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Phi as a generator: Difference between revisions

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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:Kosmorsky|Kosmorsky]] and made on <tt>2011-12-31 04:38:30 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-31 13:00:44 UTC</tt>.<br>
: The original revision id was <tt>288892033</tt>.<br>
: The original revision id was <tt>288919029</tt>.<br>
: The revision comment was: <tt></tt><br>
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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">Phi as a Generator
<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">* Phi as a Generator


Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, exponents of phi seem to approximate the fibonacci numbers with increasing accuracy, which can be put to good effect in a temperament, because if the peculiarities of the factorization of the fibonacci numbers.
Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the [[http://en.wikipedia.org/wiki/Lucas_number|Lucas numbers]], closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the [[Kleismic family#Sqrtphi|sqrtphi temperament]], a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.


Let's use the archexample of [[46edo]]. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.
Let's use the archexample of [[46edo]]. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.
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[[97ed5]] (280:347)</pre></div>
[[97ed5]] (280:347)</pre></div>
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<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;Phi as a Generator&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Phi as a Generator&lt;br /&gt;
<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;Phi as a Generator&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;ul&gt;&lt;li&gt;Phi as a Generator&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
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Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lucas_number" rel="nofollow"&gt;Lucas numbers&lt;/a&gt;, closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the &lt;a class="wiki_link" href="/Kleismic%20family#Sqrtphi"&gt;sqrtphi temperament&lt;/a&gt;, a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.&lt;br /&gt;
Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, exponents of phi seem to approximate the fibonacci numbers with increasing accuracy, which can be put to good effect in a temperament, because if the peculiarities of the factorization of the fibonacci numbers.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Let's use the archexample of &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.&lt;br /&gt;
Let's use the archexample of &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.&lt;br /&gt;
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