Defactoring: Difference between revisions

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It may be counterintuitive at first that in order to change the size of a generator we must make a change to a mapping row ''other than the one which corresponds to that generator'' (in this example, we changed the size of of the ''second'' generator by changing the ''first'' row). But there is a way to train our intuition on this effect. Think of it like transferring jobs. Remember that in order to know the size in cents of a generator, we cannot look exclusively at that generator's mapping row out of context; we need to know the entire mapping, because the mapping rows all work together to determine how the temperament works. So we can think of every mapping row like a workforce that outsources some of its work to the other rows; they're an interconnected system of workforces. And so when we subtract one row from another, we're in effect saying that the row being subtracted from is going to do less of the row being subtracted's work. So when we subtract <math>R_2</math> from <math>R_1</math>, what we're saying is that whatever work the first generator was doing for the second generator, it will no longer do that work anymore, so the second generator will need to take care of that work itself; and that's why <math>G_2</math> becomes the size of <math>G_2</math> plus <math>G_1</math>.

Conversely, if we want to ''decrease'' <math>G_a</math> by <math>G_b</math>, then we replace <math>R_b</math> with <math>R_b' = R_b</math> ''plus'' <math>R_a</math>.

=== second trick: negating a generator ===

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 It may be counterintuitive at first that in order to change the size of a generator we must make a change to a mapping row ''other than the one which corresponds to that generator'' (in this example, we changed the size of of the ''second'' generator by changing the ''first'' row). But there is a way to train our intuition on this effect. Think of it like transferring jobs. Remember that in order to know the size in cents of a generator, we cannot look exclusively at that generator's mapping row out of context; we need to know the entire mapping, because the mapping rows all work together to determine how the temperament works. So we can think of every mapping row like a workforce that outsources some of its work to the other rows; they're an interconnected system of workforces. And so when we subtract one row from another, we're in effect saying that the row being subtracted from is going to do less of the row being subtracted's work. So when we subtract <span><math>R_2</math></span> from <span><math>R_1</math></span>, what we're saying is that whatever work the first generator was doing for the second generator, it will no longer do that work anymore, so the second generator will need to take care of that work itself; and that's why <span><math>G_2</math></span> becomes the size of <span><math>G_2</math></span> plus <span><math>G_1</math></span>.
+Conversely, if we want to ''decrease'' <span><math>G_a</math></span> by <span><math>G_b</math></span>, then we replace <span><math>R_b</math></span> with <span><math>R_b' = R_b</math></span> ''plus'' <span><math>R_a</math></span>.
 === second trick: negating a generator ===