Eigenmonzo basis: Difference between revisions
Jump to navigation
Jump to search
Cmloegcmluin (talk | contribs) change link to point straight to eigenmonzo |
Cmloegcmluin (talk | contribs) m fix typo |
||
| 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 | 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 map]] of the minimax tuning and hence define the tuning. | ||
[[Category:eigenmonzo]] | [[Category:eigenmonzo]] | ||
Revision as of 20:36, 19 July 2021
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 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 map of the minimax tuning and hence define the tuning.