Diamond function
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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.
The Diamond can also be thought of as being formed by the common tone modulations of all the elements in a set. It is also known as a Lambdoma
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.
== see also ==
* [[http://en.wikipedia.org/wiki/Tonality_diamond|Tonality diamond -- Wikipedia]]Original HTML content:
<html><head><title>Diamonds</title></head><body>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 <em>tonality diamond</em>, 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 />
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The Diamond can also be thought of as being formed by the common tone modulations of all the elements in a set. It is also known as a Lambdoma<br />
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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 />
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<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-see also"></a><!-- ws:end:WikiTextHeadingRule:0 --> 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>