Pathology of enfactoring: Difference between revisions

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Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.

The tempered vector for the generator of this temperament is of course {{vector|1}}, but the JI ~~preimage of the~~ generator ~~of this temperament~~ is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.

The tempered vector for the generator of this temperament is of course {{vector|1}}, but the simplest JI pitch that maps to this generator is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.

All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!

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 Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.
-The tempered vector for the generator of this temperament is of course {{vector|1}}, but the JI preimage of the generator of this temperament is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.
+The tempered vector for the generator of this temperament is of course {{vector|1}}, but the simplest JI pitch that maps to this generator is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.
 All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!