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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | An [[eigenmonzo|eigenmonzo or unchanged-interval]] is a rational interval tuned justly by a [[regular temperament]] tuning. In other words, if a tuning is ''T'', then an eigenmonzo ''q'' satisfies {{nowrap| ''T''(''q'') {{=}} ''q'' }}. The eigenmonzos of ''T'' define a [[just intonation subgroup]], the eigenmonzo subgroup, whose basis is an '''eigenmonzo basis''' or '''unchanged-interval basis'''. |
| 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>2011-05-19 16:17:16 UTC</tt>.<br>
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| : The original revision id was <tt>230118236</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">Given a [[Abstract regular temperament|regular temperament]] tuning T, an [[Fractional monzos|eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[Just intonation subgroups|just intonation subgoup]], the eigenmonzo subgroup.
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| One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tunings|minimax tunings]] of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[Target tunings|projection map]] of the minimax tuning and hence define the tuning.</pre></div> | | One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n''</sup>} of powers of 2 is the eigenmonzo subgroup. |
| <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>Eigenmonzo subgroup</title></head><body>Given a <a class="wiki_link" href="/Abstract%20regular%20temperament">regular temperament</a> tuning T, an <a class="wiki_link" href="/Fractional%20monzos">eigenmonzo</a> is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgoup</a>, the eigenmonzo subgroup.<br />
| | The idea is most useful in connection to the [[Target tuning #Minimax tuning|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning. |
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| One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the <a class="wiki_link" href="/Target%20tunings">minimax tunings</a> of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the <a class="wiki_link" href="/Target%20tunings">projection map</a> of the minimax tuning and hence define the tuning.</body></html></pre></div>
| | == See also == |
| | * [[Projection #The unchanged-interval basis]], for a discussion of this concept in the context of other related temperament tuning objects |
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| | [[Category:Regular temperament theory]] |
| | [[Category:Terms]] |
| | [[Category:Math]] |
| | [[Category:Monzo]] |