 0 2 -1 6 | 0 1 2 3 -1 | 0 0 -3 -1 -1 }}
-<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;br /&gt;
+[[Minimax tuning]]:
-&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;
+* [[11-odd-limit]]
-Comma: hemimage 10976/10935&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;br /&gt;
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9
--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=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
-|5/8 -7/8 7/8 3/8&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
-Eigenmonzos: 2, 8/7, 6/5&lt;br /&gt;
+[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>
-&lt;br /&gt;
--limit minimax&lt;br /&gt;
+=== 13-limit  ===
-[|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;
-Eigenmonzos: 2, 10/9, 8/7&lt;br /&gt;
+Comma list: 352/351, 364/363, 540/539
-&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 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}
-Generators: 2, 3, 28/27&lt;br /&gt;
-Edos: 99, 118, 140, 316, 415, 654&lt;br /&gt;
+{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
-&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;
+Badness (Smith): 1.06 × 10<sup>-3</sup>
-&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;
+=== Tin ===
-Commas: 540/539, 1375/1372&lt;br /&gt;
+Subgroup: 2.3.5.7.11.13
-&lt;br /&gt;
-Minimax tuning&lt;br /&gt;
+Comma list: 144/143, 196/195, 729/728
-&lt;br /&gt;
-Eigenmonzos: 2, 11/9, 5/4&lt;br /&gt;
+Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}
-&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;
+{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
-Commas: 10976/10935, 441/440&lt;br /&gt;
-&lt;br /&gt;
+Badness (Smith): 1.07 × 10<sup>-3</sup>
-Minimax tuning&lt;br /&gt;
-&lt;br /&gt;
+== Turan ==
-Eigenmonzos: 2, 10/9, 11/9&lt;/body&gt;&lt;/html&gt;</pre></div>
+[[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]]