Pathology of enfactoring: Difference between revisions

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First, let's look at an defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

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 ~~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 [{{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. And so the comma basis for this temperament is [{{vector|-3 2}}].

We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is ~~tempered out~~ the comma {{vector|-3 2}} so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction.

We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is 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.

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

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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.

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.

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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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.

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 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.

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 First, let's look at an defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.
-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 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 [{{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. And so the comma basis for this temperament is [{{vector|-3 2}}].
-We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is tempered out the comma {{vector|-3 2}} so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction.
+We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is 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.
 There's a couple good ways to interpret this situation:
@@ Line 49: / Line 49: @@
 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.
+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.
 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 63: / Line 63: @@
 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.
+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 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.