Projection: Difference between revisions

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Mappings combine types (1) and (3). Generator embeddings combine types (2) and (3). Projections combine types (1) and (2). So each possible subset of two of these pieces of information is accounted for by these three objects.

One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see: [[~~https://en.xen.wiki/w/Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT~~#~~Pure_representation_of_temperament_information~~]]

One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Pure representation of temperament information]].

In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the diagram to the right, with type (1) red, type (2) blue, type (3) green.

@@ Line 1,704: / Line 1,704: @@
 Mappings combine types (1) and (3). Generator embeddings combine types (2) and (3). Projections combine types (1) and (2). So each possible subset of two of these pieces of information is accounted for by these three objects.
-One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see: [[https://en.xen.wiki/w/Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#Pure_representation_of_temperament_information]]
+One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Pure representation of temperament information]].
 In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the diagram to the right, with type (1) red, type (2) blue, type (3) green.