Wedgie error!!!

This follows from my previous post about badness space. Well, I've worked it out. It's a Euclidian space again. The vectors are weighted mappings with the mean subtracted from each element. The result is that the mean of each vector is zero and so the RMS is the same as the standard deviation.

In this space, the length of a vector is the standard deviation of the original weighted mapping, which is the simple badness of an equal temperament. And it all generalizes so that there's a scalar badness which is the equivalent of scalar complexity.

This isn't such a surprise because the badness (see equation 99 in my PDF) is of the same form as the scalar complexity, but with standard deviations instead of means of products. In general, badness is the square root of the determinant of a matrix where the (i,j)th entry is the covariance of the ith and jth weighted mappings.

So if you take these badness vectors, calculate the wedge product of a set of them, and find its magnitude, the result is the simple badness for the optimal regular temperament that unites the equal temperaments. Divide this scalar badness by the scalar complexity and the result is the TOP-RMS error. So we have an error formula in terms of wedge products.

This also means it's possible to calculate the badness using a matrix determinant, and so there's no need for inverses to get the optimal error! And more, there's the duality between mappings and unison vectors. So it's possible to calculate the error and complexity directly from unison vectors. And it means there's a geometric model for both components of the error.

I hope it's the last piece of the puzzle and I won't have to keep updating my PDF with new discoveries.

Graham