Cangwu badness hops along

OK, we have k weighted vals vi of dimension n. H is the JIP. Using the JIP if you like, you define the sum si of the elements of each vi. Now we get Cangwu badness unless I've screwed up:

B(x) = det((1+x)vi.vj/n - si*sj/n^2)

B(x) is a polynomial of degree k. I'll calculate a few and see what I see, but I wanted to post this in case Graham wants to say it isn't what he means.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> B(x) is a polynomial of degree k.

A few minutes in, and I already have two observations. The first is that the polynomial is invariant with respect to the choice of vals defining the temperament, as it must be for this business to make sense. The second is that temperaments can either cross or not cross. They cross if there is some positive x for which the badness is equal. If they don't cross, one temperament is dominant over another if its badness is always less. The relation of crossing defines a graph, but what to do with that I don't know. Out of a list of 113 rank two seven limit temperaments, septimal meantone crosses 55, dominates 58 and is dominated by none. That shows there is considerable flexibility in how this works, and suggests that temperaments which cannot be dominated are interesting--it's an idea somewhat along the lines of Herman's gold, silver and bronze.