Eigenmonzo basis: Difference between revisions
Wikispaces>genewardsmith **Imported revision 230116456 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 230116832 - 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>2011-05-19 16: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-19 16:12:39 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>230116832</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">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 subgoup]], the eigenmonzo subgroup. | <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. | ||
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 [[Targent | 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 [[Targent 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 map of the minimax tuning and hence define the tuning.</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>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="/ | <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 /> | ||
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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="/Targent% | 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="/Targent%20tuning">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 projection map of the minimax tuning and hence define the tuning.</body></html></pre></div> | ||