Diamond function
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[[toc|flat]]
=Definition=
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.
=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.
=Scales=
[[diamond5]]
[[diamond7]]
[[diamond9]]
[[diamond11]]
[[diamond13]]
[[diamond15]]
[[diamond9plus-marvel]]
=Music=
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/GWS%20Scale%20Study-ModernJazzAtTheCrystalBall%20.mp3|Modern Jazz at the Crystal Ball]] by Norbert Oldani in the 7-limit diamond.
==see also==
* [[http://en.wikipedia.org/wiki/Tonality_diamond|Tonality diamond -- Wikipedia]]Original HTML content:
<html><head><title>Diamonds</title></head><body><!-- ws:start:WikiTextTocRule:10:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Scales">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:start:WikiTextHeadingRule:0:<h1> --><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 "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. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->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 />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scales</h1>
<a class="wiki_link" href="/diamond5">diamond5</a><br />
<a class="wiki_link" href="/diamond7">diamond7</a><br />
<a class="wiki_link" href="/diamond9">diamond9</a><br />
<a class="wiki_link" href="/diamond11">diamond11</a><br />
<a class="wiki_link" href="/diamond13">diamond13</a><br />
<a class="wiki_link" href="/diamond15">diamond15</a><br />
<br />
<a class="wiki_link" href="/diamond9plus-marvel">diamond9plus-marvel</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:6 -->Music</h1>
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/GWS%20Scale%20Study-ModernJazzAtTheCrystalBall%20.mp3" rel="nofollow">Modern Jazz at the Crystal Ball</a> by Norbert Oldani in the 7-limit diamond.<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Music-see also"></a><!-- ws:end:WikiTextHeadingRule:8 -->see also</h2>
<ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonality_diamond" rel="nofollow">Tonality diamond -- Wikipedia</a></li></ul></body></html>