Eigenmonzo basis
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- This revision was by author genewardsmith and made on 2011-05-19 16:11:33 UTC.
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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 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 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 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%20subgoup">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%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 projection map of the minimax tuning and hence define the tuning.</body></html>