Eigenmonzo: Difference between revisions
Cmloegcmluin (talk | contribs) →With respect to the projection matrix: improve clarity |
Cmloegcmluin (talk | contribs) →With respect to the projection matrix: better if this point comes first, so the points in the next paragraph are less confusing |
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== With respect to the projection matrix == | == With respect to the projection matrix == | ||
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. | |||
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 "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. | ||
== See also == | == See also == | ||