Eigenmonzo basis: Difference between revisions

Line 1:

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.

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.

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.

[[Category:eigenmonzo]]

@@ Line 1: / Line 1: @@
 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.
-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.
+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.
 [[Category:eigenmonzo]]