5-limit: Difference between revisions

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<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>2011-06-~~10 01~~:11:08 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 17:27:07 UTC</tt>.<br>

: The original revision id was <tt>~~235668776~~</tt>.<br>

: The original revision id was <tt>244906775</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>

<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">The //5-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5. Some examples of 5-limit intervals are [[5_4|5/4]], [[6_5|6/5]], [[10_9|10/9]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.

<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">The //5-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [[http://en.wikipedia.org/wiki/Regular_number|regular numbers]]. Some examples of 5-limit intervals are [[5_4|5/4]], [[6_5|6/5]], [[10_9|10/9]] and [[81_80|81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.

The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [[http://en.wikipedia.org/wiki/Hexagonal_lattice|hexagonal lattice]] or as a [[http://en.wikipedia.org/wiki/Square_lattice|square lattice]]; this can be done automatically by [[http://www.huygens-fokker.org/scala/|Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[http://en.wikipedia.org/wiki/Hexagonal_tiling|hexagonal tiling]].

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[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/Do%20Wah%20Diddy.mp3|Do Wah Diddy Diddy]] by [[http://en.wikipedia.org/wiki/Jeff_Barry|Barry]] and [[http://en.wikipedia.org/wiki/Ellie_Greenwich|Greenwich]]</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"><html><head><title>5-limit</title></head><body>The <em>5-limit</em> consists of all <a class="wiki_link" href="/JustIntonation">justly tuned</a> intervals whose numerators and denominators are both products of the primes 2, 3, and 5. Some examples of 5-limit intervals are <a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/10_9">10/9</a>. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.<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>5-limit</title></head><body>The <em>5-limit</em> consists of all <a class="wiki_link" href="/JustIntonation">justly tuned</a> intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_number" rel="nofollow">regular numbers</a>. Some examples of 5-limit intervals are <a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/81_80">81/80</a>. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.<br />

<br />

The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_lattice" rel="nofollow">hexagonal lattice</a> or as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Square_lattice" rel="nofollow">square lattice</a>; this can be done automatically by <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow">Scala</a>. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_tiling" rel="nofollow">hexagonal tiling</a>.<br />

@@ 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>2011-06-10 01:11:08 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 17:27:07 UTC</tt>.<br>
-: The original revision id was <tt>235668776</tt>.<br>
+: The original revision id was <tt>244906775</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>
 <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">The //5-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5. Some examples of 5-limit intervals are [[5_4|5/4]], [[6_5|6/5]], [[10_9|10/9]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.
+<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">The //5-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [[http://en.wikipedia.org/wiki/Regular_number|regular numbers]]. Some examples of 5-limit intervals are [[5_4|5/4]], [[6_5|6/5]], [[10_9|10/9]] and [[81_80|81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.
 The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [[http://en.wikipedia.org/wiki/Hexagonal_lattice|hexagonal lattice]] or as a [[http://en.wikipedia.org/wiki/Square_lattice|square lattice]]; this can be done automatically by [[http://www.huygens-fokker.org/scala/|Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[http://en.wikipedia.org/wiki/Hexagonal_tiling|hexagonal tiling]].
@@ Line 20: / Line 20: @@
 [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/Do%20Wah%20Diddy.mp3|Do Wah Diddy Diddy]] by [[http://en.wikipedia.org/wiki/Jeff_Barry|Barry]] and [[http://en.wikipedia.org/wiki/Ellie_Greenwich|Greenwich]]</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;5-limit&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;5-limit&lt;/em&gt; consists of all &lt;a class="wiki_link" href="/JustIntonation"&gt;justly tuned&lt;/a&gt; intervals whose numerators and denominators are both products of the primes 2, 3, and 5. Some examples of 5-limit intervals are &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.&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;5-limit&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;5-limit&lt;/em&gt; consists of all &lt;a class="wiki_link" href="/JustIntonation"&gt;justly tuned&lt;/a&gt; intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_number" rel="nofollow"&gt;regular numbers&lt;/a&gt;. Some examples of 5-limit intervals are &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; and &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.&lt;br /&gt;
 &lt;br /&gt;
 The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_lattice" rel="nofollow"&gt;hexagonal lattice&lt;/a&gt; or as a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Square_lattice" rel="nofollow"&gt;square lattice&lt;/a&gt;; this can be done automatically by &lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow"&gt;Scala&lt;/a&gt;. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_tiling" rel="nofollow"&gt;hexagonal tiling&lt;/a&gt;.&lt;br /&gt;