Golden meantone
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2013-11-13 13:42:20 UTC.
- The original revision id was 468775292.
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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>