Defactoring: Difference between revisions
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'''Defactoring''' is a operation on the [[mapping]] for a [[regular temperament]] which ensures it represents the same information but without any enfactoring, or in other words, redundancies due to a common factor found in its rows. It is also defined for [[comma | '''Defactoring''' is a operation on the [[mapping]] for a [[regular temperament]] which ensures it represents the same information but without any enfactoring, or in other words, redundancies due to a common factor found in its rows. It is also defined for [[comma basis|comma bases]], the duals of mappings, where it instead checks its columns for enfactoring. | ||
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 [[defactoring#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. | 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 [[defactoring#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. | ||
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# Again, it does not have any obvious musical or mathematical meaning in this context. It's a word that was invented for RTT, so nothing else depends on it<ref>Here is the tuning list post where it was coined by [[Paul Erlich]]: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>. | # Again, it does not have any obvious musical or mathematical meaning in this context. It's a word that was invented for RTT, so nothing else depends on it<ref>Here is the tuning list post where it was coined by [[Paul Erlich]]: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>. | ||
# It was made up due to false assumptions<ref>Authors note: to be absolutely clear, I don’t care who said what or how misconceptions arose (except insofar as it helps dispel any further misconceptions, some of which certainly may be my own). I have basically infinite sympathy for anyone who gets confused over this topic. It took my good friend Dave and I months of back and forth theorization, argumentation, and diagramming before we were able to settle on an explanation we both understood and agreed upon. I am not intending to get in the business of slinging blame (or credit) around. As far as I’m concerned, as long as we can have meaningful discussion with each other, and hopefully eventually arrive at conclusions that are more musically and intellectually empowering than we had previously, then we’re doing well together. Would I have make these mistakes myself? Yes! I have literally dozens of recent emails proving that I would have gone for the same duality myself, due to a case of asymmetry-phobia.</ref>. Through researching on tuning list archives, Dave and Douglas concluded that the associated concept of "torsion" was first described in January of 2002<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2937 which is also referred to here http://tonalsoft.com/enc/t/torsion.aspx</ref>, with regards to commas used to form Fokker periodicity blocks. The concept of enfactoring was recognized in temperament mappings (though of course it did not yet go by that name), and — because torsion in lists of commas for Fokker blocks looks the same way as enfactoring looks in temperament comma | # It was made up due to false assumptions<ref>Authors note: to be absolutely clear, I don’t care who said what or how misconceptions arose (except insofar as it helps dispel any further misconceptions, some of which certainly may be my own). I have basically infinite sympathy for anyone who gets confused over this topic. It took my good friend Dave and I months of back and forth theorization, argumentation, and diagramming before we were able to settle on an explanation we both understood and agreed upon. I am not intending to get in the business of slinging blame (or credit) around. As far as I’m concerned, as long as we can have meaningful discussion with each other, and hopefully eventually arrive at conclusions that are more musically and intellectually empowering than we had previously, then we’re doing well together. Would I have make these mistakes myself? Yes! I have literally dozens of recent emails proving that I would have gone for the same duality myself, due to a case of asymmetry-phobia.</ref>. Through researching on tuning list archives, Dave and Douglas concluded that the associated concept of "torsion" was first described in January of 2002<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2937 which is also referred to here http://tonalsoft.com/enc/t/torsion.aspx</ref>, with regards to commas used to form Fokker periodicity blocks. The concept of enfactoring was recognized in temperament mappings (though of course it did not yet go by that name), and — because torsion in lists of commas for Fokker blocks looks the same way as enfactoring looks in temperament comma bases — torsion got conflated with it<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2405</ref>. But they can't truly be the same thing; the critical difference is that periodicity blocks do not involve tempering, while temperaments do. In concrete terms, while it can make sense to construct a Fokker block with {{vector|-4 4 -1}} in the middle and {{vector|-8 8 -2}} = 2{{vector|-4 4 -1}} at the edge, it does not make sense to imagine a temperament which tempers out 2{{vector|-4 4 -1}} but does not temper out {{vector|-4 4 -1}}. Unfortunately, however, this critical difference seems to have been overlooked, and so it seemed that enfactored comma bases exhibited torsion, and thus because mappings are the dual of comma bases, then enfactoring of a mapping should be the dual of torsion, and because the prefix co- or con- means "dual" (as in vectors and covectors), the term "con-torsion" was coined for it. "Torsion" already has the problem of being an obscure mathematical term that means nothing to most people, "contorsion" just compounds that problem by being made up, and it is made up in order to convey a duality which is false. So while "torsion" could be preserved as a term for the effect on periodicity blocks (though there's almost certainly something more helpful than that, but that's a battle for another day<ref>Perhaps we call it a "shredded periodicity block", due to the way how the paths that the multiple parallel generators take around the block look like shreds of paper, were the periodicity block imagined as a sheet of paper run through a paper shredder.</ref><ref>Furthermore, care should be taken to recognize the difference in behavior between, say<br><br> | ||
<math> | <math> | ||
\left[ | \left[ | ||
