Sycamore family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The head of the '''sycamore family''' is [[5-limit]] sycamore, which tempers out (25/24)<sup>6</sup>/(5/4) = {{monzo| -16 -6 11 }} = 48828125/47775744, the [[sycamore comma]]. Its [[generator]] is a [[25/24 | classic chromatic semitone]], and stacking six of these gives 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] [[support]]s sycamore, and 5\94 is recommendable 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.
: 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>312491254</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">[[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 near pure 3/2 at 702.162258 [[cent]]s 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.
== Sycamore ==
[[Subgroup]]: 2.3.5


=Sycamore=
[[Comma list]]: 48828125/47775744
Comma: 48828125/47775744


[[POTE tuning|POTE generator]]: ~25/24 = 63.779
{{Mapping|legend=1| 1 1 2 | 0 11 6 }}


Map: [&lt;1 1 2|, &lt;0 11 6|]
: mapping generators: ~2, ~25/24
EDOs: [[18edo|18]], [[19edo|19]], [[56edo|56]], [[75edo|75]], [[94edo|94]], [[207edo|207c]], [[301edo|301c]]
Badness: 0.2100


==7-limit==
[[Optimal tuning]]s:
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.
* [[WE]]: ~2 = 1200.6031{{c}}, ~25/24 = 63.8108{{c}}
: [[error map]]: {{val| +0.603 +0.567 -2.242 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 63.8234{{c}}
: error map: {{val| 0.000 +0.103 -3.373 }}


Commas: 686/675, 875/864
{{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}


[[POTE tuning|POTE generator]]: ~25/24 = 63.995
[[Badness]] (Sintel): 4.925


Map: [&lt;1 1 2 2|, &lt;0 11 6 15|]
== Septimal sycamore ==
EDOs: 18, 19, 56, 75d
{{main| Sycamore and betic }}
Badness: 0.0620


==11-limit==
The second element of the [[Normal lists #Normal interval list|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 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: 100/99, 385/384, 686/675


[[POTE tuning|POTE generator]]: ~25/24 = 64.268
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]
[[Comma list]]: 686/675, 875/864
EDOs: 18, 19, [[37edo|37]], [[56edo|56]]
Badness: 0.0559


==13-limit==
{{Mapping|legend=1| 1 1 2 2 | 0 11 6 15 }}
Commas: 91/90, 100/99, 169/168, 385/384


POTE generator: ~25/24 = 64.296
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7208, ~25/24 = 64.0334
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 64.0496


Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]
{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }}
EDOs: 18, 19, 37, 56
Badness: 0.0343


=Betic=
[[Badness]] (Sintel): 1.569
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
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 63.701
Comma list: 100/99, 385/384, 686/675


Map: [&lt;1 1 2 1|, &lt;0 11 6 34|]
Mapping: {{mapping| 1 1 2 2 4 | 0 11 6 15 -10 }}
EDOs: 19, 75, 94, [[113edo|113]], [[133edo|133]], [[320edo|320c]], [[433edo|433cd]]


11-limit
Optimal tunings:
Commas: 225/224, 385/384, 218750/216513
* WE: ~2 = 1199.4126, ~25/24 = 64.2363
* CWE: ~2 = 1200.0000, ~25/24 = 64.2505


[[POTE tuning|POTE generator]]: 63.776
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}


Map: [&lt;1 1 2 1 5|, &lt;0 11 6 34 -29|]
Badness (Sintel): 1.849
EDOs: 19, 75, 94, 207c</pre></div>
 
<h4>Original HTML content:</h4>
=== 13-limit ===
<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;
Subgroup: 2.3.5.7.11.13
&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;
Comma list: 91/90, 100/99, 169/168, 385/384
&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;
Mapping: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Sycamore"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Sycamore&lt;/h1&gt;
Optimal tunings:  
Comma: 48828125/47775744&lt;br /&gt;
* WE: ~2 = 1199.6597, ~25/24 = 64.2778
&lt;br /&gt;
* CWE: ~2 = 1200.0000, ~25/24 = 64.2853
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~25/24 = 63.779&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}
Map: [&amp;lt;1 1 2|, &amp;lt;0 11 6|]&lt;br /&gt;
 
EDOs: &lt;a class="wiki_link" href="/18edo"&gt;18&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56&lt;/a&gt;, &lt;a class="wiki_link" href="/75edo"&gt;75&lt;/a&gt;, &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/207edo"&gt;207c&lt;/a&gt;, &lt;a class="wiki_link" href="/301edo"&gt;301c&lt;/a&gt;&lt;br /&gt;
Badness (Sintel): 1.417
Badness: 0.2100&lt;br /&gt;
 
&lt;br /&gt;
== Betic ==
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Sycamore-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
{{main| Sycamore and betic }}
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;
 
&lt;br /&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 (e.g. 94edo) 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. 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.  
Commas: 686/675, 875/864&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~25/24 = 63.995&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 225/224, 1071875/1062882
Map: [&amp;lt;1 1 2 2|, &amp;lt;0 11 6 15|]&lt;br /&gt;
 
EDOs: 18, 19, 56, 75d&lt;br /&gt;
{{Mapping|legend=1| 1 1 2 1 | 0 11 6 34 }}
Badness: 0.0620&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&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;
* [[WE]]: ~2 = 1200.6891, ~25/24 = 63.7773
Commas: 100/99, 385/384, 686/675&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 63.7683
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~25/24 = 64.268&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 56d, 75, 94, 113, 320cc, 433ccd }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 2 4|, &amp;lt;0 11 6 15 -10|]&lt;br /&gt;
[[Badness]] (Sintel): 1.765
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;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Sycamore-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
 
