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

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To demonstrate these points, let’s first calculate the multicomma from a comma basis, and then confirm it by calculating the same multicomma as the dual of its dual multimap.

Here’s the comma basis for meantone: {{map|{{vector|-4 4 -1}}}}. Calculating the multicomma is almost the same as calculating the multimap. The only difference is that as a preliminary step you must transpose the matrix, or in other words, exchange rows and columns. In our bracket notation, that just looks like replacing {{map|{{vector|-4 4 -1}}}} with {{vector|{{map|-4 4 -1}}}}. Now we can see that this is just like our ET map example from the previous section: basically an identity operation, breaking the thing up into three 1×1 matrices <span><math>\begin{bmatrix}-4\end{bmatrix} \begin{bmatrix}4\end{bmatrix} \begin{bmatrix}-1\end{bmatrix}</math></span> which are their own determinants and then nesting back inside one layer of brackets because nullity is 1. So we have {{vector|-4 4 -1}}.

Here’s the comma basis for meantone: {{map|{{vector|-4 4 -1}}}}. Calculating the multicomma is almost the same as calculating the multimap. The only difference is that as a preliminary step you must transpose<ref>Or anti-transpose, if you want to be consistent with other operations discussed here; either way will work here.</ref> the matrix, or in other words, exchange rows and columns. In our bracket notation, that just looks like replacing {{map|{{vector|-4 4 -1}}}} with {{vector|{{map|-4 4 -1}}}}. Now we can see that this is just like our ET map example from the previous section: basically an identity operation, breaking the thing up into three 1×1 matrices <span><math>\begin{bmatrix}-4\end{bmatrix} \begin{bmatrix}4\end{bmatrix} \begin{bmatrix}-1\end{bmatrix}</math></span> which are their own determinants and then nesting back inside one layer of brackets because nullity is 1. So we have {{vector|-4 4 -1}}.

If we want the canonical multicomma, extracting any GCD is necessary. But unlike with the canonical multimap, we can't necessarily assume the leading term's sign will be positive. Since the canonical state is defined in terms of the canonical multimap, if we want to know the signs of the canonical multicomma, we'll have to find the canonical multimap. We can either separately find the canonical multimap, then take its dual, and that'd be our canonical multicomma. Or, we can take the dual of the multicomma we've found here, and if that has a negative leading term, then we just need to change the signs on both the multimap and the multicomma so they remain each other's duals but

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 To demonstrate these points, let’s first calculate the multicomma from a comma basis, and then confirm it by calculating the same multicomma as the dual of its dual multimap.
-Here’s the comma basis for meantone: {{map|{{vector|-4 4 -1}}}}. Calculating the multicomma is almost the same as calculating the multimap. The only difference is that as a preliminary step you must transpose the matrix, or in other words, exchange rows and columns. In our bracket notation, that just looks like replacing {{map|{{vector|-4 4 -1}}}} with {{vector|{{map|-4 4 -1}}}}. Now we can see that this is just like our ET map example from the previous section: basically an identity operation, breaking the thing up into three 1×1 matrices <span><math>\begin{bmatrix}-4\end{bmatrix} \begin{bmatrix}4\end{bmatrix} \begin{bmatrix}-1\end{bmatrix}</math></span> which are their own determinants and then nesting back inside one layer of brackets because nullity is 1. So we have {{vector|-4 4 -1}}.
+Here’s the comma basis for meantone: {{map|{{vector|-4 4 -1}}}}. Calculating the multicomma is almost the same as calculating the multimap. The only difference is that as a preliminary step you must transpose<ref>Or anti-transpose, if you want to be consistent with other operations discussed here; either way will work here.</ref> the matrix, or in other words, exchange rows and columns. In our bracket notation, that just looks like replacing {{map|{{vector|-4 4 -1}}}} with {{vector|{{map|-4 4 -1}}}}. Now we can see that this is just like our ET map example from the previous section: basically an identity operation, breaking the thing up into three 1×1 matrices <span><math>\begin{bmatrix}-4\end{bmatrix} \begin{bmatrix}4\end{bmatrix} \begin{bmatrix}-1\end{bmatrix}</math></span> which are their own determinants and then nesting back inside one layer of brackets because nullity is 1. So we have {{vector|-4 4 -1}}.
 If we want the canonical multicomma, extracting any GCD is necessary. But unlike with the canonical multimap, we can't necessarily assume the leading term's sign will be positive. Since the canonical state is defined in terms of the canonical multimap, if we want to know the signs of the canonical multicomma, we'll have to find the canonical multimap. We can either separately find the canonical multimap, then take its dual, and that'd be our canonical multicomma. Or, we can take the dual of the multicomma we've found here, and if that has a negative leading term, then we just need to change the signs on both the multimap and the multicomma so they remain each other's duals but