Defactoring algorithms: Difference between revisions

Line 46:

</math>

This is a form of 5-limit [[porcupine]], a [[rank-2]] temperament. Looking at either row, neither map has a common factor. But remember that we also need to check linear combinations of rows. If we subtract the 2nd row from the 1st row, we can produce the row {{map|3 -3 -6}}, which has a common factor of 3. So this mapping is also enfactored, even though it's not obvious from just looking at it.

This is a form of 5-limit [[porcupine]], a [[rank-2]] temperament (in fact it is the result of putting it into [[IRREF]]). Looking at either row, neither map has a common factor. But remember that we also need to check linear combinations of rows. If we subtract the 2nd row from the 1st row, we can produce the row {{map|3 -3 -6}}, which has a common factor of 3. So this mapping is also enfactored, even though it's not obvious from just looking at it.

If you're unsure why this {{map|3 -3 -6}} matters despite not being in {{ket|{{map|3 0 -1}} {{map|0 3 5}}}}, we may need to quickly review some linear algebra fundamentals. It may take some getting used to, but a mapping can be changed to another equivalent mapping (both mappings will map input vectors to the same scalars) by replacing any row with linear combinations of its rows. That is, we could replace either {{map|3 0 -1}} or {{map|0 3 5}} in our original matrix {{ket|{{map|3 0 -1}} {{map|0 3 5}}}} to get {{ket|{{map|3 -3 -6}} {{map|0 3 5}}}} or {{ket|{{map|3 0 -1}} {{map|3 -3 -6}}}} and any of these mappings represent the same temperament.

Another classic example of hidden enfactoring is the one used on [http://tonalsoft.com/enc/t/torsion.aspx Tonalsoft's article for torsion]. As a matrix, the pair of commas [[648/625]] and [[2048/2025]] for [[Joe Monzo]]'s periodicity block there would look like:

<math>

\left[

\begin{array} {r}

3 & 11 \\

4 & -4 \\

-4 & -2 \\

\end{array}

\right]

</math>

No column has a GCD > 1. And yet, if we subtract the first comma from the second, we get {{vector|11 -4 -2}} - {{vector|3 4 -4}} = {{vector|8 -8 2}}, which is clearly 2×{{vector|4 -4 2}}.

== Well-hidden enfactoring ==

@@ Line 46: / Line 46: @@
 </math>
-This is a form of 5-limit [[porcupine]], a [[rank-2]] temperament. Looking at either row, neither map has a common factor. But remember that we also need to check linear combinations of rows. If we subtract the 2nd row from the 1st row, we can produce the row {{map|3 -3 -6}}, which has a common factor of 3. So this mapping is also enfactored, even though it's not obvious from just looking at it.
+This is a form of 5-limit [[porcupine]], a [[rank-2]] temperament (in fact it is the result of putting it into [[IRREF]]). Looking at either row, neither map has a common factor. But remember that we also need to check linear combinations of rows. If we subtract the 2nd row from the 1st row, we can produce the row {{map|3 -3 -6}}, which has a common factor of 3. So this mapping is also enfactored, even though it's not obvious from just looking at it.
 If you're unsure why this {{map|3 -3 -6}} matters despite not being in {{ket|{{map|3 0 -1}} {{map|0 3 5}}}}, we may need to quickly review some linear algebra fundamentals. It may take some getting used to, but a mapping can be changed to another equivalent mapping (both mappings will map input vectors to the same scalars) by replacing any row with linear combinations of its rows. That is, we could replace either {{map|3 0 -1}} or {{map|0 3 5}} in our original matrix {{ket|{{map|3 0 -1}} {{map|0 3 5}}}} to get {{ket|{{map|3 -3 -6}} {{map|0 3 5}}}} or {{ket|{{map|3 0 -1}} {{map|3 -3 -6}}}} and any of these mappings represent the same temperament.
+Another classic example of hidden enfactoring is the one used on [http://tonalsoft.com/enc/t/torsion.aspx Tonalsoft's article for torsion]. As a matrix, the pair of commas [[648/625]] and [[2048/2025]] for [[Joe Monzo]]'s periodicity block there would look like:
+<math>
+\left[
+\begin{array} {r}
+& 11 \\
+& -4 \\
+-4 & -2 \\
+\end{array}
+\right]
+</math>
+No column has a GCD > 1. And yet, if we subtract the first comma from the second, we get {{vector|11 -4 -2}} - {{vector|3 4 -4}} = {{vector|8 -8 2}}, which is clearly 2×{{vector|4 -4 2}}.
 == Well-hidden enfactoring ==