Trivial temperament: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 300801372 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 300808810 - 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:genewardsmith|genewardsmith]] and made on <tt>2012-02-11 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-11 16:09:53 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>300808810</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> | ||
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<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">A trivial temperament is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "temperament". There are two kinds of trivial temperaments - JI, in which nothing is tempered, and **OM** temperament, in which everything is tempered. | <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">A trivial temperament is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "temperament". There are two kinds of trivial temperaments - JI, in which nothing is tempered, and **OM** temperament, in which everything is tempered. | ||
Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are tempered out is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The 2-limit version is the equal temperament [[1edo]]. The 3-limit version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix. | Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are tempered out is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The 2-limit version is the equal temperament [[1edo]]. The 3-limit version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1|, <<1||, <<<1|||... . | ||
**OM** temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is | **OM** temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, <0 0 ... 0|.</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>Trivial temperaments</title></head><body>A trivial temperament is something that fits the mathematical definition of &quot;regular temperament&quot;, but is a unique, extreme case that people might be uncomfortable calling a &quot;temperament&quot;. There are two kinds of trivial temperaments - JI, in which nothing is tempered, and <strong>OM</strong> temperament, in which everything is tempered.<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>Trivial temperaments</title></head><body>A trivial temperament is something that fits the mathematical definition of &quot;regular temperament&quot;, but is a unique, extreme case that people might be uncomfortable calling a &quot;temperament&quot;. There are two kinds of trivial temperaments - JI, in which nothing is tempered, and <strong>OM</strong> temperament, in which everything is tempered.<br /> | ||
<br /> | <br /> | ||
Just intonation is a codimension-0 &quot;temperament&quot;, which means nothing is tempered. The set of commas that are tempered out is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these &quot;temperaments&quot;, one for each subgroup of JI. The 2-limit version is the equal temperament <a class="wiki_link" href="/1edo">1edo</a>. The 3-limit version is the rank-2 temperament <a class="wiki_link" href="/pythagorean">pythagorean</a>, which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix.<br /> | Just intonation is a codimension-0 &quot;temperament&quot;, which means nothing is tempered. The set of commas that are tempered out is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these &quot;temperaments&quot;, one for each subgroup of JI. The 2-limit version is the equal temperament <a class="wiki_link" href="/1edo">1edo</a>. The 3-limit version is the rank-2 temperament <a class="wiki_link" href="/pythagorean">pythagorean</a>, which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of &lt;1|, &lt;&lt;1||, &lt;&lt;&lt;1|||... .<br /> | ||
<br /> | <br /> | ||
<strong>OM</strong> temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is | <strong>OM</strong> temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, &lt;0 0 ... 0|.</body></html></pre></div> | ||
Revision as of 16:09, 11 February 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-02-11 16:09:53 UTC.
- The original revision id was 300808810.
- 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:
A trivial temperament is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "temperament". There are two kinds of trivial temperaments - JI, in which nothing is tempered, and **OM** temperament, in which everything is tempered.
Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are tempered out is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The 2-limit version is the equal temperament [[1edo]]. The 3-limit version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1|, <<1||, <<<1|||... .
**OM** temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, <0 0 ... 0|.Original HTML content:
<html><head><title>Trivial temperaments</title></head><body>A trivial temperament is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "temperament". There are two kinds of trivial temperaments - JI, in which nothing is tempered, and <strong>OM</strong> temperament, in which everything is tempered.<br />
<br />
Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are tempered out is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The 2-limit version is the equal temperament <a class="wiki_link" href="/1edo">1edo</a>. The 3-limit version is the rank-2 temperament <a class="wiki_link" href="/pythagorean">pythagorean</a>, which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1|, <<1||, <<<1|||... .<br />
<br />
<strong>OM</strong> temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, <0 0 ... 0|.</body></html>