Mathematical theory of saturation: Difference between revisions

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To give an example, consider the matrix {{mapping| 12 19 28 34 | 26 41 60 72}} whose rows are the two vals we considered above. The Smith form itself is the 2×4 matrix {{mapping| 1 0 0 0 | 0 2 0 0}}; this does not concern us. The left-reducing matrix does not concern us either; our interest lies in the right reducing matrix, which is an invertible square integral matrix, {{mapping| -11 19 4 13 | 7 -12 -4 -10 | 0 0 1 0 | 0 0 0 1}}. Inverting this matrix gives another square integral matrix, {{mapping| 12 19 28 34 | 7 11 16 19 | 0 0 1 0 | 0 0 0 1}}. The rank of ''V'' is two, so to find a basis for the saturation of ''V'', we take the first two rows, which gives us the group generated by {{mapping| 12 19 28 34 | 7 11 16 19 }}. The [[normal lists|normal val list]] for this is {{mapping| 1 0 -4 -13 | 0 1 4 10}}, which are period and generator maps for septimal meantone, which is the saturated temperament corresponding to the contorted ''V''.

To test for saturation, we may take the wedge product of the generators. Wedging {{val| 26 41 60 72}} with {{val| 12 19 28 34 }} gives us {{multival| 2 8 20 8 26 24 }}; this is not zero, so the rank of the group these generate is two. However the coefficients have a gcd of two, and hence the group is not saturated; for saturation, the coefficients must be relatively prime, with a gcd of one.

[[Category:Math]]

[[Category:Regular temperament theory]]

{{Todo| add examples | increase applicability | reduce mathslang }}

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 To give an example, consider the matrix {{mapping| 12 19 28 34 | 26 41 60 72}} whose rows are the two vals we considered above. The Smith form itself is the 2×4 matrix {{mapping| 1 0 0 0 | 0 2 0 0}}; this does not concern us. The left-reducing matrix does not concern us either; our interest lies in the right reducing matrix, which is an invertible square integral matrix, {{mapping| -11 19 4 13 | 7 -12 -4 -10 | 0 0 1 0 | 0 0 0 1}}. Inverting this matrix gives another square integral matrix, {{mapping| 12 19 28 34 | 7 11 16 19 | 0 0 1 0 | 0 0 0 1}}. The rank of ''V'' is two, so to find a basis for the saturation of ''V'', we take the first two rows, which gives us the group generated by {{mapping| 12 19 28 34 | 7 11 16 19 }}. The [[normal lists|normal val list]] for this is {{mapping| 1 0 -4 -13 | 0 1 4 10}}, which are period and generator maps for septimal meantone, which is the saturated temperament corresponding to the contorted ''V''.
+To test for saturation, we may take the wedge product of the generators. Wedging {{val| 26 41 60 72}} with {{val| 12 19 28 34 }} gives us {{multival| 2 8 20 8 26 24 }}; this is not zero, so the rank of the group these generate is two. However the coefficients have a gcd of two, and hence the group is not saturated; for saturation, the coefficients must be relatively prime, with a gcd of one.
 [[Category:Math]]
 [[Category:Regular temperament theory]]
 {{Todo| add examples | increase applicability | reduce mathslang }}