Golden meantone: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 468774218 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 468775292 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-13 13:42:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>468775292</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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">**Golden Meantone** is based on making the relation between whole tone and diatonic semitone intervals be the [[http://en.wikipedia.org/wiki/Golden_ratio|Golden Ratio]] | <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">**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]] | [[math]] | ||
| Line 24: | Line 24: | ||
[[math]] | [[math]] | ||
cents. | cents, approximately 696.214 cents. | ||
Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal. | 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 Fibonacci | 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, 1 -> [[7edo]] | ||
1, 2 -> [[12edo]] | 1, 2 -> [[12edo]] | ||
| Line 46: | Line 46: | ||
[[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.</pre></div> | ||
<h4>Original HTML content:</h4> | <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"><html><head><title>Golden Meantone</title></head><body><strong>Golden Meantone</strong> is based on making the relation between 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 /> | <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>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 /> | <br /> | ||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
| Line 67: | Line 67: | ||
--><script type="math/tex">(9600 - 1200 \varphi) / 11</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">(9600 - 1200 \varphi) / 11</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
<br /> | <br /> | ||
cents.<br /> | cents, approximately 696.214 cents.<br /> | ||
<br /> | <br /> | ||
Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.<br /> | Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc0"><a name="x-Construction"></a><!-- ws:end:WikiTextHeadingRule:3 --> Construction </h2> | <!-- 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 Fibonacci | 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, 1 -&gt; <a class="wiki_link" href="/7edo">7edo</a><br /> | ||
1, 2 -&gt; <a class="wiki_link" href="/12edo">12edo</a><br /> | 1, 2 -&gt; <a class="wiki_link" href="/12edo">12edo</a><br /> | ||
Revision as of 13:42, 13 November 2013
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2013-11-13 13:42:20 UTC.
- The original revision id was 468775292.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
**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.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]]<br/>
\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,<br/>[[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]]<br/>
(8 - \varphi) / 11<br/>[[math]]
--><script type="math/tex">(8 - \varphi) / 11</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
octave, or<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]<br/>
(9600 - 1200 \varphi) / 11<br/>[[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:<h2> --><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 -> <a class="wiki_link" href="/7edo">7edo</a><br />
1, 2 -> <a class="wiki_link" href="/12edo">12edo</a><br />
2, 3 -> <a class="wiki_link" href="/19edo">19edo</a><br />
3, 5 -> <a class="wiki_link" href="/31edo">31edo</a><br />
5, 8 -> <a class="wiki_link" href="/50edo">50edo</a><br />
8, 13 -> <a class="wiki_link" href="/81edo">81edo</a><br />
13, 21 -> <a class="wiki_link" href="/131edo">131edo</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h2> --><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:<h2> --><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.</body></html>