Meantone 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:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2012-05-~~02 14~~:47:39 UTC</tt>.<br>

: This revision was by author [[User:clamengh|clamengh]] and made on <tt>2012-05-06 05:36:18 UTC</tt>.<br>

: The original revision id was <tt>~~328840872~~</tt>.<br>

: The original revision id was <tt>330608550</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]]

The [[5-limit]] parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

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[[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

[[@http://soonlabel.com/xenharmonic/archives/607|Scott Joplin's "The Entertainer" tuned into meanpop]]

Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]

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Badness: 0.0366

===13-limit===

Commas: 26/25, 36/35, 64/63, 66/65

POTE generator: ~3/2 = 705.496

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Badness: 0.0290

==13-limit==

Commas: 45/44, 49/48, 78/77, 81/80

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Badness: 0.0240

==Cynder==

Commas: 45/44, 81/80, 1029/1024

POTE generator: ~8/7 = 231.317

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Badness: 0.0557

===13-limit===

Commas: 45/44, 78/77, 81/80, 640/637

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Badness: 0.0416

==Lisa==

Commas: 45/44, 81/80, 343/330

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Badness: 0.0548

==13-limit==

Commas: 45/44, 81/80, 91/88, 147/143

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<div style="margin-left: 1em;"><a href="#Undevigintone">Undevigintone</a></div>

</div>

~~<br />~~

The <a class="wiki_link" href="/5-limit">5-limit</a> parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br />

The <a class="wiki_link" href="/5-limit">5-limit</a> parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br />

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<br />

<a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br />

<br />

<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/607" rel="nofollow" target="_blank">Scott Joplin's &quot;The Entertainer&quot; tuned into meanpop</a><br />

