Eigenmonzo

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An eigenmonzo (also known as an unchanged interval) is a (usually justly intoned) interval that remains unchanged when a specific tuning of a given temperament is applied to it.

A regular temperament transforms untempered intervals into tempered intervals, which changes most of their sizes. Only a small set of exceptional intervals do not change in size. This set of unchanged intervals depends on the choice of 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 "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if [2 -1 is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.

The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are eigenvectors (vectors that are not rotated, only scaled) of the tuning's projection matrix (not the temperament's mapping matrix). The etymology of "eigen" is "own" in the sense of "characteristic"; the set of unrotated vectors and their scale factors are considered to characterize the transformation represented by the matrix. However, only eigenvectors of the projection matrix with eigenvalues (scale factors) equal to 1 (scaled by 1, i.e. unchanged intervals) are considered to be eigenmonzos; eigenvectors with eigenvalues equal to anything else are not considered to be eigenmonzos. In other words, many things that are both monzos and eigenvectors are not eigenmonzos. Most notably, eigenvectors with eigenvalues equal to 0 are the vanishing commas of the temperament, being scaled to 0, but these are not eigenmonzos.

See also