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Würschmidt family: Difference between revisions

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Wikispaces>xenwolf
**Imported revision 275044630 - Original comment: removed some typos**
Wikispaces>keenanpepper
**Imported revision 275321298 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-11-14 03:10:07 UTC</tt>.<br>
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-14 15:37:12 UTC</tt>.<br>
: The original revision id was <tt>275044630</tt>.<br>
: The original revision id was <tt>275321298</tt>.<br>
: The revision comment was: <tt>removed some typos</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>
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>
<h4>Original Wikitext content:</h4>
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EDOs: [[6edo|6]], [[31edo|31]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[77edo|77]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]
EDOs: [[6edo|6]], [[31edo|31]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[77edo|77]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]
Badness: 0.0203
Badness: 0.0203
=Doppelwuerschmidt=
Doppelwuerschmidt uses an approximate 25th harmonic for the generator, (and by skipping over the 5th harmonic, avoids the jagged 128/125 they create). To arrive exactly at 6/1, (775.489)^4 is best, and yields a spanking 14-note LsssLssssLssss scale. In addition to 25/16 and 6/1, it has also good approximations other [[consonance]]s, like 7/4, 7/6, 11/9, etc. - and especially 11/8.
Commas: 390625/279963 etc.
[[POTE tuning|POTE generator]]: ~25/16 = 775 c
Map:
EDOs: (if it works for wuerschmidt it works for doppelwuerschmidt)
Badness:


==11-limit==  
==11-limit==  
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EDOs: 31, 41, 46, 58, 72, 84, 89, 99e, 108, 130, 650ce, 811ce
EDOs: 31, 41, 46, 58, 72, 84, 89, 99e, 108, 130, 650ce, 811ce
Badness: 0.0211
Badness: 0.0211
&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately &lt;/span&gt;</pre></div>
&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;
=Relationships to other temperaments=
2-Wuerschmidt, the temperament with all the same commas as Wuerschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 temperament.</pre></div>
<h4>Original HTML content:</h4>
<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;Würschmidt family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Wuerschmidt"&gt;Wuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Wurschmidt"&gt;Wurschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Hemiwuerschmidt"&gt;Hemiwuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Doppelwuerschmidt"&gt;Doppelwuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
<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;Würschmidt family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Wuerschmidt"&gt;Wuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Wurschmidt"&gt;Wurschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Hemiwuerschmidt"&gt;Hemiwuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Wuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Wuerschmidt&lt;/h1&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Wuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Wuerschmidt&lt;/h1&gt;
  The &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma, and named after José Würschmidt, Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;8 1 17|| for the wedgie. This tells us the &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt; is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Wuerschmift comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the &lt;a class="wiki_link" href="/minimax%20tuning"&gt;minimax tuning&lt;/a&gt;. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt; all possibilities.&lt;br /&gt;
  The &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma, and named after José Würschmidt, Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;8 1 17|| for the wedgie. This tells us the &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt; is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Wuerschmift comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the &lt;a class="wiki_link" href="/minimax%20tuning"&gt;minimax tuning&lt;/a&gt;. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt; all possibilities.&lt;br /&gt;
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Badness: 0.0203&lt;br /&gt;
Badness: 0.0203&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Doppelwuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Doppelwuerschmidt&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Hemiwuerschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
Doppelwuerschmidt uses an approximate 25th harmonic for the generator, (and by skipping over the 5th harmonic, avoids the jagged 128/125 they create). To arrive exactly at 6/1, (775.489)^4 is best, and yields a spanking 14-note LsssLssssLssss scale. In addition to 25/16 and 6/1, it has also good approximations other &lt;a class="wiki_link" href="/consonance"&gt;consonance&lt;/a&gt;s, like 7/4, 7/6, 11/9, etc. - and especially 11/8.&lt;br /&gt;
&lt;br /&gt;
Commas: 390625/279963 etc.&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~25/16 = 775 c&lt;br /&gt;
&lt;br /&gt;
Map:&lt;br /&gt;
EDOs: (if it works for wuerschmidt it works for doppelwuerschmidt)&lt;br /&gt;
Badness:&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Doppelwuerschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;11-limit&lt;/h2&gt;
  Commas: 243/242, 441/440, 3136/3125&lt;br /&gt;
  Commas: 243/242, 441/440, 3136/3125&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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EDOs: 31, 41, 46, 58, 72, 84, 89, 99e, 108, 130, 650ce, 811ce&lt;br /&gt;
EDOs: 31, 41, 46, 58, 72, 84, 89, 99e, 108, 130, 650ce, 811ce&lt;br /&gt;
Badness: 0.0211&lt;br /&gt;
Badness: 0.0211&lt;br /&gt;
&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately &lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Relationships to other temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Relationships to other temperaments&lt;/h1&gt;
2-Wuerschmidt, the temperament with all the same commas as Wuerschmidt but a generator of twice the size, is equivalent to &lt;a class="wiki_link" href="/skwares"&gt;skwares&lt;/a&gt; as a 2.3.7.11 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
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