# Eigenmonzo basis

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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 matrix of the minimax tuning and hence define the tuning.