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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | 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 types of trivial temperaments: [[just intonation]], which leaves all intervals [[tempering|untempered]], and [[single-pitch tuning]], which [[tempering out|tempers out]] all intervals. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-11 15:23:15 UTC</tt>.<br>
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| : The original revision id was <tt>300801372</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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.
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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 [[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 == |
| | {{Main| Just intonation }} |
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| **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.</pre></div>
| | The [[mapping]] for a [[just intonation subgroup]] of rank ''n'' is an ''n''×''n'' {{w|identity matrix}}, and transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are tempered out is {1/1}, but that is still a valid set, so just intonation still counts as valid regular temperaments. |
| <h4>Original HTML content:</h4>
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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;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 />
| | There is an infinite family of these temperaments, one for each subgroup of JI. The 2-limit version is equivalent to [[1edo|1et]]. The [[3-limit]] version, or [[pythagorean tuning]], is a rank-2 temperament, which has all the properties of any other rank-2 temperament except that it tempers out no commas. 5-limit JI is rank-3, 7-limit JI is rank-4, etc. |
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| 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 />
| | [[User:VectorGraphics|Vector]] proposes the name ''identity temperament''{{idio}} for this family of temperaments. |
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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></pre></div>
| | == Single-pitch tuning == |
| | {{Main| Single-pitch tuning }} |
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| | The single-pitch tuning 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, {{val| 0 0 … 0 }}, and its [[wedgie]] is a single entry. |
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| | As with JI, there is technically a temperament of a single pitch for every subgroup. |
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| | [[Gene Ward Smith]] proposes the name ''unison temperament'' for this family of temperaments<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. [[Keenan Pepper]] proposes the name ''Om temperament''{{idio}}. [[Wikipedia:Om|''Om'']] is a reference to that syllable's use in Hindu meditation practices, for there is only one temperament-distinct pitch in the whole system, in the same way that ''Om'' in the meditation sense is the only word you need to create the whole universe. |
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| | == Notes and references == |
| | <references /> |
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| | [[Category:Regular temperament theory]] |