Diamond function: Difference between revisions
Wikispaces>genewardsmith **Imported revision 243010985 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 243027989 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2011-07- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-27 01:37:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>243027989</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> | ||
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If S is a finite set of positive real numbers, then the diamond of S, Diamond(S), is the set {octave-reduce(u/v) | u,v in S}; that is, the set of all ratios of any two elements of S, reduced to the octave. The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers. The important special case where S is the set of odd integers less than or equal to an odd n is called the tonality diamond, and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval. | If S is a finite set of positive real numbers, then the diamond of S, Diamond(S), is the set {octave-reduce(u/v) | u,v in S}; that is, the set of all ratios of any two elements of S, reduced to the octave. The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers. The important special case where S is the set of odd integers less than or equal to an odd n is called the tonality diamond, and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval. | ||
= | =Creating scales= | ||
The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as Harry Partch did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the 11-limit diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates. | The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as Harry Partch did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the 11-limit diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates. | ||
= | =Examples of scales= | ||
[[diamond5]] | [[diamond5]] | ||
[[diamond7]] | [[diamond7]] | ||
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* [[http://en.wikipedia.org/wiki/Tonality_diamond|Tonality diamond -- Wikipedia]]</pre></div> | * [[http://en.wikipedia.org/wiki/Tonality_diamond|Tonality diamond -- Wikipedia]]</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>Diamonds</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="# | <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>Diamonds</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Creating scales">Creating scales</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Examples of scales">Examples of scales</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
<!-- ws:end:WikiTextTocRule:16 --><br /> | <!-- ws:end:WikiTextTocRule:16 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
If S is a finite set of positive real numbers, then the diamond of S, Diamond(S), is the set {octave-reduce(u/v) | u,v in S}; that is, the set of all ratios of any two elements of S, reduced to the octave. The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers. The important special case where S is the set of odd integers less than or equal to an odd n is called the tonality diamond, and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the &quot;chord of nature&quot;, or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a &quot;chord of nature&quot;, contain this interval. <br /> | If S is a finite set of positive real numbers, then the diamond of S, Diamond(S), is the set {octave-reduce(u/v) | u,v in S}; that is, the set of all ratios of any two elements of S, reduced to the octave. The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers. The important special case where S is the set of odd integers less than or equal to an odd n is called the tonality diamond, and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the &quot;chord of nature&quot;, or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a &quot;chord of nature&quot;, contain this interval. <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name=" | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Creating scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->Creating scales</h1> | ||
The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as Harry Partch did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the 11-limit diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.<br /> | The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as Harry Partch did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the 11-limit diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples of scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Examples of scales</h1> | ||
<a class="wiki_link" href="/diamond5">diamond5</a><br /> | <a class="wiki_link" href="/diamond5">diamond5</a><br /> | ||
<a class="wiki_link" href="/diamond7">diamond7</a><br /> | <a class="wiki_link" href="/diamond7">diamond7</a><br /> | ||