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.

An [[eigenmonzo|eigenmonzo or unchanged-interval]] is a rational interval tuned justly by a [[regular temperament]] tuning. In other words, if a tuning is ''T'', then an eigenmonzo ''q'' satisfies {{nowrap| ''T''(''q'') {{=}} ''q'' }}. The eigenmonzos of ''T'' define a [[just intonation subgroup]], the eigenmonzo subgroup, whose basis is an '''eigenmonzo basis''' or '''unchanged-interval basis'''.

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.

[[Category:~~eigenmonzo~~]]

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.

== See also ==

* [[Projection #The unchanged-interval basis]], for a discussion of this concept in the context of other related temperament tuning objects

[[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.
+An [[eigenmonzo|eigenmonzo or unchanged-interval]] is a rational interval tuned justly by a [[regular temperament]] tuning. In other words, if a tuning is ''T'', then an eigenmonzo ''q'' satisfies {{nowrap| ''T''(''q'') {{=}} ''q'' }}. The eigenmonzos of ''T'' define a [[just intonation subgroup]], the eigenmonzo subgroup, whose basis is an '''eigenmonzo basis''' or '''unchanged-interval basis'''.
-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.
-[[Category:eigenmonzo]]
+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.
+== See also ==
+* [[Projection #The unchanged-interval basis]], for a discussion of this concept in the context of other related temperament tuning objects
+[[Category:Regular temperament theory]]
+[[Category:Terms]]
+[[Category:Math]]
+[[Category:Monzo]]