<br />

Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]<br />

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<br />

<h3 id="toc20"><a name="Dominant-Domination-13-limit"></a>13-limit</h3>

Commas: 26/25, 36/35, 64/63, 66/65 <br />

Commas: 26/25, 36/35, 64/63, 66/65<br />

<br />

POTE generator: ~3/2 = 705.496<br />

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<br />

<h2 id="toc36"><a name="Godzilla-13-limit"></a>13-limit</h2>

Commas: 45/44, 49/48, 78/77, 81/80<br />

<br />

POTE generator: ~8/7 = 253.603<br />

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<br />

<h2 id="toc55"><a name="Mothra-Cynder"></a>Cynder</h2>

Commas: 45/44, 81/80, 1029/1024 <br />

Commas: 45/44, 81/80, 1029/1024<br />

<br />

POTE generator: ~8/7 = 231.317<br />

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<br />

<h3 id="toc56"><a name="Mothra-Cynder-13-limit"></a>13-limit</h3>

Commas: 45/44, 78/77, 81/80, 640/637<br />

<br />

POTE generator: ~8/7 = 231.293<br />

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<br />

<h2 id="toc69"><a name="Liese-Lisa"></a>Lisa</h2>

Commas: 45/44, 81/80, 343/330<br />

<br />

POTE generator: ~10/7 = 631.370<br />

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<br />

<h2 id="toc70"><a name="Liese-13-limit"></a>13-limit</h2>

Commas: 45/44, 81/80, 91/88, 147/143<br />

<br />

POTE generator: ~10/7 = 631.221<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:genewardsmith|genewardsmith]] and made on <tt>2012-05-02 14:47:39 UTC</tt>.<br>
+: This revision was by author [[User:clamengh|clamengh]] and made on <tt>2012-05-06 05:36:18 UTC</tt>.<br>
-: The original revision id was <tt>328840872</tt>.<br>
+: The original revision id was <tt>330608550</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]]
 The [[5-limit]] parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
@@ Line 95: / Line 94: @@
 [[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
+[[@http://soonlabel.com/xenharmonic/archives/607|Scott Joplin's "The Entertainer" tuned into meanpop]]
 Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
@@ Line 239: / Line 240: @@
 Badness: 0.0366
 ===13-limit===
 Commas: 26/25, 36/35, 64/63, 66/65
 POTE generator: ~3/2 = 705.496
@@ Line 397: / Line 398: @@
 Badness: 0.0290
 ==13-limit==
 Commas: 45/44, 49/48, 78/77, 81/80
@@ Line 596: / Line 597: @@
 Badness: 0.0240
 ==Cynder==
 Commas: 45/44, 81/80, 1029/1024
 POTE generator: ~8/7 = 231.317
@@ Line 605: / Line 606: @@
 Badness: 0.0557
 ===13-limit===
 Commas: 45/44, 78/77, 81/80, 640/637
@@ Line 738: / Line 739: @@
 Badness: 0.0416
 ==Lisa==
 Commas: 45/44, 81/80, 343/330
@@ Line 747: / Line 748: @@
 Badness: 0.0548
 ==13-limit==
 Commas: 45/44, 81/80, 91/88, 147/143
@@ Line 892: / Line 893: @@
 &lt;!-- ws:end:WikiTextTocRule:230 --&gt;&lt;!-- ws:start:WikiTextTocRule:231: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Undevigintone"&gt;Undevigintone&lt;/a&gt;&lt;/div&gt;
 &lt;!-- ws:end:WikiTextTocRule:231 --&gt;&lt;!-- ws:start:WikiTextTocRule:232: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:232 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:232 --&gt;The &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; parent &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt; of the &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; family is the Didymus or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow"&gt;syntonic comma&lt;/a&gt;, 81/80. This is the one they all temper out. The &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzo&lt;/a&gt; for 81/80 goes |-4 4 -1&amp;gt;, and that can be flipped around to the corresponding &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, &amp;lt;&amp;lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.&lt;br /&gt;
-The &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; parent &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt; of the &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; family is the Didymus or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow"&gt;syntonic comma&lt;/a&gt;, 81/80. This is the one they all temper out. The &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzo&lt;/a&gt; for 81/80 goes |-4 4 -1&amp;gt;, and that can be flipped around to the corresponding &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, &amp;lt;&amp;lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 696.239&lt;br /&gt;
@@ Line 980: / Line 980: @@
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/Algebraic%20generator"&gt;Algebraic generator&lt;/a&gt;: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.&lt;br /&gt;
+&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/607" rel="nofollow" target="_blank"&gt;Scott Joplin's &amp;quot;The Entertainer&amp;quot; tuned into meanpop&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;1 0 -4 -13 24|, &amp;lt;0 1 4 10 -13|]&lt;br /&gt;
@@ Line 1,125: / Line 1,127: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc20"&gt;&lt;a name="Dominant-Domination-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;13-limit&lt;/h3&gt;
-Commas: 26/25, 36/35, 64/63, 66/65 &lt;br /&gt;
+ Commas: 26/25, 36/35, 64/63, 66/65&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~3/2 = 705.496&lt;br /&gt;
@@ Line 1,283: / Line 1,285: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc36"&gt;&lt;a name="Godzilla-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;13-limit&lt;/h2&gt;
-Commas: 45/44, 49/48, 78/77, 81/80&lt;br /&gt;
+ Commas: 45/44, 49/48, 78/77, 81/80&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~8/7 = 253.603&lt;br /&gt;
@@ Line 1,482: / Line 1,484: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:110:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc55"&gt;&lt;a name="Mothra-Cynder"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:110 --&gt;Cynder&lt;/h2&gt;
-Commas: 45/44, 81/80, 1029/1024 &lt;br /&gt;
+ Commas: 45/44, 81/80, 1029/1024&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~8/7 = 231.317&lt;br /&gt;
@@ Line 1,491: / Line 1,493: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:112:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc56"&gt;&lt;a name="Mothra-Cynder-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:112 --&gt;13-limit&lt;/h3&gt;
-Commas: 45/44, 78/77, 81/80, 640/637&lt;br /&gt;
+ Commas: 45/44, 78/77, 81/80, 640/637&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~8/7 = 231.293&lt;br /&gt;
@@ Line 1,624: / Line 1,626: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:138:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc69"&gt;&lt;a name="Liese-Lisa"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:138 --&gt;Lisa&lt;/h2&gt;
-Commas: 45/44, 81/80, 343/330&lt;br /&gt;
+ Commas: 45/44, 81/80, 343/330&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~10/7 = 631.370&lt;br /&gt;
@@ Line 1,633: / Line 1,635: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:140:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc70"&gt;&lt;a name="Liese-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:140 --&gt;13-limit&lt;/h2&gt;
-Commas: 45/44, 81/80, 91/88, 147/143&lt;br /&gt;
+ Commas: 45/44, 81/80, 91/88, 147/143&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~10/7 = 631.221&lt;br /&gt;

Meantone family: Difference between revisions

Wikispaces>clamengh