Eigenmonzo basis: Difference between revisions
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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 /> | ||
<br /> | <br /> | ||
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> | ||
Revision as of 16:12, 19 May 2011
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 2011-05-19 16:12:39 UTC.
- The original revision id was 230116832.
- 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:
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 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.Original HTML content:
<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 />
<br />
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>