Projection: Difference between revisions

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*When a scaling factor is equal to 0, this means the projection scales it down to nothing. These, then, are the commas of the temperament. These are technically unrotated vectors, in the sense that rotation is no longer meaningful for something that has been vanished.

For readers familiar with the linear algebra concepts of [https://mathworld.wolfram.com/Eigenvector.html eigenvector]s and eigenvalues, this all may sound familiar: eigenvector is technical math speak for an "unrotated vector", and eigenvalue is technical math speak for its "scaling factor"; the prefix "eigen-" comes from the German word for "own", ~~and this refers~~ to ~~the fact that the vector is a member of~~ its own [[span]], or in other words, it falls along the infinite line through space we get if we extend the vector in both directions forever (~~after projecting~~ it). For those readers unfamiliar with these ideas, we recognize that this may be a lot to process at once.

For readers familiar with the linear algebra concepts of [https://mathworld.wolfram.com/Eigenvector.html eigenvector]s and eigenvalues, this all may sound familiar: eigenvector is technical math speak for an "unrotated vector", and eigenvalue is technical math speak for its "scaling factor"; the prefix "eigen-" comes from the German word for "own", which we can think of as referring to how it projects onto its own span, or in other words, its projection falls along the infinite line through space we get if we extend the original vector in both directions forever, which is just another way of saying that the projection doesn't rotate the vector off of its original span like it does for most other vectors (in actuality, though, the "eigen" part of "eigenvector" means "own" in a different, but related way: it refers to the fact that the vector is the projection's "own" vector, or in other words, that it ''characterizes'' the projection, which is why another commonly used term for a vector like this is a "characteristic vector"). For those readers unfamiliar with these ideas, we recognize that this may be a lot to process at once.

It may be helpful to visualize the projection as a distortion field across tuning space, with curvy vortices something like how we might see warm and cold fronts on a weather map, or specks of iron patterned by a magnetic field. In these sorts of visualizations, we could imagine the unrotated vectors as the arrows that points along the paths where the distortion pattern happens to come out to be perfectly straight.

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 *When a scaling factor is equal to 0, this means the projection scales it down to nothing. These, then, are the commas of the temperament. These are technically unrotated vectors, in the sense that rotation is no longer meaningful for something that has been vanished.
-For readers familiar with the linear algebra concepts of [https://mathworld.wolfram.com/Eigenvector.html eigenvector]s and eigenvalues, this all may sound familiar: eigenvector is technical math speak for an "unrotated vector", and eigenvalue is technical math speak for its "scaling factor"; the prefix "eigen-" comes from the German word for "own", and this refers to the fact that the vector is a member of its own [[span]], or in other words, it falls along the infinite line through space we get if we extend the vector in both directions forever (after projecting it). For those readers unfamiliar with these ideas, we recognize that this may be a lot to process at once.
+For readers familiar with the linear algebra concepts of [https://mathworld.wolfram.com/Eigenvector.html eigenvector]s and eigenvalues, this all may sound familiar: eigenvector is technical math speak for an "unrotated vector", and eigenvalue is technical math speak for its "scaling factor"; the prefix "eigen-" comes from the German word for "own", which we can think of as referring to how it projects onto its own span, or in other words, its projection falls along the infinite line through space we get if we extend the original vector in both directions forever, which is just another way of saying that the projection doesn't rotate the vector off of its original span like it does for most other vectors (in actuality, though, the "eigen" part of "eigenvector" means "own" in a different, but related way: it refers to the fact that the vector is the projection's "own" vector, or in other words, that it ''characterizes'' the projection, which is why another commonly used term for a vector like this is a "characteristic vector"). For those readers unfamiliar with these ideas, we recognize that this may be a lot to process at once.
 It may be helpful to visualize the projection as a distortion field across tuning space, with curvy vortices something like how we might see warm and cold fronts on a weather map, or specks of iron patterned by a magnetic field. In these sorts of visualizations, we could imagine the unrotated vectors as the arrows that points along the paths where the distortion pattern happens to come out to be perfectly straight.