Defactoring: Difference between revisions

Defactoring a matrix means to perform an operation on it which ensures that it is not "enfactored". And a matrix is considered to be "enfactored" if linear combinations of its rows can produce another row whose elements have a common factor (other than 1). This definition includes matrices whose rows already include a common factor, such as {{map|24 38 56}} which has a common factor of 2. Being enfactored is a bad thing. Enfactored matrices — those in the RTT domain, at least — are sick, in a way<ref>According to [[saturation]], "...if [an RTT matrix] isn't saturated the supposed temperament it defines may be regarded as pathological..." </ref>; it's no accident that "enfactored" sounds sort of like "infected". We'll discuss this pathology in detail in [[canonical_form#the_pathology_of_enfactoredness|a later section of this article]]. Fortunately, the remedy is simple: all one has to do is "defactor" it — identify and divide out the common factor — to produce a healthy mapping.

== unenfactored case ==

First, let's look at an unenfactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.  

If you compare this lattice of an enfactored mapping with the previous lattice for a healthy, unenfactored mapping, they should look almost the same. They have the same comma and tube circumference. And the generator follows the same path through that tube/swath. The key difference is how far the generator moves with each step along that path.  

Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma-basis. In the unenfactored situation, if our comma-basis was {{map|{{vector|-3 2}}}}, then 2-enfactoring it produces 2×{{map|{{vector|-3 2}}}} = {{map|{{vector|-6 4}}}}.

And so our lattice for an enfactored comma-basis looks almost identical to the original unenfactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.

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.

# Everything is still the same now, ''except HNF''. Example: enfactored meantone, e.g. {{vector|{{map|5 8 12}} {{map|14 22 32}}}} causes the HNF to be {{vector|{{map|1 0 -4}} {{map|0 2 8}}}}. This case, like the corresponding unenfactored state, is also quite rare.

ones where there are no common factors to be found in sums or differences of the rows, but are to be found in sums or differences of ''multiples of'' these rows. In order to solve this problem, SAD defactoring would need to be reworked to check every possible combination of integer multiples of each row, potentially up to the largest absolute value of integer in the given mapping, which would cause it to become intractably slow to compute.

Fortunately this did not turn out to be necessary. The solution here was to do one pass of HNF before doing the defactoring (and still do the HNF pass at the end too, when going for a canonical form). This pre-pass of HNF doesn't manage to remove the common factor, but it does at least reveal it. For example, in the case of {{vector|{{map|6 5 -4}} {{map|4 -4 1}}}}, the HNF is {{vector|{{map|2 9 -5}} {{map|0 22 -11}}}}, so that hidden common factor of 11 has now been revealed, put in a form where SAD is capable of removing it.