View source for Eigenmonzo basis

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.

== 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]]