Defactoring terminology proposal: Difference between revisions

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# To "saturate" in everyday use means to ''add'' to something, whereas in this mathematical sense it means to ''remove'' something, which can be very confusing. We suggest an effective way to look at it is that saturating is accomplished not by adding more information into the container but rather by shrinking the container itself so that it has no wasted capacity, and thereby attains a saturated state, but still feel that this is a bit of a stretch.

# 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.

# 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>.

# 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>

== Enfactoring, to replace torsion and contorsion ==

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

# There is an argument that using torsion in this way is an abuse of the term, which was originally applied to periodicity blocks, not temperaments<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2937 which is also referred to here http://tonalsoft.com/enc/t/torsion.aspx</ref>. Both periodicity blocks and temperaments can be defined by lists of commas. And either way, these lists can be saturated or unsaturated. But in the case of periodicity blocks — where commas ''are not made to vanish'' — there is an audible difference in the choice between a saturated and unsaturated list, whereas with a temperament — where commas ''are made to vanish'' — there is no audible difference. 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 — which leads to a pitch system with 24 pitches instead of 12 where half of the pitches are a copy of the other half but offset by a fixed amount — it does not make sense to imagine a temperament which makes 2{{vector|-4 4 -1}} vanish but does not make {{vector|-4 4 -1}} vanish. And so the conflation<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2405</ref> of the two situations by using the same term is misleading, so the authors of this article believe that the term torsion should not be used in RTT. While it is ''theoretically possible'' to interpret RTT using mathematical structures like quotient subgroups, lattices, and free abelian groups, or in other words, as if a temperament looked like a periodicity block, in which case one can imagine a reality where e.g. (81/80)² is made to vanish while 81/80 is not, this is not how temperaments actually sound from a musical point of view in our physical reality; in this case, the inherently projective approach to linear algebra, where a (81/80)² and a 81/80 that both have been made to vanish map to the same tempered lattice node, models this problem better. So "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>Furthermore, care should be taken to recognize the difference in behavior between, say<br><br>

# There is an argument that using torsion in this way is an abuse of the term, which was originally applied to periodicity blocks, not temperaments.<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2937 which is also referred to here http://tonalsoft.com/enc/t/torsion.aspx</ref> Both periodicity blocks and temperaments can be defined by lists of commas. And either way, these lists can be saturated or unsaturated. But in the case of periodicity blocks — where commas ''are not made to vanish'' — there is an audible difference in the choice between a saturated and unsaturated list, whereas with a temperament — where commas ''are made to vanish'' — there is no audible difference. 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 — which leads to a pitch system with 24 pitches instead of 12 where half of the pitches are a copy of the other half but offset by a fixed amount — it does not make sense to imagine a temperament which makes 2{{vector|-4 4 -1}} vanish but does not make {{vector|-4 4 -1}} vanish. And so the conflation<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2405</ref> of the two situations by using the same term is misleading, so the authors of this article believe that the term torsion should not be used in RTT. While it is ''theoretically possible'' to interpret RTT using mathematical structures like quotient subgroups, lattices, and free abelian groups, or in other words, as if a temperament looked like a periodicity block, in which case one can imagine a reality where e.g. (81/80)² is made to vanish while 81/80 is not, this is not how temperaments actually sound from a musical point of view in our physical reality; in this case, the inherently projective approach to linear algebra, where a (81/80)² and a 81/80 that both have been made to vanish map to the same tempered lattice node, models this problem better. So "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>Furthermore, care should be taken to recognize the difference in behavior between, say<br><br>

<math>

\left[

@@ Line 17: / Line 17: @@
 # To "saturate" in everyday use means to ''add'' to something, whereas in this mathematical sense it means to ''remove'' something, which can be very confusing. We suggest an effective way to look at it is that saturating is accomplished not by adding more information into the container but rather by shrinking the container itself so that it has no wasted capacity, and thereby attains a saturated state, but still feel that this is a bit of a stretch.
 # 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.
-# 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>.
+# 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>
 == Enfactoring, to replace torsion and contorsion ==
@@ Line 24: / Line 24: @@
 # Again, it does not have any obvious musical or mathematical meaning in this context.
-# There is an argument that using torsion in this way is an abuse of the term, which was originally applied to periodicity blocks, not temperaments<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2937 which is also referred to here http://tonalsoft.com/enc/t/torsion.aspx</ref>. Both periodicity blocks and temperaments can be defined by lists of commas. And either way, these lists can be saturated or unsaturated. But in the case of periodicity blocks — where commas ''are not made to vanish'' — there is an audible difference in the choice between a saturated and unsaturated list, whereas with a temperament —  where commas ''are made to vanish'' — there is no audible difference. 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 — which leads to a pitch system with 24 pitches instead of 12 where half of the pitches are a copy of the other half but offset by a fixed amount — it does not make sense to imagine a temperament which makes 2{{vector|-4 4 -1}} vanish but does not make {{vector|-4 4 -1}} vanish. And so the conflation<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2405</ref> of the two situations by using the same term is misleading, so the authors of this article believe that the term torsion should not be used in RTT. While it is ''theoretically possible'' to interpret RTT using mathematical structures like quotient subgroups, lattices, and free abelian groups, or in other words, as if a temperament looked like a periodicity block, in which case one can imagine a reality where e.g. (81/80)² is made to vanish while 81/80 is not, this is not how temperaments actually sound from a musical point of view in our physical reality; in this case, the inherently projective approach to linear algebra, where a (81/80)² and a 81/80 that both have been made to vanish map to the same tempered lattice node, models this problem better. So "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>Furthermore, care should be taken to recognize the difference in behavior between, say<br><br>
+# There is an argument that using torsion in this way is an abuse of the term, which was originally applied to periodicity blocks, not temperaments.<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2937 which is also referred to here http://tonalsoft.com/enc/t/torsion.aspx</ref> Both periodicity blocks and temperaments can be defined by lists of commas. And either way, these lists can be saturated or unsaturated. But in the case of periodicity blocks — where commas ''are not made to vanish'' — there is an audible difference in the choice between a saturated and unsaturated list, whereas with a temperament —  where commas ''are made to vanish'' — there is no audible difference. 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 — which leads to a pitch system with 24 pitches instead of 12 where half of the pitches are a copy of the other half but offset by a fixed amount — it does not make sense to imagine a temperament which makes 2{{vector|-4 4 -1}} vanish but does not make {{vector|-4 4 -1}} vanish. And so the conflation<ref>See: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2405</ref> of the two situations by using the same term is misleading, so the authors of this article believe that the term torsion should not be used in RTT. While it is ''theoretically possible'' to interpret RTT using mathematical structures like quotient subgroups, lattices, and free abelian groups, or in other words, as if a temperament looked like a periodicity block, in which case one can imagine a reality where e.g. (81/80)² is made to vanish while 81/80 is not, this is not how temperaments actually sound from a musical point of view in our physical reality; in this case, the inherently projective approach to linear algebra, where a (81/80)² and a 81/80 that both have been made to vanish map to the same tempered lattice node, models this problem better. So "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>Furthermore, care should be taken to recognize the difference in behavior between, say<br><br>
 <math>
 \left[