Tetracot family: Difference between revisions
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Wikispaces>keenanpepper **Imported revision 288794657 - 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: | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-29 22:06:30 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>288794657</tt>.<br> | ||
: The revision comment was: <tt></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> | ||
<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">[[toc|flat]] | <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">[[toc|flat]] | ||
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. | 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. | ||
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EDOs: 14c, 27, 34, 75, 109, 470b, 579b | EDOs: 14c, 27, 34, 75, 109, 470b, 579b | ||
==Seven limit children== | ==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. | 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 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 val|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. | 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 val|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. | 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. | ||
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 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 bunya. 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. | 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 bunya. 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. | ||
=Monkey= | =Monkey= | ||
Commas: 5120/5103, 875/864 | Commas: 5120/5103, 875/864 | ||
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Badness: 0.0734 | Badness: 0.0734 | ||
==11-limit== | ==11-limit== | ||
Commas: 243/242, 385/384, 100/99 | Commas: 243/242, 385/384, 100/99 | ||
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Badness: 0.0284 | Badness: 0.0284 | ||
=Bunya= | =Bunya= | ||
Commas: 225/224, 15625/15309 | Commas: 225/224, 15625/15309 | ||
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Badness: 0.0629 | Badness: 0.0629 | ||
==11-limit== | ==11-limit== | ||
Commas: 100/99, 225/224, 1344/1331 | Commas: 100/99, 225/224, 1344/1331 | ||
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Badness: 0.0313 | Badness: 0.0313 | ||
==13-limit== | ==13-limit== | ||
Commas: 100/99, 144/143, 225/224, 243/242 | Commas: 100/99, 144/143, 225/224, 243/242 | ||
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Badness: 0.0249 | Badness: 0.0249 | ||
=Octacot= | =Octacot= | ||
Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It has wedgie <<8 18 11 10 -5 -25|| and may also be described as 41&68. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11/150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them. | Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It has wedgie <<8 18 11 10 -5 -25|| and may also be described as 41&68. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11/150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them. | ||
Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving <<8 18 11 20 -4 ...|| as the octave part of the wedgie. Generators of 3/41, 8/109 and 11/150 (88 cents) are all good choices for the 7, 11 and 13 limits. | Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving <<8 18 11 20 -4 ...|| as the octave part of the wedgie. Generators of 3/41, 8/109 and 11/150 (88 cents) are all good choices for the 7, 11 and 13 limits. | ||
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EDOs: 14c, 27, 41, 68, 109 | EDOs: 14c, 27, 41, 68, 109 | ||
==11-limit== | ==11-limit== | ||
Commas: 100/99, 243/242, 245/242 | Commas: 100/99, 243/242, 245/242 | ||
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EDOs: 27e, 41, 109e, 150e, 191e | EDOs: 27e, 41, 109e, 150e, 191e | ||
==13-limit== | See also: [[Chords of octacot]] | ||
==13-limit== | |||
Commas: 100/99, 144/143, 196/195, 243/242 | Commas: 100/99, 144/143, 196/195, 243/242 | ||
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<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"><html><head><title>Tetracot family</title></head><body><!-- ws:start:WikiTextTocRule:22:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Octacot">Octacot</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <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>Tetracot family</title></head><body><!-- ws:start:WikiTextTocRule:22:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Octacot">Octacot</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | ||
<!-- ws:end:WikiTextTocRule:34 --> | <!-- ws:end:WikiTextTocRule:34 -->The parent of the <strong>tetracot family</strong> is <strong>tetracot</strong>, the 5-limit temperament <a class="wiki_link" href="/tempering%20out">tempering out</a> 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 /> | ||
The parent of the <strong>tetracot family</strong> is <strong>tetracot</strong>, the 5-limit temperament <a class="wiki_link" href="/tempering%20out">tempering out</a> 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 /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 176.160<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 176.160<br /> | ||
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<!-- 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> | <!-- 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 /> | 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 /> | <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> | <!-- 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 <a class="wiki_link" href="/Patent%20val">patent vals</a>, 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 /> | 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 <a class="wiki_link" href="/Patent%20val">patent vals</a>, 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 /> | <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.<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.<br /> | ||
<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 /> | 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 /> | <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 bunya. 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 /> | 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 bunya. 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Monkey"></a><!-- ws:end:WikiTextHeadingRule:4 -->Monkey</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Monkey"></a><!-- ws:end:WikiTextHeadingRule:4 -->Monkey</h1> | ||
Commas: 5120/5103, 875/864<br /> | Commas: 5120/5103, 875/864<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.659<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.659<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Monkey-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Monkey-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> | ||
Commas: 243/242, 385/384, 100/99<br /> | Commas: 243/242, 385/384, 100/99<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.570<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.570<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Bunya"></a><!-- ws:end:WikiTextHeadingRule:10 -->Bunya</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Bunya"></a><!-- ws:end:WikiTextHeadingRule:10 -->Bunya</h1> | ||
Commas: 225/224, 15625/15309<br /> | Commas: 225/224, 15625/15309<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.741<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.741<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Bunya-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Bunya-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> | ||
Commas: 100/99, 225/224, 1344/1331<br /> | Commas: 100/99, 225/224, 1344/1331<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.777<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.777<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Bunya-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Bunya-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> | ||
Commas: 100/99, 144/143, 225/224, 243/242<br /> | Commas: 100/99, 144/143, 225/224, 243/242<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.886<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.886<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule:16 -->Octacot</h1> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule:16 -->Octacot</h1> | ||
Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It has wedgie &lt;&lt;8 18 11 10 -5 -25|| and may also be described as 41&amp;68. <a class="wiki_link" href="/68edo">68edo</a> or <a class="wiki_link" href="/109edo">109edo</a> can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is <a class="wiki_link" href="/150edo">150edo</a>, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the <a class="wiki_link" href="/88cET">88cET</a> non-octave temperament, which like <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a> arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them. <br /> | Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It has wedgie &lt;&lt;8 18 11 10 -5 -25|| and may also be described as 41&amp;68. <a class="wiki_link" href="/68edo">68edo</a> or <a class="wiki_link" href="/109edo">109edo</a> can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is <a class="wiki_link" href="/150edo">150edo</a>, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the <a class="wiki_link" href="/88cET">88cET</a> non-octave temperament, which like <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a> arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them.<br /> | ||
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Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving &lt;&lt;8 18 11 20 -4 ...|| as the octave part of the wedgie. Generators of 3/41, 8/109 and 11/150 (88 cents) are all good choices for the 7, 11 and 13 limits.<br /> | Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving &lt;&lt;8 18 11 20 -4 ...|| as the octave part of the wedgie. Generators of 3/41, 8/109 and 11/150 (88 cents) are all good choices for the 7, 11 and 13 limits.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Octacot-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Octacot-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | ||
Commas: 100/99, 243/242, 245/242<br /> | Commas: 100/99, 243/242, 245/242<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 87.975<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 87.975<br /> | ||
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Map: [&lt;1 1 1 2 2|, &lt;0 8 18 11 20|]<br /> | Map: [&lt;1 1 1 2 2|, &lt;0 8 18 11 20|]<br /> | ||
EDOs: 27e, 41, 109e, 150e, 191e<br /> | EDOs: 27e, 41, 109e, 150e, 191e<br /> | ||
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See also: <a class="wiki_link" href="/Chords%20of%20octacot">Chords of octacot</a><br /> | |||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Octacot-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Octacot-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> | ||
Commas: 100/99, 144/143, 196/195, 243/242<br /> | Commas: 100/99, 144/143, 196/195, 243/242<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~22/21 = 88.106<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~22/21 = 88.106<br /> | ||