Commas: 91/90, 100/99, 169/168, 385/384&lt;br /&gt;
Comma list: 225/224, 385/384, 218750/216513
&lt;br /&gt;
 
POTE generator: ~25/24 = 64.296&lt;br /&gt;
Mapping: {{mapping| 1 1 2 1 5 | 0 11 6 34 -29 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 2 4|, &amp;lt;0 11 6 15 -10|]&lt;br /&gt;
Optimal tunings:
EDOs: 18, 19, 37, 56&lt;br /&gt;
* WE: ~2 = 1200.4466, ~25/24 = 63.7993
Badness: 0.0343&lt;br /&gt;
* CWE: ~2 = 1200.0000, ~25/24 = 63.7796
&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;
{{Optimal ET sequence|legend=0| 19, 75, 94, 207c }}
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;
 
&lt;br /&gt;
Badness (Sintel): 1.880
Commas: 225/224, 1071875/1062882&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 63.701&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 1|, &amp;lt;0 11 6 34|]&lt;br /&gt;
Comma list: 225/224, 325/324, 385/384, 1875/1859
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;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }}
11-limit&lt;br /&gt;
 
Commas: 225/224, 385/384, 218750/216513&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1200.3946, ~25/24 = 63.7867
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 63.776&lt;br /&gt;
* CWE: ~2 = 1200.0000, ~25/24 = 63.7702
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 1 5|, &amp;lt;0 11 6 34 -29|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }}
EDOs: 19, 75, 94, 207c&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Badness (Sintel): 1.342
 
[[Category:Temperament families]]
[[Category:Sycamore family ]] <!-- main article -->
[[Category:Sycamore| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 22:49, 22 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The head of the sycamore family is 5-limit sycamore, which tempers out (25/24)6/(5/4) = [-16 -6 11⟩ = 48828125/47775744, the sycamore comma. Its generator is a classic chromatic semitone, and stacking six of these gives 5/4 (and hence five 6/5) and eleven give 3/2. 94edo supports sycamore, and 5\94 is recommendable 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 near pure 3/2 at 702.162258 cents 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

Subgroup: 2.3.5

Comma list: 48828125/47775744

Mapping: [⟨1 1 2], ⟨0 11 6]]

mapping generators: ~2, ~25/24

Optimal tunings:

  • WE: ~2 = 1200.6031 ¢, ~25/24 = 63.8108 ¢
error map: ⟨+0.603 +0.567 -2.242]
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 63.8234 ¢
error map: ⟨0.000 +0.103 -3.373]

Optimal ET sequence: 18, 19, 56, 75, 94, 207c, 301c

Badness (Sintel): 4.925

Septimal sycamore

Main article: Sycamore and betic

The second element of the 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 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.

Subgroup: 2.3.5.7

Comma list: 686/675, 875/864

Mapping: [⟨1 1 2 2], ⟨0 11 6 15]]

Optimal tunings:

  • WE: ~2 = 1200.7208, ~25/24 = 64.0334
  • CWE: ~2 = 1200.0000, ~25/24 = 64.0496

Optimal ET sequence: 18, 19, 56, 75d

Badness (Sintel): 1.569

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 686/675

Mapping: [⟨1 1 2 2 4], ⟨0 11 6 15 -10]]

Optimal tunings:

  • WE: ~2 = 1199.4126, ~25/24 = 64.2363
  • CWE: ~2 = 1200.0000, ~25/24 = 64.2505

Optimal ET sequence: 18, 19, 37, 56

Badness (Sintel): 1.849

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 169/168, 385/384

Mapping: [⟨1 1 2 2 4 3], ⟨0 11 6 15 -10 13]]

Optimal tunings:

  • WE: ~2 = 1199.6597, ~25/24 = 64.2778
  • CWE: ~2 = 1200.0000, ~25/24 = 64.2853

Optimal ET sequence: 18, 19, 37, 56

Badness (Sintel): 1.417

Betic

Main article: Sycamore and 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 (e.g. 94edo) 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. 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.

Subgroup: 2.3.5.7

Comma list: 225/224, 1071875/1062882

Mapping: [⟨1 1 2 1], ⟨0 11 6 34]]

Optimal tunings:

  • WE: ~2 = 1200.6891, ~25/24 = 63.7773
  • CWE: ~2 = 1200.0000, ~25/24 = 63.7683

Optimal ET sequence: 19, 56d, 75, 94, 113, 320cc, 433ccd

Badness (Sintel): 1.765

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 218750/216513

Mapping: [⟨1 1 2 1 5], ⟨0 11 6 34 -29]]

Optimal tunings:

  • WE: ~2 = 1200.4466, ~25/24 = 63.7993
  • CWE: ~2 = 1200.0000, ~25/24 = 63.7796

Optimal ET sequence: 19, 75, 94, 207c

Badness (Sintel): 1.880

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1875/1859

Mapping: [⟨1 1 2 1 5 2], ⟨0 11 6 34 -29 32]]

Optimal tunings:

  • WE: ~2 = 1200.3946, ~25/24 = 63.7867
  • CWE: ~2 = 1200.0000, ~25/24 = 63.7702

Optimal ET sequence: 19, 75, 94, 113, 207c

Badness (Sintel): 1.342

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