Diamond function: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 239205485 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 239286913 - 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: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-29 04:38:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239286913</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">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. | <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;white-space: pre-wrap ! important" class="old-revision-html">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 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 | ||
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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>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 &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 /> | <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>If S is a finite set of positive real numbers, then the diamond of S, Diamond(S), is the set {<a class="wiki_link" href="/octave-reduce">octave-reduce</a>(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 &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 /> | ||
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 /> | 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 /> | ||
Revision as of 04:38, 29 June 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2011-06-29 04:38:18 UTC.
- The original revision id was 239286913.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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 {<a class="wiki_link" href="/octave-reduce">octave-reduce</a>(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 />
<br />
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 />
<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 />
<!-- 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>