Defactoring: Difference between revisions

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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>.

# 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>. ~~More on this in the next subsection.~~

# 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

# A word with the same spelling was also coined with 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>

# Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.

~~In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored.~~

~~=== deeper dive on concerns re: contorsion concept; enfactorization illustrated with lattices to demonstrate pathology of temperoids ===~~

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

<math>

\left[

Line 72:

Line 63:

</math>

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>), the term "contorsion" must be banished from the RTT community altogether.

# A word with the same spelling was also coined with 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>

# Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.

In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored.

== 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-bases with common factors.

=== unenfactored case ===

[[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]

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.

This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{vector|{{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 {{map|{{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.

There's a couple good ways to interpret this situation:

# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.

# We've rolled up space, so that the line from 1/1 to 9/8 ₋ the width of our swath ₋ is like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).

Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.

You can see that in the grey lattice underneath, coordinates have 2 values, e.g. {{vector|-1 2}}. That's because the JI lattice essentially had two generators: the octave and the tritave. But tempering has helped us simplify things by reducing us to a single generator, so here, in the new blue-colored tempered lattice, the coordinates have only 1 value, e.g. {{vector|8}}, and they simply indicate how many iterations of the single generator here that we've taken to reach the given point.

We've superimposed the tempered lattice atop the former JI lattice so we can see which JI intervals map to which tempered intervals. For example, {{vector|-1 2}}, AKA 9/2, maps to {{vector|8}}. That tells us that if we want to use the temperament's approximation of the JI interval 9/2, then we want the tempered pitch arrived at by moving by the generator 8 times.

Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.

The tempered vector for the generator of this temperament is of course {{vector|1}}, but the JI preimage of the generator of this temperament is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.

All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!

=== enfactored mapping ===

[[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]]

We are now comparing the previous diagram, which had the mapping {{vector|{{map|2 3}}}}, with the 2-enfactored version of it, i.e. the mapping 2×{{vector|{{map|2 3}}}} = {{vector|{{map|4 6}}}}, AKA 4-ET.

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.

Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.

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-bases, we'll see why things get pretty pathological.

=== enfactored comma-bases ===

[[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-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}}}}.

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-basis {{vector|-6 4}}, whether RTT-wise or musically in general. It is garbage.

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.

=== 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]]

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).

The key difference is that the former (temperaments) tempers the commas out, while the latter (periodicity blocks) does not.

Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector|-6 4}} was because the tempering: if {{vector|-6 4}} is tempered out, then {{vector|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: 2048/2025 and 625/324. The value at the fourth corner, 12800/6561, has vector 2×{{vector|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.

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-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-bases ===

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 thinkers have suggested these cases are meaningfully independent<ref>such as in [[Kite Giedraitis]]'s writings on [[Color_notation/Temperament_Names|color notation]], which read "it's possible that there is both torsion and contorsion"</ref>.

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.

By the way, DCF is not only for mappings. Comma-bases may also be put into DCF, as long as they are first antitransposed[20], and then antitransposed again at the end, or in other words, you sandwich the defactoring and HNF operations between antitransposes.

=== conclusion ===

So due to the high likelihood for confusion when conceptualizing enfactored temperaments, we believe that using HNF as the unique ID for temperaments, i.e. treating temperoids as true temperaments, is a dangerous and unhelpful road. Instead, DCF should be used, essentially HNF but with defactoring built in, so that practitioners of RTT can focus on working with true temperaments.

== identifying enfactored mappings ==

@@ Line 52: / Line 52: @@
 # 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>. More on this in the next subsection.
+# 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>
-# A word with the same spelling was also coined with 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>
-# Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.
-In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored.
-=== deeper dive on concerns re: contorsion concept; enfactorization illustrated with lattices to demonstrate pathology of temperoids ===
-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>
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 \left[
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 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>), the term "contorsion" must be banished from the RTT community altogether.
+# A word with the same spelling was also coined with 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>
+# Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.
+In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored.
+== 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-bases with common factors.
+=== unenfactored case ===
+[[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]
+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.
+This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{vector|{{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 {{map|{{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.
+There's a couple good ways to interpret this situation:
+# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.
+# We've rolled up space, so that the line from 1/1 to 9/8 ₋ the width of our swath ₋ is like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).
+Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.
+You can see that in the grey lattice underneath, coordinates have 2 values, e.g. {{vector|-1 2}}. That's because the JI lattice essentially had two generators: the octave and the tritave. But tempering has helped us simplify things by reducing us to a single generator, so here, in the new blue-colored tempered lattice, the coordinates have only 1 value, e.g. {{vector|8}}, and they simply indicate how many iterations of the single generator here that we've taken to reach the given point.
+We've superimposed the tempered lattice atop the former JI lattice so we can see which JI intervals map to which tempered intervals. For example, {{vector|-1 2}}, AKA 9/2, maps to {{vector|8}}. That tells us that if we want to use the temperament's approximation of the JI interval 9/2, then we want the tempered pitch arrived at by moving by the generator 8 times.
+Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.
+The tempered vector for the generator of this temperament is of course {{vector|1}}, but the JI preimage of the generator of this temperament is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.
+All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!
+=== enfactored mapping ===
+[[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]]
+We are now comparing the previous diagram, which had the mapping {{vector|{{map|2 3}}}}, with the 2-enfactored version of it, i.e. the mapping 2×{{vector|{{map|2 3}}}} = {{vector|{{map|4 6}}}}, AKA 4-ET.
+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.
+Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.
+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-bases, we'll see why things get pretty pathological.
+=== enfactored comma-bases ===
+[[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-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}}}}.
+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-basis {{vector|-6 4}}, whether RTT-wise or musically in general. It is garbage.
+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.
+=== 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]]
+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).
+The key difference is that the former (temperaments) tempers the commas out, while the latter (periodicity blocks) does not.
+Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector|-6 4}} was because the tempering: if {{vector|-6 4}} is tempered out, then {{vector|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.
+Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: 2048/2025 and 625/324. The value at the fourth corner, 12800/6561, has vector 2×{{vector|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.
+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-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-bases ===
+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 thinkers have suggested these cases are meaningfully independent<ref>such as in [[Kite Giedraitis]]'s writings on [[Color_notation/Temperament_Names|color notation]], which read "it's possible that there is both torsion and contorsion"</ref>.
+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.
+By the way, DCF is not only for mappings. Comma-bases may also be put into DCF, as long as they are first antitransposed[20], and then antitransposed again at the end, or in other words, you sandwich the defactoring and HNF operations between antitransposes.
+=== conclusion ===
+So due to the high likelihood for confusion when conceptualizing enfactored temperaments, we believe that using HNF as the unique ID for temperaments, i.e. treating temperoids as true temperaments, is a dangerous and unhelpful road. Instead, DCF should be used, essentially HNF but with defactoring built in, so that practitioners of RTT can focus on working with true temperaments.
 == identifying enfactored mappings ==