Eigenmonzo

A regular temperament transforms a set of untempered intervals into a set of tempered intervals, which changes the size of all of these intervals except for a few exceptions; which of these intervals that are the exceptions and do not change depends on the choice of tuning (of the temperament's generators), and thus each of these is an unchanged interval, or eigenmonzo, of the tuning.

A popular example of an eigenmonzo is the JI interval [0 0 1⟩, or 1:5, when it is mapped by quarter-comma meantone; because this temperament tuning's generator is defined as exactly one-quarter the size of the interval 1:5, it remains justly tuned.

For any pure-octave temperament tuning, [1⟩, aka 1:2, is an eigenmonzo.

A rank-n temperament can have up to n different eigenmonzos — one for each generator.

With respect to the projection matrix

The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are eigenvectors of the tuning's projection matrix (not the temperament's mapping matrix). Only eigenvectors of the projection matrix with eigenvalue equal to 1 are considered eigenmonzos, while those with eigenvalue equal to 0 are the vanishing commas of the temperament; in other words, a vector that is a monzo and an eigenvector is not necessarily an eigenmonzo.

The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form, e.g. [2 -1⟩; for example, 4/3 may be called an eigenmonzo.

See also

• fractional monzo: for more mathematical information
• eigenmonzo subgroup