Golden meantone: Difference between revisions

Revision as of 18:24, 24 December 2013

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This revision was by author spt3125 and made on 2013-12-24 18:24:53 UTC.

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**Golden Meantone** is based on making the relation between the whole tone and diatonic semitone intervals be the [[http://en.wikipedia.org/wiki/Golden_ratio|Golden Ratio]]

[[math]]
\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,
[[math]]

This makes the Golden fifth exactly

[[math]]
(8 - \varphi) / 11
[[math]]

octave, or

[[math]]
(9600 - 1200 \varphi) / 11
[[math]]

cents, approximately 696.214 cents.

Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.

==Construction== 
If you use two neighboring numbers from the Fibonacci series 1 1 2 3 5 8 13 21... you get the following approximations:
1, 1 -> [[7edo]]
1, 2 -> [[12edo]]
2, 3 -> [[19edo]]
3, 5 -> [[31edo]]
5, 8 -> [[50edo]]
8, 13 -> [[81edo]]
13, 21 -> [[131edo]]

==Evaluation== 

Graham Breed [[http://x31eq.com/meantone.htm|writes]]: //I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.//

==Listening== 

[[http://www.io.com/~hmiller/midi/canon-golden.mid|An acoustic experience]] - Kornerup himself had no chance to have it - is contained in the [[Warped canon]] collection.


==Additional reading== 

[[@http://www.tonalsoft.com/enc/g/golden.aspx|Golden meantone - Tonalsoft encyclopedia]]

Original HTML content:

<html><head><title>Golden Meantone</title></head><body><strong>Golden Meantone</strong> is based on making the relation between the whole tone and diatonic semitone intervals be the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow">Golden Ratio</a><br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,&lt;br/&gt;[[math]]
 --><script type="math/tex">\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
This makes the Golden fifth exactly<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
(8 - \varphi) / 11&lt;br/&gt;[[math]]
 --><script type="math/tex">(8 - \varphi) / 11</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
octave, or<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
(9600 - 1200 \varphi) / 11&lt;br/&gt;[[math]]
 --><script type="math/tex">(9600 - 1200 \varphi) / 11</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
cents, approximately 696.214 cents.<br />
<br />
Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc0"><a name="x-Construction"></a><!-- ws:end:WikiTextHeadingRule:3 -->Construction</h2>
 If you use two neighboring numbers from the Fibonacci series 1 1 2 3 5 8 13 21... you get the following approximations:<br />
1, 1 -&gt; <a class="wiki_link" href="/7edo">7edo</a><br />
1, 2 -&gt; <a class="wiki_link" href="/12edo">12edo</a><br />
2, 3 -&gt; <a class="wiki_link" href="/19edo">19edo</a><br />
3, 5 -&gt; <a class="wiki_link" href="/31edo">31edo</a><br />
5, 8 -&gt; <a class="wiki_link" href="/50edo">50edo</a><br />
8, 13 -&gt; <a class="wiki_link" href="/81edo">81edo</a><br />
13, 21 -&gt; <a class="wiki_link" href="/131edo">131edo</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc1"><a name="x-Evaluation"></a><!-- ws:end:WikiTextHeadingRule:5 -->Evaluation</h2>
 <br />
Graham Breed <a class="wiki_link_ext" href="http://x31eq.com/meantone.htm" rel="nofollow">writes</a>: <em>I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.</em><br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc2"><a name="x-Listening"></a><!-- ws:end:WikiTextHeadingRule:7 -->Listening</h2>
 <br />
<a class="wiki_link_ext" href="http://www.io.com/~hmiller/midi/canon-golden.mid" rel="nofollow">An acoustic experience</a> - Kornerup himself had no chance to have it - is contained in the <a class="wiki_link" href="/Warped%20canon">Warped canon</a> collection.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc3"><a name="x-Additional reading"></a><!-- ws:end:WikiTextHeadingRule:9 -->Additional reading</h2>
 <br />
<a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/g/golden.aspx" rel="nofollow" target="_blank">Golden meantone - Tonalsoft encyclopedia</a></body></html>

