Eigenmonzo basis: Difference between revisions

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Given a [[~~Abstract_regular_temperament|~~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_subgroups|~~just intonation subgroup]], the eigenmonzo subgroup.

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^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 matrix]] of the minimax tuning and hence define the tuning.

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:~~eigenmonzo~~]]

[[Category:Regular temperament theory]]

[[Category:Terms]]

[[Category:Math]]

[[Category:Monzo]]

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-Given a [[Abstract_regular_temperament|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_subgroups|just intonation subgroup]], the eigenmonzo subgroup.
+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^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 matrix]] of the minimax tuning and hence define the tuning.
+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:eigenmonzo]]
+[[Category:Regular temperament theory]]
+[[Category:Terms]]
+[[Category:Math]]
+[[Category:Monzo]]