Wilson norm: Difference between revisions

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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.

We can likewise keep the <math>p</math> and <math>1/p</math> weighting, but change things so that we have a weighted math>\ell_2</math> norm instead of a weighted <math>\ell_1</math>. We can call this the '''Wilson-Euclidean''' norm, and likewise use it to create metrics similar to the [[Tenney-Euclidean_metrics]], including a Wilson-weighted version of the [[Cangwu_badness]].

We can likewise keep the <math>p</math> and <math>1/p</math> weighting, but change things so that we have a weighted <math>\ell_2</math> norm instead of a weighted <math>\ell_1</math>. We can call this the '''Wilson-Euclidean''' norm, and likewise use it to create metrics similar to the [[Tenney-Euclidean_metrics]], including a Wilson-weighted version of the [[Cangwu_badness]].

== Wilson Height and Tenney Height: A Psychoacoustic Comparison ==

@@ Line 41: / Line 41: @@
 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.
-We can likewise keep the <math>p</math> and <math>1/p</math> weighting, but change things so that we have a weighted math>\ell_2</math> norm instead of a weighted <math>\ell_1</math>. We can call this the '''Wilson-Euclidean''' norm, and likewise use it to create metrics similar to the [[Tenney-Euclidean_metrics]], including a Wilson-weighted version of the [[Cangwu_badness]].
+We can likewise keep the <math>p</math> and <math>1/p</math> weighting, but change things so that we have a weighted <math>\ell_2</math> norm instead of a weighted <math>\ell_1</math>. We can call this the '''Wilson-Euclidean''' norm, and likewise use it to create metrics similar to the [[Tenney-Euclidean_metrics]], including a Wilson-weighted version of the [[Cangwu_badness]].
 == Wilson Height and Tenney Height: A Psychoacoustic Comparison ==