Douglas Blumeyer's RTT How-To: Difference between revisions

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[[File:Mapping to tempered vector.png|400px|thumb|right|'''Figure 5b.''' Converting from a JI interval vector to a tempered interval vector, with one less rank, conflating intervals related by the tempered out comma.]]

Sometimes it may be more helpful to imagine slicing your mapping matrix the other way, by columns (vectors) corresponding to the different primes, rather than rows (covectors) corresponding to generators. Meaning we can look at {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} as a matrix of three vectors, {{bra|{{vector|1 0}} {{vector|0 1}} {{vector|-4 4}}}} which tells us that 2/1 is {{vector|1 0}}, 3/1 is {{vector|0 1}}, and 5/1 is {{vector|-4 4}}}}.

Sometimes it may be more helpful to imagine slicing your mapping matrix the other way, by columns (vectors) corresponding to the different primes, rather than rows (covectors) corresponding to generators. Meaning we can look at {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} as a matrix of three vectors, {{bra|{{vector|1 0}} {{vector|0 1}} {{vector|-4 4}}}} which tells us that 2/1 is {{vector|1 0}}, 3/1 is {{vector|0 1}}, and 5/1 is {{vector|-4 4}}.

And so we can see that tempering has reduced the dimensionality of our lattice by 1. Or in other words, the dimensionality of our lattice was always the rank; it’s just that in JI, the rank was equal to the dimensionality. And what’s happened by reducing this rank is that we eliminated one of the primes in a sense, by making it so we can only express things in terms of it via combinations of the other remaining primes.

@@ Line 968: / Line 968: @@
 [[File:Mapping to tempered vector.png|400px|thumb|right|'''Figure 5b.''' Converting from a JI interval vector to a tempered interval vector, with one less rank, conflating intervals related by the tempered out comma.]]
-Sometimes it may be more helpful to imagine slicing your mapping matrix the other way, by columns (vectors) corresponding to the different primes, rather than rows (covectors) corresponding to generators. Meaning we can look at {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} as a matrix of three vectors, {{bra|{{vector|1 0}} {{vector|0 1}} {{vector|-4 4}}}} which tells us that 2/1 is {{vector|1 0}}, 3/1 is {{vector|0 1}}, and 5/1 is {{vector|-4 4}}}}.
+Sometimes it may be more helpful to imagine slicing your mapping matrix the other way, by columns (vectors) corresponding to the different primes, rather than rows (covectors) corresponding to generators. Meaning we can look at {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} as a matrix of three vectors, {{bra|{{vector|1 0}} {{vector|0 1}} {{vector|-4 4}}}} which tells us that 2/1 is {{vector|1 0}}, 3/1 is {{vector|0 1}}, and 5/1 is {{vector|-4 4}}.
 And so we can see that tempering has reduced the dimensionality of our lattice by 1. Or in other words, the dimensionality of our lattice was always the rank; it’s just that in JI, the rank was equal to the dimensionality. And what’s happened by reducing this rank is that we eliminated one of the primes in a sense, by making it so we can only express things in terms of it via combinations of the other remaining primes.