Defactoring: Difference between revisions

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The conclusion we arrive at here is that because enfactored comma-bases don't make any sense, or at least don't represent any legitimately new musical information of any kind that their unenfactored version doesn't already represent, it is not generally useful to think of enfactored mappings and enfactored comma-bases as independent phenomena. It only makes sense to speak of enfactored temperaments. Of course, one will often use the term "enfactored mapping" because enfactored mappings are the kind which do have some musical purpose, and often the enfactored mapping will be being used to represent the enfactored temperament — or temperoid, that is.

By the way, canonical form is not only for mappings. Comma-bases may also be put into canonical form, as long as they are first antitransposed[20], and then antitransposed again at the end, or in other words, you sandwich the defactoring and HNF operations between antitransposes.

== conclusion ==

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The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref>In the doubly-enfactored case of {{vector|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of 33 = 3 × 11, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref>It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with -3 of the 2nd row, i.e. 2{{map|6 5 -4}} + -3{{map|4 -4 1}} = {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{vector|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{map|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: -1{{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} = {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.

= canonical comma-bases =

Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[User:Cmloegcmluin/RTT How-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.

For example, suppose we have the comma-basis for septimal meantone:

<math>

\left[ \begin{array} {rrr}

-4 & 1 \\

4 & 2 \\

-1 & -3 \\

0 & 1 \\

\end{array} \right]

</math>

Note that the interval vectors are columns when put into matrix form like this. So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left). This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves.<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>

<math>

\begin{array} {l}

\left[ \begin{array} {rrr}

\colorbox{skyblue}{-4} & \colorbox{yellow}{1} \\

\colorbox{yellow}{4} & \colorbox{pink}{2} \\

\colorbox{springgreen}{-1} & \colorbox{pink}{-3} \\

\colorbox{springgreen}{0} & \colorbox{pink}{1} \\

\end{array} \right]

& → &

\left[ \begin{array} {rrr}

\colorbox{pink}{1} & \colorbox{pink}{-3} & \colorbox{pink}{2} & \colorbox{yellow}{1} \\

\colorbox{springgreen}{0} & \colorbox{springgreen}{-1} & \colorbox{yellow}{4} & \colorbox{skyblue}{-4} \\

\end{array} \right]

\end{array}

</math>

Now we can defactor and HNF this as if it were a mapping.

<math>

\left[ \begin{array} {rrr}

1 & 0 & -10 & -13 \\

0 & 1 & -4 & 4 \\

\end{array} \right]

</math>

Finally we antitranspose again:

<math>

\begin{array} {l}

\left[ \begin{array} {rrr}

\colorbox{pink}{1} & \colorbox{pink}{0} & \colorbox{pink}{-10} & \colorbox{yellow}{-13} \\

\colorbox{springgreen}{0} & \colorbox{springgreen}{1} & \colorbox{yellow}{-4} & \colorbox{skyblue}{4} \\

\end{array} \right]

& → &

\left[ \begin{array} {rrr}

\colorbox{skyblue}{4} & \colorbox{yellow}{13} \\

\colorbox{yellow}{-4} & \colorbox{pink}{-10} \\

\colorbox{springgreen}{1} & \colorbox{pink}{0} \\

\colorbox{springgreen}{0} & \colorbox{pink}{1} \\

\end{array} \right]

\end{array}

</math>

And there's our canonical comma-basis.

= other details to report =

@@ Line 158: / Line 158: @@
 The conclusion we arrive at here is that because enfactored comma-bases don't make any sense, or at least don't represent any legitimately new musical information of any kind that their unenfactored version doesn't already represent, it is not generally useful to think of enfactored mappings and enfactored comma-bases as independent phenomena. It only makes sense to speak of enfactored temperaments. Of course, one will often use the term "enfactored mapping" because enfactored mappings are the kind which do have some musical purpose, and often the enfactored mapping will be being used to represent the enfactored temperament — or temperoid, that is.
-By the way, canonical form is not only for mappings. Comma-bases may also be put into canonical form, as long as they are first antitransposed[20], and then antitransposed again at the end, or in other words, you sandwich the defactoring and HNF operations between antitransposes.
 == conclusion ==
@@ Line 868: / Line 866: @@
 The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref>In the doubly-enfactored case of {{vector|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of 33 = 3 × 11, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref>It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with -3 of the 2nd row, i.e. 2{{map|6 5 -4}} + -3{{map|4 -4 1}} = {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{vector|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{map|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: -1{{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} = {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.
+= canonical comma-bases =
+Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[User:Cmloegcmluin/RTT How-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.
+For example, suppose we have the comma-basis for septimal meantone:
+<math>
+\left[ \begin{array} {rrr}
+-4 & 1 \\
+& 2 \\
+-1 & -3 \\
+& 1 \\
+\end{array} \right]
+</math>
+Note that the interval vectors are columns when put into matrix form like this. So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left). This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves.<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>
+<math>
+\begin{array} {l}
+\left[ \begin{array} {rrr}
+\colorbox{skyblue}{-4} & \colorbox{yellow}{1} \\
+\colorbox{yellow}{4} & \colorbox{pink}{2} \\
+\colorbox{springgreen}{-1} & \colorbox{pink}{-3} \\
+\colorbox{springgreen}{0} & \colorbox{pink}{1} \\
+\end{array} \right]
+& → &
+\left[ \begin{array} {rrr}
+\colorbox{pink}{1} & \colorbox{pink}{-3} & \colorbox{pink}{2} & \colorbox{yellow}{1} \\
+\colorbox{springgreen}{0} & \colorbox{springgreen}{-1} & \colorbox{yellow}{4} & \colorbox{skyblue}{-4} \\
+\end{array} \right]
+\end{array}
+</math>
+Now we can defactor and HNF this as if it were a mapping.
+<math>
+\left[ \begin{array} {rrr}
+& 0 & -10 & -13 \\
+& 1 & -4 & 4 \\
+\end{array} \right]
+</math>
+Finally we antitranspose again:
+<math>
+\begin{array} {l}
+\left[ \begin{array} {rrr}
+\colorbox{pink}{1} & \colorbox{pink}{0} & \colorbox{pink}{-10} & \colorbox{yellow}{-13} \\
+\colorbox{springgreen}{0} & \colorbox{springgreen}{1} & \colorbox{yellow}{-4} & \colorbox{skyblue}{4} \\
+\end{array} \right]
+& → &
+\left[ \begin{array} {rrr}
+\colorbox{skyblue}{4} & \colorbox{yellow}{13} \\
+\colorbox{yellow}{-4} & \colorbox{pink}{-10} \\
+\colorbox{springgreen}{1} & \colorbox{pink}{0} \\
+\colorbox{springgreen}{0} & \colorbox{pink}{1} \\
+\end{array} \right]
+\end{array}
+</math>
+And there's our canonical comma-basis.
 = other details to report =