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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 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.