Defactoring algorithms: Difference between revisions

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This article discusses how to identify enfactoring and then defactor it.

A major use case for defactoring is to enable a [[canonical form]] for temperament mappings, or in other words, to achieve a unique ID for temperaments in the form of a matrix. Previously this was only available by using lists of minor determinants AKA wedge products of mapping rows, which by virtue of reducing the information down to a single list of numbers, could be checked for enfactoring by simply checking the single row's GCD.

A major use case for defactoring is to enable a [[canonical form]] for temperament mappings, or in other words, to achieve a unique ID for temperaments in the form of a matrix. Previously this was only available by using lists of minor determinants AKA wedge products of mapping rows, which by virtue of reducing the information down to a single list of numbers, could be checked for enfactoring by simply checking the single row's GCD<ref>At the time Dave and Douglas began their investigation into Exterior Algebra (EA), most of the math involved in RTT could be handled using only Linear Algebra (LA), a relatively basic and commonplace subject that many people get a chance to learn in high school or university along with subjects like calculus or trigonometry. But there was one crucial task which LA hadn't proven able to handle yet: providing a "fingerprint" — a unique mathematical representation — for each distinct temperament, to allow it to be recognized as the same temperament even though it might be derived in different ways, or in other words, a canonical form for them. For many years, EA had provided this service for RTT, using a structure called a "[[wedgie]]".

Dave and Douglas began their investigations with the hypothesis that canonicalization via wedgies was the primary reason it was important for RTT beginners to learn EA, and that if a canonical form could be developed using only LA, then EA could be reframed as an advanced topic. Gene himself, upon introducing the wedgie (which he initially called a "wedge invariant"), dismissed it as a bad idea to use for identifying temperaments: "Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament." (see: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545)

Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the minor determinants of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding minor (this idea is discussed in more detail in a later section of this article). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.

Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either didn't realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.</ref>.

= Identifying enfactored mappings =

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 This article discusses how to identify enfactoring and then defactor it.
-A major use case for defactoring is to enable a [[canonical form]] for temperament mappings, or in other words, to achieve a unique ID for temperaments in the form of a matrix. Previously this was only available by using lists of minor determinants AKA wedge products of mapping rows, which by virtue of reducing the information down to a single list of numbers, could be checked for enfactoring by simply checking the single row's GCD.
+A major use case for defactoring is to enable a [[canonical form]] for temperament mappings, or in other words, to achieve a unique ID for temperaments in the form of a matrix. Previously this was only available by using lists of minor determinants AKA wedge products of mapping rows, which by virtue of reducing the information down to a single list of numbers, could be checked for enfactoring by simply checking the single row's GCD<ref>At the time Dave and Douglas began their investigation into Exterior Algebra (EA), most of the math involved in RTT could be handled using only Linear Algebra (LA), a relatively basic and commonplace subject that many people get a chance to learn in high school or university along with subjects like calculus or trigonometry. But there was one crucial task which LA hadn't proven able to handle yet: providing a "fingerprint" — a unique mathematical representation — for each distinct temperament, to allow it to be recognized as the same temperament even though it might be derived in different ways, or in other words, a canonical form for them. For many years, EA had provided this service for RTT, using a structure called a "[[wedgie]]".
+<br><br>
+Dave and Douglas began their investigations with the hypothesis that canonicalization via wedgies was the primary reason it was important for RTT beginners to learn EA, and that if a canonical form could be developed using only LA, then EA could be reframed as an advanced topic. Gene himself, upon introducing the wedgie (which he initially called a "wedge invariant"), dismissed it as a bad idea to use for identifying temperaments: "Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament." (see: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545)
+<br><br>
+Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the minor determinants of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding minor (this idea is discussed in more detail in a later section of this article). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.
+<br><br>
+Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either didn't realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.</ref>.
 = Identifying enfactored mappings =