Eigenmonzo basis: Difference between revisions
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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<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. | 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. | |||
== See also == | == See also == | ||
* [[Projection #The unchanged-interval basis]] | |||
* [[unchanged-interval basis]] | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
Revision as of 07:10, 22 February 2025
An 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 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 {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.