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

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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 exterior 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.

Take each combination of <math>r</math> primes where <math>r</math> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <math>(2,3,5)</math>, <math>(2,3,7)</math>, <math>(2,5,7)</math>, and <math>(3,5,7)</math>. ~~Now convert~~ each of those combinations to a square <math>r×r</math> matrix by slicing a column for each prime out of the mapping and putting them together. ~~Now take~~ each matrix's determinant. Extract the GCD from the ~~resulting sequence of scalars, then set them~~ inside <math>r</math> brackets~~, and you've got your multimap.~~

# Take each combination of <math>r</math> primes where <math>r</math> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <math>(2,3,5)</math>, <math>(2,3,7)</math>, <math>(2,5,7)</math>, and <math>(3,5,7)</math>.

# Convert each of those combinations to a square <math>r×r</math> matrix by slicing a column for each prime out of the mapping and putting them together.

# Take each matrix's determinant.

# Flip the sign of every result if the first result is negative.

# Extract the GCD from these results.

# Set the results inside <math>r</math> brackets

Let’s work through the meantone ~~example~~.

And you've got your multimap!

Let’s work through these steps for the example of meantone.

We have rank <math>r</math> = 2, so we’re looking for every combination of two primes. That’s out of the three total primes we have in the 5-limit: 2, 3, and 5. So those combinations are <math>(2,3)</math>, <math>(2,5)</math>, and <math>(3,5)</math>. Those are already in lexicographic order, or in other words, just like how alphabetic order works, but generalized to work for size of numbers too (so that 11 comes after 2, not before).

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And so our results are <math>-2</math>, <math>3</math>, <math>1</math>, <math>-11</math>. ~~At first glance we don’t have a~~ GCD~~, but there is a slight trick here~~. We prefer for the first term to be positive~~. It~~ doesn’t make a difference, but is ~~a way to normalize~~ things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). So, ~~we can simply~~ think of this as extracting a GCD of -1, and our list ends up as <math>2</math>, <math>-3</math>, <math>-1</math>, <math>11</math>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}.

And so our results are <math>-2</math>, <math>3</math>, <math>1</math>, <math>-11</math>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). And so we flip the signs<ref>If it helps you, you could think of this sign-flipping step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <math>2</math>, <math>-3</math>, <math>-1</math>, <math>11</math>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}.

As for getting from the multimap back to the mapping, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never have the multimap without the mapping anyway.

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 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 exterior 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.
-Take each combination of <span><math>r</math></span> primes where <span><math>r</math></span> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <span><math>(2,3,5)</math></span>, <span><math>(2,3,7)</math></span>, <span><math>(2,5,7)</math></span>, and <span><math>(3,5,7)</math></span>. Now convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the mapping and putting them together. Now take each matrix's determinant. Extract the GCD from the resulting sequence of scalars, then set them inside <span><math>r</math></span> brackets, and you've got your multimap.
+# Take each combination of <span><math>r</math></span> primes where <span><math>r</math></span> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <span><math>(2,3,5)</math></span>, <span><math>(2,3,7)</math></span>, <span><math>(2,5,7)</math></span>, and <span><math>(3,5,7)</math></span>.
+# Convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the mapping and putting them together.
+# Take each matrix's determinant.
+# Flip the sign of every result if the first result is negative.
+# Extract the GCD from these results.
+# Set the results inside <span><math>r</math></span> brackets
-Let’s work through the meantone example.
+And you've got your multimap!
+Let’s work through these steps for the example of meantone.
 We have rank <span><math>r</math></span> = 2, so we’re looking for every combination of two primes. That’s out of the three total primes we have in the 5-limit: 2, 3, and 5. So those combinations are <span><math>(2,3)</math></span>, <span><math>(2,5)</math></span>, and <span><math>(3,5)</math></span>. Those are already in lexicographic order, or in other words, just like how alphabetic order works, but generalized to work for size of numbers too (so that 11 comes after 2, not before).
@@ Line 1,232: / Line 1,239: @@
 |}
-And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. At first glance we don’t have a GCD, but there is a slight trick here. We prefer for the first term to be positive. It doesn’t make a difference, but is a way to normalize things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). So, we can simply think of this as extracting a GCD of -1, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}.
+And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). And so we flip the signs<ref>If it helps you, you could think of this sign-flipping step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}.
 As for getting from the multimap back to the mapping, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never have the multimap without the mapping anyway.