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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:genewardsmith|genewardsmith]] and made on <tt> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-19 13:17:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>312491254</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">The head of the sycamore family is [[5-limit]] sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11> = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], <<11 6 -16||, which tells us that six chromatic semitone [[generator]]s give 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. [[MOS]] of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous. | <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 head of the sycamore family is [[5-limit]] sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11> = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], <<11 6 -16||, which tells us that six chromatic semitone [[generator]]s give 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. [[MOS]] of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous. | |||
Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as [[Carlos Beta]]. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves. | Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as [[Carlos Beta]]. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves. | ||
[[POTE tuning|POTE generator]]: 63.779 | =Sycamore= | ||
Comma: 48828125/47775744 | |||
[[POTE tuning|POTE generator]]: ~25/24 = 63.779 | |||
Map: [<1 1 2|, <0 11 6|] | Map: [<1 1 2|, <0 11 6|] | ||
EDOs: [[18edo|18]], [[19edo|19]], [[56edo|56]], [[75edo|75]], [[94edo|94]], [[207edo|207c]], [[301edo|301c]] | EDOs: [[18edo|18]], [[19edo|19]], [[56edo|56]], [[75edo|75]], [[94edo|94]], [[207edo|207c]], [[301edo|301c]] | ||
Badness: 0.2100 | |||
== | ==7-limit== | ||
The second element of the [[Normal lists|normal comma list]] for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has <<11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&56 temperament. This may also be used as the name for the temperament obtained by adding 100/99 to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. [[75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version. | The second element of the [[Normal lists|normal comma list]] for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has <<11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&56 temperament. This may also be used as the name for the temperament obtained by adding 100/99 to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. [[75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version. | ||
Commas: 686/675, 875/864 | Commas: 686/675, 875/864 | ||
[[POTE tuning|POTE generator]]: 63.995 | [[POTE tuning|POTE generator]]: ~25/24 = 63.995 | ||
Map: [<1 1 2 2|, <0 11 6 15|] | Map: [<1 1 2 2|, <0 11 6 15|] | ||
EDOs: 18, 19, 56, 75d | EDOs: 18, 19, 56, 75d | ||
Badness: 0.0620 | |||
==11-limit== | |||
11-limit | |||
Commas: 100/99, 385/384, 686/675 | Commas: 100/99, 385/384, 686/675 | ||
[[POTE tuning|POTE generator]]: 64.268 | [[POTE tuning|POTE generator]]: ~25/24 = 64.268 | ||
Map: [<1 1 2 2 4|, <0 11 6 15 -10|] | Map: [<1 1 2 2 4|, <0 11 6 15 -10|] | ||
EDOs: 18, 19, [[37edo|37]], [[56edo|56]] | EDOs: 18, 19, [[37edo|37]], [[56edo|56]] | ||
Badness: 0.0559 | |||
==13-limit== | |||
Commas: 91/90, 100/99, 169/168, 385/384 | |||
POTE generator: ~25/24 = 64.296 | |||
Map: [<1 1 2 2 4|, <0 11 6 15 -10|] | |||
EDOs: 18, 19, 37, 56 | |||
Badness: 0.0343 | |||
=Betic= | |||
Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (eg 94et) or exactly those of Carlos Beta, we get the 19&94 temperament, betic, for the 7-limit. This adds 225/224 to the sycamore comma, and has <<11 6 34 -16 23 62|| as a wedgie. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of 385/384 or 540/539 to the list of commas, which means it supports both 7 and 11-limit marvel. The wedgie starts <<11 6 34 -29 ...||. | Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (eg 94et) or exactly those of Carlos Beta, we get the 19&94 temperament, betic, for the 7-limit. This adds 225/224 to the sycamore comma, and has <<11 6 34 -16 23 62|| as a wedgie. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of 385/384 or 540/539 to the list of commas, which means it supports both 7 and 11-limit marvel. The wedgie starts <<11 6 34 -29 ...||. | ||
