Tetracot family: Difference between revisions

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<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:~~keenanpepper~~|~~keenanpepper~~]] and made on <tt>~~2011~~-12-~~29 22~~:06:30 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-04 19:01:00 UTC</tt>.<br>

: The original revision id was <tt>~~288794657~~</tt>.<br>

: The original revision id was <tt>289647435</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>

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EDOs: 34d, 41, 75e, 116ef

Badness: 0.0249

=Modus=

Commas: 64/63, 4375/4374

POTE generator: ~10/9 = 177.203

Map: [<1 1 1 4|, <0 4 9 -8|]

EDOs: 7, 27, 61d, 88bcd

Badness: 0.0682

==11-limit==

Commas: 64/63, 100/99, 243/242

POTE generator: ~10/9 = 177.053

Map: [<1 1 1 4 2|, <0 4 9 -8 10|]

EDOs: 7, 20ce, 27e, 34d, 61de

Badness: 0.0351

==13-limit==

Commas: 64/63, 78/77, 100/99, 144/143

POTE generator: ~10/9 = 176.953

Map: [<1 1 1 4 2 4|, <0 4 9 -8 10 -2|]

EDOs: 7, 27e, 34d, 61de

Badness: 0.0238

=Octacot=

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Badness: 0.0233</pre></div>

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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tetracot family</title></head><body> | <a href="#Monkey">Monkey</a> | <a href="#Bunya">Bunya</a> | <a href="#Octacot">Octacot</a>

<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> | <a href="#Monkey">Monkey</a> | <a href="#Bunya">Bunya</a> | <a href="#Modus">Modus</a> | <a href="#Octacot">Octacot</a>

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 />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 176.160<br />

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Badness: 0.0249<br />

<br />

<h1 id="toc8"><a name="Octacot"></a>Octacot</h1>

<h1 id="toc8"><a name="Modus"></a>Modus</h1>

Commas: 64/63, 4375/4374<br />

<br />

POTE generator: ~10/9 = 177.203<br />

<br />

Map: [&lt;1 1 1 4|, &lt;0 4 9 -8|]<br />

EDOs: 7, 27, 61d, 88bcd<br />

Badness: 0.0682<br />

<br />

<h2 id="toc9"><a name="Modus-11-limit"></a>11-limit</h2>

Commas: 64/63, 100/99, 243/242<br />

<br />

POTE generator: ~10/9 = 177.053<br />

<br />

Map: [&lt;1 1 1 4 2|, &lt;0 4 9 -8 10|]<br />

EDOs: 7, 20ce, 27e, 34d, 61de<br />

Badness: 0.0351<br />

<br />

<h2 id="toc10"><a name="Modus-13-limit"></a>13-limit</h2>

Commas: 64/63, 78/77, 100/99, 144/143<br />

<br />

POTE generator: ~10/9 = 176.953<br />

<br />

Map: [&lt;1 1 1 4 2 4|, &lt;0 4 9 -8 10 -2|]<br />

EDOs: 7, 27e, 34d, 61de<br />

Badness: 0.0238<br />

<br />

<h1 id="toc11"><a name="Octacot"></a>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 />

<br />

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EDOs: 14c, 27, 41, 68, 109<br />

<br />

<h2 id="~~toc9~~"><a name="Octacot-11-limit"></a>11-limit</h2>

<h2 id="toc12"><a name="Octacot-11-limit"></a>11-limit</h2>

Commas: 100/99, 243/242, 245/242<br />

<br />

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See also: <a class="wiki_link" href="/Chords%20of%20octacot">Chords of octacot</a><br />

