Eigenmonzo basis: Difference between revisions
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m FloraC moved page Eigenmonzo subgroup to Eigenmonzo basis over redirect: Align with "comma basis" |
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Given a [[ | Given a [[regular temperament]] tuning T, an [[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 subgroup]], 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 | 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. 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. | ||
[[Category: | [[Category:Regular temperament theory]] | ||
[[Category:Terms]] | |||
[[Category:Math]] | |||
[[Category:Monzo]] | |||
Revision as of 15:29, 3 August 2022
Given a regular temperament tuning T, an 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 subgroup, 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 {2n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the 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.