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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.
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. The term "eigenmonzo" does not imply that the interval is notated in monzo form, so that if 2.3 {{Monzo|2 -1}} is an eigenmonzo, then we may also refer to the same interval written as a ratio, 4/3, as an eigenmonzo.


A popular example of an eigenmonzo is the JI interval {{monzo|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.
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.  


For any pure-octave temperament tuning, {{monzo|1}}, aka 1:2, is an eigenmonzo.
A popular example of an unchanged interval is the JI interval 5/1, 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 5/1, it remains justly tuned.


A [[rank]]-n temperament can have up to n different eigenmonzos — one for each [[generator]].
For any pure-octave temperament tuning, 2/1 is an unchanged interval.


== With respect to the projection matrix ==
A [[rank]]-''n'' temperament can have up to ''n'' linearly independent unchanged intervals—one for each [[generator]].


The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors eigenvectors] of the tuning's [[projection matrix]] (not the [[Temperament_mapping_matrices|temperament's mapping matrix]]). Only eigenvectors of the projection matrix with [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors 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 term "eigenmonzo" here comes from the [[linear algebra formalism]], where intervals are often represented as vectors corresponding to their [[monzos]] (and thus instances of "vector" are often replaced with "monzo"). An [[wikipedia: Eigenvalues and eigenvectors|eigenvector]] is a vector that is not rotated (only scaled) by a 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. In this case, the transformation matrix is the [[projection]] corresponding to the tuning of the regular temperament, which gives the conflations of the just bases with [[Radical interval|radical intervals]], such as 3/2 to 5^(1/4). Note that this is ''not'' the matrix corresponding to the [[mapping]], which cannot specify a precise tuning.


The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form, e.g. {{monzo|2 -1}}; for example, 4/3 may be called an eigenmonzo.
However, the definition of eigenmonzo is more precise; along with corresponding to an eigenvector, it must be scaled by a factor of 1 (i.e. left unchanged). In other words, there are many intervals that correspond to eigenvectors of a projection but that are not unchanged intervals in the corresponding tuning (most notably, any comma tempered out by a temperament is scaled to 0, but that is obviously not unchanged).


== See also ==
== See also ==
* [[Eigenmonzo subgroup]]


* [[fractional monzo]]: for more mathematical information
[[Category:Regular temperament theory]]
* [[eigenmonzo subgroup]]
[[Category:Terms]]
[[Category:Math]]
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