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\right] | \right] | ||
</math><br><br> | </math><br><br> | ||
when it is used as a list of 5-limit commas defining a periodicity block versus when it is used as a [[comma | when it is used as a list of 5-limit commas defining a periodicity block versus when it is used as a [[comma basis]] for a temperament, namely, that in the first case the fact that the first column has a common factor of 2 and the second column has a common factor of 3 is meaningful, i.e. the 2-enfactorment will affect one dimension of the block and the 3-enfactorment will affect a different dimension of the block, or in other words, we can say that the commas here are individually enfactored rather than the entire list being enfactored, while in the second case there is no such meaning to the individual columns' factors of 2 and 3, respectively, because it would be equivalent of any form where the product of all the column factors was 6, or in other words, all that matters is that the comma basis as a whole is 6-enfactored here. So perhaps it would be best if, for periodicity blocks, the term "enfactored" was avoided altogether, and instead commas were described as "2-torted".</ref><ref>The explanation for "why 'torsion' in the first place?" is interesting. It comes from group theory (see: https://en.wikipedia.org/wiki/Group_(mathematics)#Uniqueness_of_identity_element). In group theory, to have torsion, a group must have an element that comes back to zero after being chained 2 or more times. The number of times before coming back to zero is called the "order" of the element, sometimes also called the "period length" or "period". When the order is greater than 1 (and less than infinity), the element is said to have torsion, or to be a torsion element, and so the group it is an identity element of is said to have torsion. See also: https://en.wikipedia.org/wiki/Order_(group_theory). Clearly we can't use period (length) because period has another firmly established meaning in xenharmonics. But we could refer to torsion as "finite order greater than one", but that's quite the mouthful while still nearly as obscure.</ref>), they feel it would be better to banish the term "contorsion" from the RTT community altogether. | ||
# A word with the same spelling was also coined with a different mathematical meaning outside of RTT, in the field of differential geometry: https://en.wikipedia.org/wiki/Contorsion_tensor<ref>In this field, it does definitely represent twisting, like in a Möbius strip. Also, DG contorsion is related to DG torsion by subtraction, not duality.</ref> | # A word with the same spelling was also coined with a different mathematical meaning outside of RTT, in the field of differential geometry: https://en.wikipedia.org/wiki/Contorsion_tensor<ref>In this field, it does definitely represent twisting, like in a Möbius strip. Also, DG contorsion is related to DG torsion by subtraction, not duality.</ref> | ||
# It is prone to spelling confusion. People commonly refer to temperaments with contorsion as "contorted". But contorted is the adjective form of a different word, contortion, with a t, not an s. The proper adjective form of contorsion would be contorsioned. Would you use "torted" instead of torsioned? Or would people prefer "torsional" and "contorsional", even though that suggests only of or pertaining to in general rather than having the effect applied.<ref>If it was meant to most strongly evoke duality with torsion, it should have been spelled "cotorsion". Naming it "contorsion" is an annoying step toward "contortion" but stopping halfway there. But this isn't a strong point, because duality with torsion was the false assumption mentioned above.</ref> | # It is prone to spelling confusion. People commonly refer to temperaments with contorsion as "contorted". But contorted is the adjective form of a different word, contortion, with a t, not an s. The proper adjective form of contorsion would be contorsioned. Would you use "torted" instead of torsioned? Or would people prefer "torsional" and "contorsional", even though that suggests only of or pertaining to in general rather than having the effect applied.<ref>If it was meant to most strongly evoke duality with torsion, it should have been spelled "cotorsion". Naming it "contorsion" is an annoying step toward "contortion" but stopping halfway there. But this isn't a strong point, because duality with torsion was the false assumption mentioned above.</ref> | ||
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= The pathology of enfactoredness = | = The pathology of enfactoredness = | ||
In this section, we will use lattices to visualize enfactored temperaments, to demonstrate the musical implications of mappings with common factors, and the lack of musical implications of comma | In this section, we will use lattices to visualize enfactored temperaments, to demonstrate the musical implications of mappings with common factors, and the lack of musical implications of comma bases with common factors. | ||
== Defactored case == | == Defactored case == | ||
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First, let's look at an defactored 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. | First, let's look at an defactored 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. | ||
This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{ket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament tempers out a single comma, whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma | This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{ket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament tempers out a single comma, whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is {{bra|{{vector|-3 2}}}}. | ||
We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is tempered out the comma {{vector|-3 2}} so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction. | We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is tempered out the comma {{vector|-3 2}} so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction. | ||
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And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's not JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but has no meaning with respect to tempering JI, or in other words, no RTT purpose. | And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's not JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but has no meaning with respect to tempering JI, or in other words, no RTT purpose. | ||
