Eigenmonzo: Difference between revisions

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== With respect to the projection matrix ==

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]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged) are considered to be eigenmonzos, while those with eigenvalue equal to 0 — which are the vanishing commas of the temperament, being that they are scaled by 0 — are ''not'' considered to be eigenmonzos. In other words, some things that are both monzos and eigenvectors are not eigenmonzos, most notably any vanishing comma of any temperament.

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.

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 == With respect to the projection matrix ==
-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]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged) are considered to be eigenmonzos, while those with eigenvalue equal to 0 — which are the vanishing commas of the temperament, being that they are scaled by 0 — are ''not'' considered to be eigenmonzos. In other words, some things that are both monzos and eigenvectors are not eigenmonzos, most notably any vanishing comma of any temperament.
 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.