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EDOs: 19, 75, 94, 207c</pre></div> | EDOs: 19, 75, 94, 207c</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"><html><head><title>Sycamore family</title></head><body>The head of the sycamore family is <a class="wiki_link" href="/5-limit">5-limit</a> sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11&gt; = 48828125/47775744. The dual of the <a class="wiki_link" href="/monzo">monzo</a> is the <a class="wiki_link" href="/wedgie">wedgie</a>, &lt;&lt;11 6 -16||, which tells us that six chromatic semitone <a class="wiki_link" href="/generator">generator</a>s give 5/4 (and hence five 6/5) and eleven give 3/2. <a class="wiki_link" href="/94edo">94edo</a> supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&amp;94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. <a class="wiki_link" href="/MOS">MOS</a> of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.<br /> | <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>Sycamore family</title></head><body><!-- ws:start:WikiTextTocRule:10:&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:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Sycamore">Sycamore</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Betic">Betic</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
<!-- ws:end:WikiTextTocRule:16 --><br /> | |||
The head of the sycamore family is <a class="wiki_link" href="/5-limit">5-limit</a> sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11&gt; = 48828125/47775744. The dual of the <a class="wiki_link" href="/monzo">monzo</a> is the <a class="wiki_link" href="/wedgie">wedgie</a>, &lt;&lt;11 6 -16||, which tells us that six chromatic semitone <a class="wiki_link" href="/generator">generator</a>s give 5/4 (and hence five 6/5) and eleven give 3/2. <a class="wiki_link" href="/94edo">94edo</a> supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&amp;94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. <a class="wiki_link" href="/MOS">MOS</a> of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.<br /> | |||
<br /> | <br /> | ||
Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as <a class="wiki_link" href="/Carlos%20Beta">Carlos Beta</a>. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves.<br /> | Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as <a class="wiki_link" href="/Carlos%20Beta">Carlos Beta</a>. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves.<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 63.779<br /> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Sycamore"></a><!-- ws:end:WikiTextHeadingRule:0 -->Sycamore</h1> | ||
Comma: 48828125/47775744<br /> | |||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/24 = 63.779<br /> | |||
<br /> | <br /> | ||
Map: [&lt;1 1 2|, &lt;0 11 6|]<br /> | Map: [&lt;1 1 2|, &lt;0 11 6|]<br /> | ||
EDOs: <a class="wiki_link" href="/18edo">18</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/56edo">56</a>, <a class="wiki_link" href="/75edo">75</a>, <a class="wiki_link" href="/94edo">94</a>, <a class="wiki_link" href="/207edo">207c</a>, <a class="wiki_link" href="/301edo">301c</a><br /> | EDOs: <a class="wiki_link" href="/18edo">18</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/56edo">56</a>, <a class="wiki_link" href="/75edo">75</a>, <a class="wiki_link" href="/94edo">94</a>, <a class="wiki_link" href="/207edo">207c</a>, <a class="wiki_link" href="/301edo">301c</a><br /> | ||
Badness: 0.2100<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Sycamore-7-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit</h2> | ||
The second element of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has &lt;&lt;11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&amp;56 temperament. This may also be used as the name for the temperament obtained by adding 100/99 to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. <a class="wiki_link" href="/75edo">75edo</a> is an excellent tuning for 7-limit sycamore, and <a class="wiki_link" href="/56edo">56edo</a> for the 11-limit version.<br /> | The second element of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has &lt;&lt;11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&amp;56 temperament. This may also be used as the name for the temperament obtained by adding 100/99 to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. <a class="wiki_link" href="/75edo">75edo</a> is an excellent tuning for 7-limit sycamore, and <a class="wiki_link" href="/56edo">56edo</a> for the 11-limit version.<br /> | ||
<br /> | <br /> | ||