<br />

<h2 id="~~toc10~~"><a name="Octacot-13-limit"></a>13-limit</h2>

<h2 id="toc13"><a name="Octacot-13-limit"></a>13-limit</h2>

Commas: 100/99, 144/143, 196/195, 243/242<br />

<br />

@@ 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:keenanpepper|keenanpepper]] and made on <tt>2011-12-29 22:06:30 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-04 19:01:00 UTC</tt>.<br>
-: The original revision id was <tt>288794657</tt>.<br>
+: The original revision id was <tt>289647435</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>
@@ Line 79: / Line 79: @@
 EDOs: 34d, 41, 75e, 116ef
 Badness: 0.0249
+=Modus=
+Commas: 64/63, 4375/4374
+POTE generator: ~10/9 = 177.203
+Map: [&lt;1 1 1 4|, &lt;0 4 9 -8|]
+EDOs: 7, 27, 61d, 88bcd
+Badness: 0.0682
+==11-limit==
+Commas: 64/63, 100/99, 243/242
+POTE generator: ~10/9 = 177.053
+Map: [&lt;1 1 1 4 2|, &lt;0 4 9 -8 10|]
+EDOs: 7, 20ce, 27e, 34d, 61de
+Badness: 0.0351
+==13-limit==
+Commas: 64/63, 78/77, 100/99, 144/143
+POTE generator: ~10/9 = 176.953
+Map: [&lt;1 1 1 4 2 4|, &lt;0 4 9 -8 10 -2|]
+EDOs: 7, 27e, 34d, 61de
+Badness: 0.0238
 =Octacot=
@@ Line 111: / Line 138: @@
 Badness: 0.0233</pre></div>
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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;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;&lt;!-- ws:start:WikiTextTocRule:22:&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Monkey"&gt;Monkey&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Bunya"&gt;Bunya&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Octacot"&gt;Octacot&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;
+<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;&lt;!-- ws:start:WikiTextTocRule:28:&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:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Monkey"&gt;Monkey&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Bunya"&gt;Bunya&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#Modus"&gt;Modus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt; | &lt;a href="#Octacot"&gt;Octacot&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:34 --&gt;The parent of the &lt;strong&gt;tetracot family&lt;/strong&gt; is &lt;strong&gt;tetracot&lt;/strong&gt;, the 5-limit temperament &lt;a class="wiki_link" href="/tempering%20out"&gt;tempering out&lt;/a&gt; 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;!-- ws:end:WikiTextTocRule:43 --&gt;The parent of the &lt;strong&gt;tetracot family&lt;/strong&gt; is &lt;strong&gt;tetracot&lt;/strong&gt;, the 5-limit temperament &lt;a class="wiki_link" href="/tempering%20out"&gt;tempering out&lt;/a&gt; 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;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 176.160&lt;br /&gt;
@@ Line 185: / Line 212: @@
 Badness: 0.0249&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Octacot"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Octacot&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Modus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Modus&lt;/h1&gt;
+Commas: 64/63, 4375/4374&lt;br /&gt;
+&lt;br /&gt;
+POTE generator: ~10/9 = 177.203&lt;br /&gt;
+&lt;br /&gt;
+Map: [&amp;lt;1 1 1 4|, &amp;lt;0 4 9 -8|]&lt;br /&gt;
+EDOs: 7, 27, 61d, 88bcd&lt;br /&gt;
+Badness: 0.0682&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Modus-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
+Commas: 64/63, 100/99, 243/242&lt;br /&gt;
+&lt;br /&gt;
+POTE generator: ~10/9 = 177.053&lt;br /&gt;
+&lt;br /&gt;
+Map: [&amp;lt;1 1 1 4 2|, &amp;lt;0 4 9 -8 10|]&lt;br /&gt;
+EDOs: 7, 20ce, 27e, 34d, 61de&lt;br /&gt;
+Badness: 0.0351&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Modus-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
+Commas: 64/63, 78/77, 100/99, 144/143&lt;br /&gt;
+&lt;br /&gt;
+POTE generator: ~10/9 = 176.953&lt;br /&gt;
+&lt;br /&gt;
+Map: [&amp;lt;1 1 1 4 2 4|, &amp;lt;0 4 9 -8 10 -2|]&lt;br /&gt;
+EDOs: 7, 27e, 34d, 61de&lt;br /&gt;
+Badness: 0.0238&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Octacot"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Octacot&lt;/h1&gt;
   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 &amp;lt;&amp;lt;8 18 11 10 -5 -25|| and may also be described as 41&amp;amp;68. &lt;a class="wiki_link" href="/68edo"&gt;68edo&lt;/a&gt; or &lt;a class="wiki_link" href="/109edo"&gt;109edo&lt;/a&gt; can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is &lt;a class="wiki_link" href="/150edo"&gt;150edo&lt;/a&gt;, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the &lt;a class="wiki_link" href="/88cET"&gt;88cET&lt;/a&gt; non-octave temperament, which like &lt;a class="wiki_link" href="/Carlos%20Alpha"&gt;Carlos Alpha&lt;/a&gt; arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 197: / Line 251: @@
 EDOs: 14c, 27, 41, 68, 109&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Octacot-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Octacot-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
   Commas: 100/99, 243/242, 245/242&lt;br /&gt;
 &lt;br /&gt;
@@ Line 207: / Line 261: @@
 See also: &lt;a class="wiki_link" href="/Chords%20of%20octacot"&gt;Chords of octacot&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Octacot-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Octacot-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
   Commas: 100/99, 144/143, 196/195, 243/242&lt;br /&gt;
 &lt;br /&gt;

Tetracot family: Difference between revisions

Wikispaces>genewardsmith