Defactoring: Difference between revisions

Line 34:

# It has another unrelated meaning within xenharmonics that it would conflict with: https://en.xen.wiki/w/Anomalous_saturated_suspension

# The most common everyday usage of that word is for "saturated fats", which are the bad kind of fats, so it has negative associations, despite "saturation" being the ''good'' state for a matrix to be in.

# Research on the tuning list archives suggests that [[Gene Ward Smith]] chose the word "saturation" because it was used in the mathematical software he was using at the time, Sage<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18026.html and https://doc.sagemath.org/html/en/reference/search.html?q=index_in_saturation. There's also https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense_saturation.html, which refers to an "index in saturation" which appears to be the common factor:

# Research on the tuning list archives suggests that [[Gene Ward Smith]] chose the word "saturation" because it was used in the mathematical software he was using at the time, Sage. However, no information about saturation indices has been identified elsewhere online. Certainly, if this mathematical concept of saturation is established outside of this particular piece of software, it is not well-established enough yet such that it's worth deferring to<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18026.html and https://doc.sagemath.org/html/en/reference/search.html?q=index_in_saturation. There's also https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense_saturation.html, which refers to an "index in saturation" which appears to be the common factor:

sage: C

Line 46:

sage: S.index_in_saturation()

1

Indeed 18 is the common factor in {{vector|{{map|11 16 21}} {{map|19 26 33}}}}. No information about saturation indices is readily available elsewhere online. Certainly, if this mathematical concept of saturation is established outside of this particular piece of software, it is not well-established enough yet such that it gets any presence.</ref>. We think we should avoid propagating Sage's decision to overload matrix saturation with a second meaning.

Indeed 18 is the common factor in {{vector|{{map|11 16 21}} {{map|19 26 33}}}}.</ref>. We think we should avoid propagating Sage's decision to overload matrix saturation with a second meaning.

# Furthermore, there is another common but conflicting sense of saturation for matrices which clamps entry values to between -1 and 1<ref>See https://math.stackexchange.com/questions/1964814/linear-transformation-of-a-saturated-vector and https://faculty.uml.edu//thu/tcs01-june.pdf</ref>.

As for the term "contorsion", the concerns with it are:

# Again, it does not have any obvious musical or mathematical meaning in this context.

# 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's a word that was invented for RTT ~~and has no meaning outside of RTT~~<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 64:

Line 72:

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

~~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 ===~~

== identifying enfactored mappings ==

@@ Line 34: / Line 34: @@
 # It has another unrelated meaning within xenharmonics that it would conflict with: https://en.xen.wiki/w/Anomalous_saturated_suspension
 # The most common everyday usage of that word is for "saturated fats", which are the bad kind of fats, so it has negative associations, despite "saturation" being the ''good'' state for a matrix to be in.
-# Research on the tuning list archives suggests that [[Gene Ward Smith]] chose the word "saturation" because it was used in the mathematical software he was using at the time, Sage<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18026.html and https://doc.sagemath.org/html/en/reference/search.html?q=index_in_saturation. There's also https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense_saturation.html, which refers to an "index in saturation" which appears to be the common factor:<br>
+# Research on the tuning list archives suggests that [[Gene Ward Smith]] chose the word "saturation" because it was used in the mathematical software he was using at the time, Sage. However, no information about saturation indices has been identified elsewhere online. Certainly, if this mathematical concept of saturation is established outside of this particular piece of software, it is not well-established enough yet such that it's worth deferring to<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18026.html and https://doc.sagemath.org/html/en/reference/search.html?q=index_in_saturation. There's also https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense_saturation.html, which refers to an "index in saturation" which appears to be the common factor:<br>
 <span style="font-family: monospace;"><br>
 sage: C<br>
@@ Line 46: / Line 46: @@
 sage: S.index_in_saturation()<br>
 <br></span><br>
-Indeed 18 is the common factor in {{vector|{{map|11 16 21}} {{map|19 26 33}}}}. No information about saturation indices is readily available elsewhere online. Certainly, if this mathematical concept of saturation is established outside of this particular piece of software, it is not well-established enough yet such that it gets any presence.</ref>. We think we should avoid propagating Sage's decision to overload matrix saturation with a second meaning.
+Indeed 18 is the common factor in {{vector|{{map|11 16 21}} {{map|19 26 33}}}}.</ref>. We think we should avoid propagating Sage's decision to overload matrix saturation with a second meaning.
 # Furthermore, there is another common but conflicting sense of saturation for matrices which clamps entry values to between -1 and 1<ref>See https://math.stackexchange.com/questions/1964814/linear-transformation-of-a-saturated-vector and https://faculty.uml.edu//thu/tcs01-june.pdf</ref>.
 As for the term "contorsion", the concerns with it are:
-# Again, it does not have any obvious musical or mathematical meaning in this context.
+# 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's a word that was invented for RTT and has no meaning outside of RTT<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>
 <math>
 \left[
@@ Line 64: / Line 72: @@
 </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 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>), the term "contorsion" must be banished from the RTT community altogether.
-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 ===
 == identifying enfactored mappings ==