Defactoring algorithms: Difference between revisions

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

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<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]]".

A major use case for defactoring is to enable a [[canonical form]] for [[regular 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 [[wedgie|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 Keenan]] and [[Douglas Blumeyer]] 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)

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-This article discusses how to identify enfactoring and then defactor it.
+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<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]]".
+A major use case for defactoring is to enable a [[canonical form]] for [[regular 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 [[wedgie|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 Keenan]] and [[Douglas Blumeyer]] 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]]".
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 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)