Commas: 686/675, 875/864<br /> | Commas: 686/675, 875/864<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 63.995<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/24 = 63.995<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 1 2 2|, &lt;0 11 6 15|]<br /> | Map: [&lt;1 1 2 2|, &lt;0 11 6 15|]<br /> | ||
EDOs: 18, 19, 56, 75d<br /> | EDOs: 18, 19, 56, 75d<br /> | ||
Badness: 0.0620<br /> | |||
<br /> | <br /> | ||
< | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Sycamore-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> | ||
11-limit< | |||
Commas: 100/99, 385/384, 686/675<br /> | Commas: 100/99, 385/384, 686/675<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 64.268<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/24 = 64.268<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]<br /> | Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]<br /> | ||
EDOs: 18, 19, <a class="wiki_link" href="/37edo">37</a>, <a class="wiki_link" href="/56edo">56</a><br /> | EDOs: 18, 19, <a class="wiki_link" href="/37edo">37</a>, <a class="wiki_link" href="/56edo">56</a><br /> | ||
Badness: 0.0559<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Sycamore-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2> | |||
Commas: 91/90, 100/99, 169/168, 385/384<br /> | |||
<br /> | |||
POTE generator: ~25/24 = 64.296<br /> | |||
<br /> | |||
Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]<br /> | |||
EDOs: 18, 19, 37, 56<br /> | |||
Badness: 0.0343<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Betic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Betic</h1> | ||
Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (eg 94et) or exactly those of Carlos Beta, we get the 19&amp;94 temperament, betic, for the 7-limit. This adds 225/224 to the sycamore comma, and has &lt;&lt;11 6 34 -16 23 62|| as a wedgie. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of 385/384 or 540/539 to the list of commas, which means it supports both 7 and 11-limit marvel. The wedgie starts &lt;&lt;11 6 34 -29 ...||.<br /> | |||
<br /> | <br /> | ||
Commas: 225/224, 1071875/1062882<br /> | Commas: 225/224, 1071875/1062882<br /> | ||
Revision as of 13:17, 19 March 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-03-19 13:17:19 UTC.
- The original revision id was 312491254.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[toc|flat]] The head of the sycamore family is [[5-limit]] sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11> = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], <<11 6 -16||, which tells us that six chromatic semitone [[generator]]s give 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. [[MOS]] of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous. Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as [[Carlos Beta]]. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves. =Sycamore= Comma: 48828125/47775744 [[POTE tuning|POTE generator]]: ~25/24 = 63.779 Map: [<1 1 2|, <0 11 6|] EDOs: [[18edo|18]], [[19edo|19]], [[56edo|56]], [[75edo|75]], [[94edo|94]], [[207edo|207c]], [[301edo|301c]] Badness: 0.2100 ==7-limit== The second element of the [[Normal lists|normal comma list]] for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has <<11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&56 temperament. This may also be used as the name for the temperament obtained by adding 100/99 to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. [[75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version. Commas: 686/675, 875/864 [[POTE tuning|POTE generator]]: ~25/24 = 63.995 Map: [<1 1 2 2|, <0 11 6 15|] EDOs: 18, 19, 56, 75d Badness: 0.0620 ==11-limit== Commas: 100/99, 385/384, 686/675 [[POTE tuning|POTE generator]]: ~25/24 = 64.268 Map: [<1 1 2 2 4|, <0 11 6 15 -10|] EDOs: 18, 19, [[37edo|37]], [[56edo|56]] Badness: 0.0559 ==13-limit== Commas: 91/90, 100/99, 169/168, 385/384 POTE generator: ~25/24 = 64.296 Map: [<1 1 2 2 4|, <0 11 6 15 -10|] EDOs: 18, 19, 37, 56 Badness: 0.0343 =Betic= Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (eg 94et) or exactly those of Carlos Beta, we get the 19&94 temperament, betic, for the 7-limit. This adds 225/224 to the sycamore comma, and has <<11 6 34 -16 23 62|| as a wedgie. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of 385/384 or 540/539 to the list of commas, which means it supports both 7 and 11-limit marvel. The wedgie starts <<11 6 34 -29 ...||. Commas: 225/224, 1071875/1062882 [[POTE tuning|POTE generator]]: 63.701 Map: [<1 1 2 1|, <0 11 6 34|] EDOs: 19, 75, 94, [[113edo|113]], [[133edo|133]], [[320edo|320c]], [[433edo|433cd]] 11-limit Commas: 225/224, 385/384, 218750/216513 [[POTE tuning|POTE generator]]: 63.776 Map: [<1 1 2 1 5|, <0 11 6 34 -29|] EDOs: 19, 75, 94, 207c
