Generator form manipulation: Difference between revisions
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A [[canonical mapping form]] is an important standard to have as community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]]. | A [[User:Cmloegcmluin/Defactored_canonical_form|canonical mapping form]] is an important standard to have as community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]]. | ||
If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches. | If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches. | ||
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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. | 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. | 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, as you can see in the diagram to the right. | ||
== mingen form == | == mingen form == | ||