Tetracot family: Difference between revisions

Revision as of 18:54, 3 July 2010

IMPORTED REVISION FROM WIKISPACES

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This revision was by author genewardsmith and made on 2010-07-03 18:54:12 UTC.

The original revision id was 151464697.

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Original Wikitext content:

The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = |5 -9 4>, the minimal diesis or tetracot comma. The dual of this comma is the wedgie <<4 9 5||, which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)^4 = 20000/19683 * 3/2. We also have (10/9)^9 = (20000/19683)^2 * 5/2. From this it is evident we should flatten the generator a bit, and [[34edo]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.

Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey, <<4 9 -15 10 ...|| and 11-limit banya, <<4 9 26 10...||. Again, [[41edo]] can be used as a tuning, making the two identical, which is also the case if we turn to the {2,3,5,11} temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the (14)^(1/26) generator supplies, or even sharper yet, as for instance by the val <355 563 823 997 1230|, with a 52/355 generator. 

Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us <<4 9 -15 10 -2 ...|| for 13-limit monkey and <<4 9 26 10 -2 ...|| for 13-limit banya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. 

==Seven limit children==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.

===Monkey and Bunya===
Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie <<4 9 -15 5 -35 -60||. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the 7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34&41 temperament, if the vals in question are taken to be patent vals, meaning that n*log2(prime) rounded to the nearest integer gives the mapping. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.

Bunya, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&75 temperament. It has <<4 9 26 5 30 35|| as a wedgie, and [[41edo]] can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is (14)^(1/26) as a generator, giving just 7s and an improved value for 5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.

Original HTML content:

<html><head><title>Tetracot family</title></head><body>The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = |5 -9 4&gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &lt;&lt;4 9 5||, which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)^4 = 20000/19683 * 3/2. We also have (10/9)^9 = (20000/19683)^2 * 5/2. From this it is evident we should flatten the generator a bit, and <a class="wiki_link" href="/34edo">34edo</a> does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.<br />
<br />
Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey, &lt;&lt;4 9 -15 10 ...|| and 11-limit banya, &lt;&lt;4 9 26 10...||. Again, <a class="wiki_link" href="/41edo">41edo</a> can be used as a tuning, making the two identical, which is also the case if we turn to the {2,3,5,11} temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the (14)^(1/26) generator supplies, or even sharper yet, as for instance by the val &lt;355 563 823 997 1230|, with a 52/355 generator. <br />
<br />
Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us &lt;&lt;4 9 -15 10 -2 ...|| for 13-limit monkey and &lt;&lt;4 9 26 10 -2 ...|| for 13-limit banya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2>
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Seven limit children-Monkey and Bunya"></a><!-- ws:end:WikiTextHeadingRule:2 -->Monkey and Bunya</h3>
Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie &lt;&lt;4 9 -15 5 -35 -60||. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the 7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34&amp;41 temperament, if the vals in question are taken to be patent vals, meaning that n*log2(prime) rounded to the nearest integer gives the mapping. <a class="wiki_link" href="/41edo">41edo</a> is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.<br />
<br />
Bunya, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&amp;75 temperament. It has &lt;&lt;4 9 26 5 30 35|| as a wedgie, and <a class="wiki_link" href="/41edo">41edo</a> can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is (14)^(1/26) as a generator, giving just 7s and an improved value for 5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-03 01:21:19 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-03 18:54:12 UTC</tt>.<br>
-: The original revision id was <tt>151425695</tt>.<br>
+: The original revision id was <tt>151464697</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">The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = |5 -9 4&gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &lt;&lt;4 9 5||, which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)^4 = 20000/19683 * 3/2. We also have (10/9)^9 = (20000/19683)^2 * 5/2. From this it is evident we should flatten the generator a bit, and [[34edo]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.
+Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey, &lt;&lt;4 9 -15 10 ...|| and 11-limit banya, &lt;&lt;4 9 26 10...||. Again, [[41edo]] can be used as a tuning, making the two identical, which is also the case if we turn to the {2,3,5,11} temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the (14)^(1/26) generator supplies, or even sharper yet, as for instance by the val &lt;355 563 823 997 1230|, with a 52/355 generator.
+Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us &lt;&lt;4 9 -15 10 -2 ...|| for 13-limit monkey and &lt;&lt;4 9 26 10 -2 ...|| for 13-limit banya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp.
 ==Seven limit children==
-The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.</pre></div>
+The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.
+===Monkey and Bunya===
+Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie &lt;&lt;4 9 -15 5 -35 -60||. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the 7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34&amp;41 temperament, if the vals in question are taken to be patent vals, meaning that n*log2(prime) rounded to the nearest integer gives the mapping. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.
+Bunya, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&amp;75 temperament. It has &lt;&lt;4 9 26 5 30 35|| as a wedgie, and [[41edo]] can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is (14)^(1/26) as a generator, giving just 7s and an improved value for 5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.
+</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;Tetracot family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = |5 -9 4&amp;gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &amp;lt;&amp;lt;4 9 5||, which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)^4 = 20000/19683 * 3/2. We also have (10/9)^9 = (20000/19683)^2 * 5/2. From this it is evident we should flatten the generator a bit, and &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.&lt;br /&gt;
+&lt;br /&gt;
+Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey, &amp;lt;&amp;lt;4 9 -15 10 ...|| and 11-limit banya, &amp;lt;&amp;lt;4 9 26 10...||. Again, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; can be used as a tuning, making the two identical, which is also the case if we turn to the {2,3,5,11} temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the (14)^(1/26) generator supplies, or even sharper yet, as for instance by the val &amp;lt;355 563 823 997 1230|, with a 52/355 generator. &lt;br /&gt;
+&lt;br /&gt;
+Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us &amp;lt;&amp;lt;4 9 -15 10 -2 ...|| for 13-limit monkey and &amp;lt;&amp;lt;4 9 26 10 -2 ...|| for 13-limit banya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
-The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.&lt;/body&gt;&lt;/html&gt;</pre></div>
+The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Seven limit children-Monkey and Bunya"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Monkey and Bunya&lt;/h3&gt;
+Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie &amp;lt;&amp;lt;4 9 -15 5 -35 -60||. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the 7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34&amp;amp;41 temperament, if the vals in question are taken to be patent vals, meaning that n*log2(prime) rounded to the nearest integer gives the mapping. &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.&lt;br /&gt;
+&lt;br /&gt;
+Bunya, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&amp;amp;75 temperament. It has &amp;lt;&amp;lt;4 9 26 5 30 35|| as a wedgie, and &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is (14)^(1/26) as a generator, giving just 7s and an improved value for 5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.&lt;/body&gt;&lt;/html&gt;</pre></div>

Tetracot family: Difference between revisions

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