5-limit: Difference between revisions
Wikispaces>xenwolf **Imported revision 215742060 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 232438890 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-27 13:04:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>232438890</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">Some examples | <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. | ||
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]]. | |||
Edos which do relatively well in approximating the 5-limit are [[5edo]], [[7edo]], [[12edo]], [[19edo]], [[34edo]], [[53edo]], [[65edo]], [[118edo]], [[171do]]. | |||
See [[Harmonic Limit]].</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>5-limit</title></head><body>Some examples | <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 /> | ||
<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 /> | |||
<br /> | |||
Edos which do relatively well in approximating the 5-limit are <a class="wiki_link" href="/5edo">5edo</a>, <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/53edo">53edo</a>, <a class="wiki_link" href="/65edo">65edo</a>, <a class="wiki_link" href="/118edo">118edo</a>, <a class="wiki_link" href="/171do">171do</a>.<br /> | |||
<br /> | <br /> | ||
See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a>.</body></html></pre></div> | |||