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. | 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 {{ | 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}}]. | ||
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 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. | ||
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[[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]] | [[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]] | ||
We are now comparing the previous diagram, which had the mapping {{ | We are now comparing the previous diagram, which had the mapping {{rket|{{map|2 3}}}}, with the 2-enfactored version of it, i.e. the mapping 2×{{rket|{{map|2 3}}}} = {{rket|{{map|4 6}}}}, AKA 4-ET. | ||
If you compare this lattice of an enfactored mapping with the previous lattice for a healthy, defactored mapping, they should look almost the same. They have the same comma and tube circumference. And the generator follows the same path through that tube/swath. The key difference is how far the generator moves with each step along that path. | If you compare this lattice of an enfactored mapping with the previous lattice for a healthy, defactored mapping, they should look almost the same. They have the same comma and tube circumference. And the generator follows the same path through that tube/swath. The key difference is how far the generator moves with each step along that path. | ||
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[[File:2-enfactored comma-basis.png|365px|thumb|left|enfactored comma bases are garbage]] | [[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 | 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}}]. | ||
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. | 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. | ||