Trivial temperament
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- This revision was by author genewardsmith and made on 2012-02-11 15:23:15 UTC.
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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.
**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 an nxn zero matrix.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 />
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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.<br />
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<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 an nxn zero matrix.</body></html>