Eigenmonzo: Difference between revisions

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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 ==

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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 "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.

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

== See also ==

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 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 ==
@@ Line 13: / Line 17: @@
 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, e.g. {{monzo| 2 -1 }}; for example, 4/3 may be called an eigenmonzo.
+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. But of course it must be able to specify the position in the JI lattice; can't be as 498¢.
 == See also ==