Diamond function: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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<h4>Original Wikitext content:</h4>

=Definition=

Given a collection of notes S, the //diamond// of S, diamond(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. 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.

=Examples of scales=

&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;

Given a collection of notes S, the &lt;em&gt;diamond&lt;/em&gt; of S, diamond(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. 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 &amp;quot;chord of nature&amp;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 &amp;quot;chord of nature&amp;quot;, contain this interval. &lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Creating scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Creating scales&lt;/h1&gt;

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.&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Examples of scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Examples of scales&lt;/h1&gt;