Das Goldene Tonsystem
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**[[http://d-nb.info/361092458|Das Goldene Tonsystem]]**
als Fundament der Theoretischen Akustik
is a book of the danish music theoretician (music reformer and visionary) Thorvald Kornerup, written in German and published in Copenhagen in 1935.
The system is based on the paradigm that the relation between whole and half tone intervals should 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]]
Thus some edo systems - the 12-step too - could be considered as approximations to this ideal.
== Construction ==
If you use two neighboring numbers from 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>Das Goldene Tonsystem</title></head><body><strong><a class="wiki_link_ext" href="http://d-nb.info/361092458" rel="nofollow">Das Goldene Tonsystem</a></strong><br />
als Fundament der Theoretischen Akustik<br />
<br />
is a book of the danish music theoretician (music reformer and visionary) Thorvald Kornerup, written in German and published in Copenhagen in 1935.<br />
<br />
The system is based on the paradigm that the relation between whole and half tone intervals should 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 />
Thus some edo systems - the 12-step too - could be considered as approximations to this ideal.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:1:<h2> --><h2 id="toc0"><a name="x-Construction"></a><!-- ws:end:WikiTextHeadingRule:1 --> Construction </h2>
If you use two neighboring numbers from 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:3:<h2> --><h2 id="toc1"><a name="x-Evaluation"></a><!-- ws:end:WikiTextHeadingRule:3 --> 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:5:<h2> --><h2 id="toc2"><a name="x-Listening"></a><!-- ws:end:WikiTextHeadingRule:5 --> 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>