Sycamore family: Difference between revisions

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<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&gt; = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], &lt;&lt;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&amp;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.

Comma: 48828125/47775744

[[POTE tuning|POTE generator]]: ~25/24 = 63.779

Map: [&lt;1 1 2|, &lt;0 11 6|]

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 &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. [[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: [&lt;1 1 2 2|, &lt;0 11 6 15|]

Commas: 100/99, 385/384, 686/675

[[POTE tuning|POTE generator]]: ~25/24 = 64.268

Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]

EDOs: 18, 19, [[37edo|37]], [[56edo|56]]

==13-limit==

POTE generator: ~25/24 = 64.296

Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]

EDOs: 18, 19, 37, 56

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 ...||.

Commas: 225/224, 1071875/1062882

[[POTE tuning|POTE generator]]: 63.701

Map: [&lt;1 1 2 1|, &lt;0 11 6 34|]

Commas: 225/224, 385/384, 218750/216513

[[POTE tuning|POTE generator]]: 63.776

Map: [&lt;1 1 2 1 5|, &lt;0 11 6 34 -29|]

<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;Sycamore family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&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:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Sycamore"&gt;Sycamore&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Betic"&gt;Betic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;

&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;br /&gt;

The head of the sycamore family is &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11&amp;gt; = 48828125/47775744. The dual of the &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is the &lt;a class="wiki_link" href="/wedgie"&gt;wedgie&lt;/a&gt;, &amp;lt;&amp;lt;11 6 -16||, which tells us that six chromatic semitone &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt;s give 5/4 (and hence five 6/5) and eleven give 3/2. &lt;a class="wiki_link" href="/94edo"&gt;94edo&lt;/a&gt; supports sycamore, and 5\94 is reommendable as a generator. It can be described as the 19&amp;amp;94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt; of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.&lt;br /&gt;

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 &lt;a class="wiki_link" href="/Carlos%20Beta"&gt;Carlos Beta&lt;/a&gt;. 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.&lt;br /&gt;

Comma: 48828125/47775744&lt;br /&gt;

Map: [&amp;lt;1 1 2|, &amp;lt;0 11 6|]&lt;br /&gt;

The second element of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; for septimal sycamore is 875/864, the keema, and it also tempers out 686/675, the senga, and 3136/3125, hemimean. It has &amp;lt;&amp;lt;11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&amp;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. &lt;a class="wiki_link" href="/75edo"&gt;75edo&lt;/a&gt; is an excellent tuning for 7-limit sycamore, and &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt; for the 11-limit version.&lt;br /&gt;

Commas: 686/675, 875/864&lt;br /&gt;

Map: [&amp;lt;1 1 2 2|, &amp;lt;0 11 6 15|]&lt;br /&gt;

Badness: 0.0620&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Sycamore-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;

Commas: 100/99, 385/384, 686/675&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~25/24 = 64.268&lt;br /&gt;

Map: [&amp;lt;1 1 2 2 4|, &amp;lt;0 11 6 15 -10|]&lt;br /&gt;

EDOs: 18, 19, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56&lt;/a&gt;&lt;br /&gt;

Badness: 0.0559&lt;br /&gt;

Commas: 91/90, 100/99, 169/168, 385/384&lt;br /&gt;

POTE generator: ~25/24 = 64.296&lt;br /&gt;

Map: [&amp;lt;1 1 2 2 4|, &amp;lt;0 11 6 15 -10|]&lt;br /&gt;

EDOs: 18, 19, 37, 56&lt;br /&gt;

Badness: 0.0343&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Betic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Betic&lt;/h1&gt;

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;amp;94 temperament, betic, for the 7-limit. This adds 225/224 to the sycamore comma, and has &amp;lt;&amp;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 &amp;lt;&amp;lt;11 6 34 -29 ...||.&lt;br /&gt;

Commas: 225/224, 1071875/1062882&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 63.701&lt;br /&gt;

Map: [&amp;lt;1 1 2 1|, &amp;lt;0 11 6 34|]&lt;br /&gt;

EDOs: 19, 75, 94, &lt;a class="wiki_link" href="/113edo"&gt;113&lt;/a&gt;, &lt;a class="wiki_link" href="/133edo"&gt;133&lt;/a&gt;, &lt;a class="wiki_link" href="/320edo"&gt;320c&lt;/a&gt;, &lt;a class="wiki_link" href="/433edo"&gt;433cd&lt;/a&gt;&lt;br /&gt;

Commas: 225/224, 385/384, 218750/216513&lt;br /&gt;

&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 63.776&lt;br /&gt;

Map: [&amp;lt;1 1 2 1 5|, &amp;lt;0 11 6 34 -29|]&lt;br /&gt;

EDOs: 19, 75, 94, 207c&lt;/body&gt;&lt;/html&gt;</pre></div>