@@ 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>2013-11-13 13:42:20 UTC</tt>.<br>
+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2013-12-24 18:24:53 UTC</tt>.<br>
-: The original revision id was <tt>468775292</tt>.<br>
+: The original revision id was <tt>479295880</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 28: / Line 28: @@
 Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.
-== Construction ==
+==Construction==
 If you use two neighboring numbers from the Fibonacci series 1 1 2 3 5 8 13 21... you get the following approximations:
 , 1 -&gt; [[7edo]]
 , 2 -&gt; [[12edo]]
 , 3 -&gt; [[19edo]]
 , 5 -&gt; [[31edo]]
 , 8 -&gt; [[50edo]]
 , 13 -&gt; [[81edo]]
 , 21 -&gt; [[131edo]]
-== Evaluation ==
+==Evaluation==
 Graham Breed [[http://x31eq.com/meantone.htm|writes]]: //I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.//
-== Listening ==
+==Listening==
-[[http://www.io.com/~hmiller/midi/canon-golden.mid|An acoustic experience]] - Kornerup himself had no chance to have it - is contained in the [[Warped canon]] collection.</pre></div>
+[[http://www.io.com/~hmiller/midi/canon-golden.mid|An acoustic experience]] - Kornerup himself had no chance to have it - is contained in the [[Warped canon]] collection.
+==Additional reading==
+[[@http://www.tonalsoft.com/enc/g/golden.aspx|Golden meantone - Tonalsoft encyclopedia]]</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Golden Meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Golden Meantone&lt;/strong&gt; is based on making the relation between the whole tone and diatonic semitone intervals be the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow"&gt;Golden Ratio&lt;/a&gt;&lt;br /&gt;
@@ Line 71: / Line 76: @@
 Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Construction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt; Construction &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Construction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Construction&lt;/h2&gt;
-If you use two neighboring numbers from the Fibonacci series 1 1 2 3 5 8 13 21... you get the following approximations:&lt;br /&gt;
+ If you use two neighboring numbers from the Fibonacci series 1 1 2 3 5 8 13 21... you get the following approximations:&lt;br /&gt;
 , 1 -&amp;gt; &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;&lt;br /&gt;
 , 2 -&amp;gt; &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;br /&gt;
 , 3 -&amp;gt; &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;&lt;br /&gt;
 , 5 -&amp;gt; &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;&lt;br /&gt;
 , 8 -&amp;gt; &lt;a class="wiki_link" href="/50edo"&gt;50edo&lt;/a&gt;&lt;br /&gt;
 , 13 -&amp;gt; &lt;a class="wiki_link" href="/81edo"&gt;81edo&lt;/a&gt;&lt;br /&gt;
 , 21 -&amp;gt; &lt;a class="wiki_link" href="/131edo"&gt;131edo&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Evaluation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt; Evaluation &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Evaluation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Evaluation&lt;/h2&gt;
+ &lt;br /&gt;
+Graham Breed &lt;a class="wiki_link_ext" href="http://x31eq.com/meantone.htm" rel="nofollow"&gt;writes&lt;/a&gt;: &lt;em&gt;I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-Graham Breed &lt;a class="wiki_link_ext" href="http://x31eq.com/meantone.htm" rel="nofollow"&gt;writes&lt;/a&gt;: &lt;em&gt;I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.&lt;/em&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Listening"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Listening&lt;/h2&gt;
+ &lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://www.io.com/~hmiller/midi/canon-golden.mid" rel="nofollow"&gt;An acoustic experience&lt;/a&gt; - Kornerup himself had no chance to have it - is contained in the &lt;a class="wiki_link" href="/Warped%20canon"&gt;Warped canon&lt;/a&gt; collection.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Listening"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt; Listening &lt;/h2&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.io.com/~hmiller/midi/canon-golden.mid" rel="nofollow"&gt;An acoustic experience&lt;/a&gt; - Kornerup himself had no chance to have it - is contained in the &lt;a class="wiki_link" href="/Warped%20canon"&gt;Warped canon&lt;/a&gt; collection.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Additional reading"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Additional reading&lt;/h2&gt;
+ &lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/g/golden.aspx" rel="nofollow" target="_blank"&gt;Golden meantone - Tonalsoft encyclopedia&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Golden meantone: Difference between revisions

Revision as of 18:24, 24 December 2013

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