Eigenmonzo: Difference between revisions
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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. 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 [[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 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. | |||
For any pure-octave temperament tuning, 2/1 is an unchanged interval. | |||
A [[rank]]-''n'' temperament can have up to ''n'' linearly independent unchanged intervals—one for each [[generator]]. | |||
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. | |||
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 == | |||
* [[Eigenmonzo subgroup]] | |||
[[Category:Regular temperament theory]] | |||
[[Category:Terms]] | |||
[[Category:Math]] | |||
Latest revision as of 23:59, 1 January 2026
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 [2 -1⟩ is an eigenmonzo, then we may also refer to the same interval written as a ratio, 4/3, as an eigenmonzo.
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 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.
For any pure-octave temperament tuning, 2/1 is an unchanged interval.
A rank-n temperament can have up to n linearly independent unchanged intervals—one for each generator.
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 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 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.
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).