Diamond function: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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-~~27 01~~:37:47 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-28 12:33:57 UTC</tt>.<br>

: The original revision id was <tt>~~243027989~~</tt>.<br>

: The original revision id was <tt>243257651</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>

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=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.

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=

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<br />

<h1 id="toc0"><a name="Definition"></a>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 />

Given a collection of notes S, the <em>diamond</em> 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 &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 />

<h1 id="toc1"><a name="Creating scales"></a>Creating scales</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-07-27 01:37:47 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-28 12:33:57 UTC</tt>.<br>
-: The original revision id was <tt>243027989</tt>.<br>
+: The original revision id was <tt>243257651</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>
@@ Line 9: / Line 9: @@
 =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.
+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=
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 &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;
-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 &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;
+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;
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 &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;