Reduced mapping: Difference between revisions

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== Standard reduced mapping ==

~~{{todo|inline=1|correct|comment=this is only unique in the rank 2 case atm}}~~

To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is

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|617.610

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We start with the mapping above, which must be ~~an upper triangular matrix~~. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]

We start with the mapping above, which must be in [[Hermite normal form]]. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]

Then we mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 |" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> |".) Thus we have ⟨2 | 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]

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 == Standard reduced mapping ==
-{{todo|inline=1|correct|comment=this is only unique in the rank 2 case atm}}
 To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is
@@ Line 30: / Line 28: @@
 |617.610
 |}
-We start with the mapping above, which must be an upper triangular matrix. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]
+We start with the mapping above, which must be in [[Hermite normal form]]. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]
 Then we mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 |" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> |".) Thus we have ⟨2 | 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]