Generator embedding optimization: Difference between revisions

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Two of these four specialized methods were briefly discussed in D&D's guide, along with the general method, because these specialized methods are actually even quicker and easier than the general method. These two are the only held-intervals method, and the pseudoinverse method. But there's still plenty more insight to be had into how and why exactly these methods work, in particular for the pseudoinverse method, so we'll be doing a much deeper dive into it in this article here than was done in D&D's guide.

The other two of these four specialized ~~methods — the~~ zero-damage method, and the coinciding-damage method—are significantly more challenging to understand than the general method. Most students of RTT would not gain enough musical insight by familiarizing themselves with them to have justified the investment. This is why these two methods were not discussed in D&D's guide. However, if you feel compelled to understand the nuts and bolts of these methods anyway, then those sections of the article may well appeal to you.

The other two of these four specialized methods—the zero-damage method, and the coinciding-damage method—are significantly more challenging to understand than the general method. Most students of RTT would not gain enough musical insight by familiarizing themselves with them to have justified the investment. This is why these two methods were not discussed in D&D's guide. However, if you feel compelled to understand the nuts and bolts of these methods anyway, then those sections of the article may well appeal to you.

This article is titled "Generator embedding optimization" because of a key feature these four specialized methods share: they can all give their solutions as [[generator embedding matrix|generator embedding]]s, i.e. lists of [[prime-count vector]]s, one for each generator, where typically these prime-count vectors have non-integer entries (and are thus not [[JI]]). This is different from the general method, which can only give [[generator tuning map]]s, i.e. sizes in cents for each generator. As we'll see, a tuning optimization method's ability to give solutions as generator embeddings is equivalent to its ability to give solutions that are exact.

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 Two of these four specialized methods were briefly discussed in D&D's guide, along with the general method, because these specialized methods are actually even quicker and easier than the general method. These two are the only held-intervals method, and the pseudoinverse method. But there's still plenty more insight to be had into how and why exactly these methods work, in particular for the pseudoinverse method, so we'll be doing a much deeper dive into it in this article here than was done in D&D's guide.
-The other two of these four specialized methods — the zero-damage method, and the coinciding-damage method&mdash;are significantly more challenging to understand than the general method. Most students of RTT would not gain enough musical insight by familiarizing themselves with them to have justified the investment. This is why these two methods were not discussed in D&D's guide. However, if you feel compelled to understand the nuts and bolts of these methods anyway, then those sections of the article may well appeal to you.
+The other two of these four specialized methods&mdash;the zero-damage method, and the coinciding-damage method&mdash;are significantly more challenging to understand than the general method. Most students of RTT would not gain enough musical insight by familiarizing themselves with them to have justified the investment. This is why these two methods were not discussed in D&D's guide. However, if you feel compelled to understand the nuts and bolts of these methods anyway, then those sections of the article may well appeal to you.
 This article is titled "Generator embedding optimization" because of a key feature these four specialized methods share: they can all give their solutions as [[generator embedding matrix|generator embedding]]s, i.e. lists of [[prime-count vector]]s, one for each generator, where typically these prime-count vectors have non-integer entries (and are thus not [[JI]]). This is different from the general method, which can only give [[generator tuning map]]s, i.e. sizes in cents for each generator. As we'll see, a tuning optimization method's ability to give solutions as generator embeddings is equivalent to its ability to give solutions that are exact.