And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma | And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma bases, we'll see why things get pretty pathological. | ||
== Enfactored comma | == Enfactored comma bases == | ||
[[File:2-enfactored comma | [[File:2-enfactored comma basis.png|365px|thumb|left|enfactored comma bases are garbage]] | ||
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 | 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 defactored situation, if our comma basis was {{bra|{{vector|-3 2}}}}, then 2-enfactoring it produces 2×{{bra|{{vector|-3 2}}}} = {{bra|{{vector|-6 4}}}}. | ||
We know that in the original diagram, the large-labelled {{vector|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector|-6 4}} is that point instead. Fine. | We know that in the original diagram, the large-labelled {{vector|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector|-6 4}} is that point instead. Fine. | ||
But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is tempered out, then so is {{vector|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to temper out {{vector|-6 4}} without also tempering out {{vector|-3 2}}. And so there is no meaning or purpose to the comma | But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is tempered out, then so is {{vector|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to temper out {{vector|-6 4}} without also tempering out {{vector|-3 2}}. And so there is no meaning or purpose to the comma basis {{vector|-6 4}}, whether RTT-wise or musically in general. It is garbage. | ||
And so our lattice for an enfactored comma | And so our lattice for an enfactored comma basis looks almost identical to the original defactored 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. | ||
== Enfactored comma | == Enfactored comma bases vs. periodicity blocks with torsion == | ||
[[File:Torsion.png|400px|thumb|right|a reworking of the classic torsion example from Tonalsoft to reveal the twinned generator paths]] | [[File:Torsion.png|400px|thumb|right|a reworking of the classic torsion example from Tonalsoft to reveal the twinned generator paths]] | ||
And now we're prepared to confront the key difference between the enfactored comma | And now we're prepared to confront the key difference between the enfactored comma basis of a temperament, and torsion of a periodicity block. | ||
What they have in common is that both take the form of a common factor found somewhere in linear combinations of entries in a list of commas defining a pitch structure, and that these commas can be visualized by slicing the JI lattice into swaths of "periodicity" (that's just a fancy word for the effect we've already been observing for temperaments, where nodes outside the swath related by the size of that comma are considered equivalent and therefore redundant, or repetitions of the same pitch class). | What they have in common is that both take the form of a common factor found somewhere in linear combinations of entries in a list of commas defining a pitch structure, and that these commas can be visualized by slicing the JI lattice into swaths of "periodicity" (that's just a fancy word for the effect we've already been observing for temperaments, where nodes outside the swath related by the size of that comma are considered equivalent and therefore redundant, or repetitions of the same pitch class). | ||
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The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset. | The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset. | ||
So we can see how tempting the duality can be here. In the case of a 2-enfactored mapping, the generator path reaches ''twice'' as many nodes as there were JI nodes. But in the case of a 2-enfactored comma | So we can see how tempting the duality can be here. In the case of a 2-enfactored mapping, the generator path reaches ''twice'' as many nodes as there were JI nodes. But in the case of a 2-enfactored comma basis — if we could legitimately extend the width of the block, as we do in untempered periodicity blocks! — we would reach ''half'' as many nodes. But this duality just is not musically, audibly real. | ||
== Enfactored mappings vs. enfactored comma | == Enfactored mappings vs. enfactored comma bases == | ||
One may pose the question: what is the relationship between an enfactored mapping and an enfactored comma | One may pose the question: what is the relationship between an enfactored mapping and an enfactored comma basis? Can you have one but not the other? Must you? Or must you not? Or does the question even make sense? Certainly at least some have suggested these cases are meaningfully independent<ref>such as the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>. | ||
The conclusion we arrive at here is that because enfactored comma | 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 defactored 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. | ||
== Conclusion == | == Conclusion == | ||
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==== Inversion by hand ==== | ==== Inversion by hand ==== | ||
Now let's take a look at how to do inversion by hand. The first thing to note is that this process only works for rectangular matrices. So you will not be using on non-trivial mappings or comma | Now let's take a look at how to do inversion by hand. The first thing to note is that this process only works for rectangular matrices. So you will not be using on non-trivial mappings or comma bases directly. But, as you know, there is a useful application for this process on the unimodular matrix which is the other result of the Hermite decomposition than the HNF, and unimodular matrices are always square. | ||
Here's a random example matrix (well, one I stole from a quick web search, anyway): | Here's a random example matrix (well, one I stole from a quick web search, anyway): | ||
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* [[MADAM defactoring]] | * [[MADAM defactoring]] | ||
= Canonical comma | = Canonical comma bases = | ||
Canonical form is not only for mappings; comma | 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: [[Douglas_Blumeyer%27s_RTT_How-To#Null-space]]</ref>once at the beginning, and then antitransposed again at the end. | ||
For example, suppose we have the comma | For example, suppose we have the comma basis for septimal meantone: | ||
<math> | <math> | ||
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</math> | </math> | ||
And there's our canonical comma | And there's our canonical comma basis. | ||
= References = | = References = | ||