Pathology of enfactoring: Difference between revisions

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First, let's look at a [[defactoring|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 {{rket|{{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 makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector|-3 2}}].

This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{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 makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8, the 3-limit major 2nd. And so the comma basis for this temperament is [{{vector|-3 2}}]. The generator is both 4/3 and 3/2. In musical terms, both the fourth and the fifth are so heavily tempered that they each become a half-octave.

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 made the comma {{vector|-3 2}} vanish 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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[[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 defactored situation, if our comma basis was [{{vector|-3 2}}], then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}].

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 [{{vector|-3 2}}] = 9/8, then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}] = 81/64, the 3-limit major 3rd.

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 made to vanish, 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 make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. 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.

But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is made to vanish, 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 make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. In musical terms, the 3-limit major 3rd is the sum of two 3-limit major 2nds. If the major 3rd vanishes, the major 2nd must also. 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 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.

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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 made to vanish, 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]] ({{vector|11 -4 -4}}) and 625/324 ({{vector|-2 -4 4}}). 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.

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]] ({{vector|11 -4 -2}}) and 625/324 ({{vector|-2 -4 4}}). The value at the fourth corner, 12800/6561, is the product of those two and is the octave complement of 6561/6400 which has vector {{vector|-8 8 -2}}. This is the meantone comma squared, (81/80)^2, and its vector is 2-enfactored, {{vector|-4 4 -1}}×2. You can see that the node at the very center of this block is 160/81, the octave complement of the meantone 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.

@@ Line 7: / Line 7: @@
 First, let's look at a [[defactoring|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 {{rket|{{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 makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector|-3 2}}].
+This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{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 makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8, the 3-limit major 2nd. And so the comma basis for this temperament is [{{vector|-3 2}}]. The generator is both 4/3 and 3/2. In musical terms, both the fourth and the fifth are so heavily tempered that they each become a half-octave.
 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 made the comma {{vector|-3 2}} vanish 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.
@@ Line 45: / Line 45: @@
 [[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 defactored situation, if our comma basis was [{{vector|-3 2}}], then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}].
+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 [{{vector|-3 2}}] = 9/8, then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}] = 81/64, the 3-limit major 3rd.
 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 made to vanish, 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 make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. 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.
+But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is made to vanish, 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 make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. In musical terms, the 3-limit major 3rd is the sum of two 3-limit major 2nds. If the major 3rd vanishes, the major 2nd must also. 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 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.
@@ Line 65: / Line 65: @@
 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 made to vanish, 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]] ({{vector|11 -4 -4}}) and 625/324 ({{vector|-2 -4 4}}). 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.
+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]] ({{vector|11 -4 -2}}) and 625/324 ({{vector|-2 -4 4}}). The value at the fourth corner, 12800/6561, is the product of those two and is the octave complement of 6561/6400 which has vector {{vector|-8 8 -2}}. This is the meantone comma squared, (81/80)^2, and its vector is 2-enfactored, {{vector|-4 4 -1}}×2. You can see that the node at the very center of this block is 160/81, the octave complement of the meantone 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.