Douglas Blumeyer's RTT How-To: Difference between revisions

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Why care about multimaps? Well, a key reason is that they can serve the same purpose as the canonical form of a temperament’s map basis: the process for converting a map basis to a multimap will convert any equivalent map basis to the same exact multimap. In other words, a multimap can serve as a unique identifier for its temperament.

Alright, then, sounds great! But how do I convert a map basis to a multimap? The process is doable. It’s closely related to the wedge product (hence the name “wedgie”), which we write it with the symbol ∧.

Alright, then, sounds great! But how do I convert a map basis to a multimap? The process is doable. It’s closely related to the wedge product<ref>The wedge product is also known as the exterior product. Exterior and interior products (∧ and ⨼) should not be confused with outer and inner products, which in RTT are ordinary matrix products between a vector and a covector. The outer product gives a matrix and the inner product a scalar. e.g. [1 2⟩⟨3 4] = ⟨[3 4⟩ [6 8⟩] and ⟨3 4][1 2⟩ = 11. In our application, the inner product is the same as the dot product used for maps and intervals.</ref> (hence the name “wedgie”), which we write it with the symbol ∧.

First I’ll list the steps. Don’t worry if it doesn’t all make sense the first time. We’ll work through an example and go into more detail as we do. To be clear, what we're doing here is both more and less and different ways from the strict definition of the wedge product as you may see it elsewhere; I'm specifically here describing the process for finding the multimap in the form you're going to be interested in for RTT purposes.

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 Why care about multimaps? Well, a key reason is that they can serve the same purpose as the canonical form of a temperament’s map basis: the process for converting a map basis to a multimap will convert any equivalent map basis to the same exact multimap. In other words, a multimap can serve as a unique identifier for its temperament.
-Alright, then, sounds great! But how do I convert a map basis to a multimap? The process is doable. It’s closely related to the wedge product (hence the name “wedgie”), which we write it with the symbol ∧.
+Alright, then, sounds great! But how do I convert a map basis to a multimap? The process is doable. It’s closely related to the wedge product<ref>The wedge product is also known as the exterior product. Exterior and interior products (∧ and ⨼) should not be confused with outer and inner products, which in RTT are ordinary matrix products between a vector and a covector. The outer product gives a matrix and the inner product a scalar. e.g. [1 2⟩⟨3 4] = ⟨[3 4⟩ [6 8⟩] and ⟨3 4][1 2⟩ = 11. In our application, the inner product is the same as the dot product used for maps and intervals.</ref> (hence the name “wedgie”), which we write it with the symbol ∧.
 First I’ll list the steps. Don’t worry if it doesn’t all make sense the first time. We’ll work through an example and go into more detail as we do. To be clear, what we're doing here is both more and less and different ways from the strict definition of the wedge product as you may see it elsewhere; I'm specifically here describing the process for finding the multimap in the form you're going to be interested in for RTT purposes.