Eigenmonzo: Difference between revisions
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A [[rank]]-''n'' temperament can have up to ''n'' different eigenmonzos — one for each [[generator]]. | A [[rank]]-''n'' temperament can have up to ''n'' different eigenmonzos — one for each [[generator]]. | ||
== With respect to the projection matrix == | == With respect to the projection matrix == | ||
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The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors|eigenvectors]] of the tuning's [[projection matrix]] (not the [[mapping|temperament's mapping matrix]]). Only eigenvectors of the projection matrix with [[wikipedia: 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 "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors|eigenvectors]] of the tuning's [[projection matrix]] (not the [[mapping|temperament's mapping matrix]]). Only eigenvectors of the projection matrix with [[wikipedia: 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 "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form, | The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo. | ||
== See also == | == See also == | ||
Revision as of 16:39, 16 September 2022
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 "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. 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.
See also
- Fractional monzo: for more mathematical information
- Eigenmonzo subgroup