Eigenmonzo basis
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2011-05-19 16:17:16 UTC.
- The original revision id was 230118236.
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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 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 [[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.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 />
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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>