Wilson norm: Difference between revisions

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The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by

<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>

<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = 2\cdot|e_2| + 3\cdot|e_3| + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>

which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. Both of these norms can be extended to the exterior algebra, so that we can use it as a measure of the complexity of a temperament.

@@ Line 37: / Line 37: @@
 The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by
-<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>
+<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = 2\cdot|e_2| + 3\cdot|e_3| + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>
 which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. Both of these norms can be extended to the exterior algebra, so that we can use it as a measure of the complexity of a temperament.