Original HTML content:
<html><head><title>Sycamore family</title></head><body><!-- ws:start:WikiTextTocRule:10:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Sycamore">Sycamore</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Betic">Betic</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> <!-- ws:end:WikiTextTocRule:16 --><br /> The head of the sycamore family is <a class="wiki_link" href="/5-limit">5-limit</a> sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11> = 48828125/47775744. The dual of the <a class="wiki_link" href="/monzo">monzo</a> is the <a class="wiki_link" href="/wedgie">wedgie</a>, <<11 6 -16||, which tells us that six chromatic semitone <a class="wiki_link" href="/generator">generator</a>s give 5/4 (and hence five 6/5) and eleven give 3/2. <a class="wiki_link" href="/94edo">94edo</a> supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. <a class="wiki_link" href="/MOS">MOS</a> of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.<br /> <br /> Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as <a class="wiki_link" href="/Carlos%20Beta">Carlos Beta</a>. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Sycamore"></a><!-- ws:end:WikiTextHeadingRule:0 -->Sycamore</h1> Comma: 48828125/47775744<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/24 = 63.779<br /> <br /> Map: [<1 1 2|, <0 11 6|]<br /> EDOs: <a class="wiki_link" href="/18edo">18</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/56edo">56</a>, <a class="wiki_link" href="/75edo">75</a>, <a class="wiki_link" href="/94edo">94</a>, <a class="wiki_link" href="/207edo">207c</a>, <a class="wiki_link" href="/301edo">301c</a><br /> Badness: 0.2100<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Sycamore-7-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit</h2> The second element of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has <<11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&56 temperament. This may also be used as the name for the temperament obtained by adding 100/99 to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. <a class="wiki_link" href="/75edo">75edo</a> is an excellent tuning for 7-limit sycamore, and <a class="wiki_link" href="/56edo">56edo</a> for the 11-limit version.<br /> <br /> Commas: 686/675, 875/864<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/24 = 63.995<br /> <br /> Map: [<1 1 2 2|, <0 11 6 15|]<br /> EDOs: 18, 19, 56, 75d<br /> Badness: 0.0620<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Sycamore-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> Commas: 100/99, 385/384, 686/675<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/24 = 64.268<br /> <br /> Map: [<1 1 2 2 4|, <0 11 6 15 -10|]<br /> EDOs: 18, 19, <a class="wiki_link" href="/37edo">37</a>, <a class="wiki_link" href="/56edo">56</a><br /> Badness: 0.0559<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Sycamore-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2> Commas: 91/90, 100/99, 169/168, 385/384<br /> <br /> POTE generator: ~25/24 = 64.296<br /> <br /> Map: [<1 1 2 2 4|, <0 11 6 15 -10|]<br /> EDOs: 18, 19, 37, 56<br /> Badness: 0.0343<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Betic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Betic</h1> Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (eg 94et) or exactly those of Carlos Beta, we get the 19&94 temperament, betic, for the 7-limit. This adds 225/224 to the sycamore comma, and has <<11 6 34 -16 23 62|| as a wedgie. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of 385/384 or 540/539 to the list of commas, which means it supports both 7 and 11-limit marvel. The wedgie starts <<11 6 34 -29 ...||.<br /> <br /> Commas: 225/224, 1071875/1062882<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 63.701<br /> <br /> Map: [<1 1 2 1|, <0 11 6 34|]<br /> EDOs: 19, 75, 94, <a class="wiki_link" href="/113edo">113</a>, <a class="wiki_link" href="/133edo">133</a>, <a class="wiki_link" href="/320edo">320c</a>, <a class="wiki_link" href="/433edo">433cd</a><br /> <br /> 11-limit<br /> Commas: 225/224, 385/384, 218750/216513<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 63.776<br /> <br /> Map: [<1 1 2 1 5|, <0 11 6 34 -29|]<br /> EDOs: 19, 75, 94, 207c</body></html>