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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.

From the scale's or the instrument's point of view, eigenmonzos have two sides of information.

  • Specify the interval in the scale. (1/4-comma meantone: specifying "M3", which tempers together ~5/4~81/64~6561/5120~...) Then,
  • Tune it to the specified ratio. (1/4-comma meantone: tuning M3 to just 5/4.)

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. But of course it must be able to specify the position in the JI lattice; can't be as 498¢.

See also