Defactoring: Difference between revisions

Line 267:

=== beyond rank-2 ===

These two tricks should enable us to attain any valid generator sizes we may wish for a given temperament. And these tricks work for any rank<ref>past 1, anyway; of course, rank 1 temperaments are somewhat inflexible in their single generator's size.</ref>, not only rank 2 like we've looked at thus far with 5-limit meantone examples. To be clear, for a rank <math>r</math> temperament, subtracting <math>R_a</math> from ~~<math>R_b</math>~~ will only affect the ~~sizes~~ of <math>G_a~~</math> and <math>G_b~~</math>.

These two tricks should enable us to attain any valid generator sizes we may wish for a given temperament. And these tricks work for any rank<ref>past 1, anyway; of course, rank 1 temperaments are somewhat inflexible in their single generator's size.</ref>, not only rank 2 like we've looked at thus far with 5-limit meantone examples. To be clear, for a rank <math>r</math> temperament, adding or subtracting <math>R_a</math> from another generator will only affect the size of <math>G_a</math>.

For example, [[Marvel_family#Marvel|7-limit marvel]]'s canonical form is {{vector|{{map|1 0 0 -5}} {{map|0 1 0 2}} {{map|0 0 1 2}}}}, with generators of an octave, tritave, and [[pentave]] (5/1), in that order. We can change that second generator from a tritave to a perfect fifth by decreasing <math>G_2</math> by <math>G_1</math>, which we know by the first trick means we add <math>R_2</math> to <math>R_1</math>, producing {{vector|{{map|1 1 0 -3}} {{map|0 1 0 2}} {{map|0 0 1 2}}}}. Helpfully, this trick has no effect on any other generators that were not involved, which in this case is just the size of the pentave, which was the one remaining generator out of the three in this temperament.

The fact that these tricks have isolated effects on the generator sizes like this makes it straightforward to compose sequences of them, applied one after the other, to attain an incredible variety of valid generator sizes.

@@ Line 267: / Line 267: @@
 === beyond rank-2 ===
-These two tricks should enable us to attain any valid generator sizes we may wish for a given temperament. And these tricks work for any rank<ref>past 1, anyway; of course, rank 1 temperaments are somewhat inflexible in their single generator's size.</ref>, not only rank 2 like we've looked at thus far with 5-limit meantone examples. To be clear, for a rank <span><math>r</math></span> temperament, subtracting <span><math>R_a</math></span> from <span><math>R_b</math></span> will only affect the sizes of <span><math>G_a</math></span> and <span><math>G_b</math></span>.
+These two tricks should enable us to attain any valid generator sizes we may wish for a given temperament. And these tricks work for any rank<ref>past 1, anyway; of course, rank 1 temperaments are somewhat inflexible in their single generator's size.</ref>, not only rank 2 like we've looked at thus far with 5-limit meantone examples. To be clear, for a rank <span><math>r</math></span> temperament, adding or subtracting <span><math>R_a</math></span> from another generator will only affect the size of <span><math>G_a</math></span>.
 For example, [[Marvel_family#Marvel|7-limit marvel]]'s canonical form is {{vector|{{map|1 0 0 -5}} {{map|0 1 0 2}} {{map|0 0 1 2}}}}, with generators of an octave, tritave, and [[pentave]] (5/1), in that order. We can change that second generator from a tritave to a perfect fifth by decreasing <span><math>G_2</math></span> by <span><math>G_1</math></span>, which we know by the first trick means we add <span><math>R_2</math></span> to <span><math>R_1</math></span>, producing {{vector|{{map|1 1 0 -3}} {{map|0 1 0 2}} {{map|0 0 1 2}}}}. Helpfully, this trick has no effect on any other generators that were not involved, which in this case is just the size of the pentave, which was the one remaining generator out of the three in this temperament.
 The fact that these tricks have isolated effects on the generator sizes like this makes it straightforward to compose sequences of them, applied one after the other, to attain an incredible variety of valid generator sizes.