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 | 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 == | |||
* [[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]] | |||
Latest revision as of 16:13, 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.
See also
- Projection #The unchanged-interval basis, for a discussion of this concept in the context of other related temperament tuning objects