Wedgie badness of commas

Given a monzo, the Hodge dual or complement or whatever you like calling it is the (n-1)-val obtained by reversing the elements of the monzo and alternating signs. If we start with a weighted monzo, the corresponding weighted (n-1)-val will be the same alternating sign thing, divided through by a scaling factor which is the product of the log base 2 of the primes: log2(2)log2(3)...log2(p). Wedging with the JIP (we don't care about sign, so in which order doesn't matter) is the same as expanding a determinant one row (or column, if you write vals that way) consists of 1s, and so is an alternating sum of minors, the minors being the coefficients of the (n-1)-val. The two sign alterations cancel out, and the norm of the wedge product is simply the absolute value of the log base 2 of the rational number corresponding to the monzo. This is the simple badness of the monzo.

The error is simple badness divided by complexity; the log2 scaling factors cancel out and you simply get:

error(q) = |q|/||q||

where for a rational number q, ||q|| is the root mean square of the weighted monzo for q.

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

>The two sign alterations cancel out

Sorry, this isn't true. I really shouldn't spout off until I've thought things through, but I've always been like that. Considered bad form in mathematics, I'm sorry to say,

> The error is simple badness divided by complexity; the log2 scaling factors cancel out and you simply get:
>
> error(q) = |q|/||q||

So, I guess not. Darn.

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

> >The two sign alterations cancel out
>
> Sorry, this isn't true.

Sorry again, it is true. I'll see if I can track down what is really giving me the bum answers numerically when I test things.

After explaining that log2(|q|) is the simple badness, I forgot to put in the log. Everything else seems to be right.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Given a monzo, the Hodge dual or complement or whatever you like calling it is the (n-1)-val obtained by reversing the elements of the monzo and alternating signs. If we start with a weighted monzo, the corresponding weighted (n-1)-val will be the same alternating sign thing, divided through by a scaling factor which is the product of the log base 2 of the primes: log2(2)log2(3)...log2(p). Wedging with the JIP (we don't care about sign, so in which order doesn't matter) is the same as expanding a determinant one row (or column, if you write vals that way) consists of 1s, and so is an alternating sum of minors, the minors being the coefficients of the (n-1)-val. The two sign alterations cancel out, and the norm of the wedge product is simply the absolute value of the log base 2 of the rational number corresponding to the monzo. This is the simple badness of the monzo.
>
> The error is simple badness divided by complexity; the log2 scaling factors cancel out and you simply get:
>
> error(q) = |q|/||q||

Should be error(q) = log2(|q|)/||q||

> where for a rational number q, ||q|| is the root mean square of the weighted monzo for q.

I should add, you get logflat badness too:

logflat(q) = log2(|q|)*||q||^(n-1)