The Weil norm

First off, the thing I was calling "Farey Height" is now "Weil
height." Again, the Weil height of any rational number is
log(max(n,d)), or max(n,d) for the arithmetic version.

As expressed in my last post, the Weil height of any rational is
subject to the following identity:

log(max(n,d)) = (log(n*d) + |log(n/d)|)/2 = (complexity + span)/2

where complexity is Tenney height, and span is the log-frequency size
of the interval expressed in the same units as Tenney height. For a
monzo with weighted coordinates |a b c d e ... >, the Weil height W
becomes

W(|a b c d e ... >) = (|a| + |b| + |c| + |d| + |e| + ... + |a+b+c+d+e+...|)/2

Note that the /2 term at the end doesn't really do anything, since
multiplication by a constant term doesn't really change anything. So
we can drop it and get

W(|a b c d e ... >) = |a| + |b| + |c| + |d| + |e| + ... + |a+b+c+d+e+...|

Both of these functions have the structure of a norm on interval space
which represents the Weil height of any interval. The first one, with
the /2 at the end, is what I'm calling the "normalized Weil norm,"
corresponding to log(max(n,d)), and the second is the "unnormalized
Weil norm," corresponding to log(max(n,d)^2). As far as I'm concerned,
the term "Weil norm" can refer to either; for now, there's really no
practical distinction between the two. However, the normalization
distinction will become important when we start looking at generalized
Weil norms, which I'll call Wp norms, so it's good to note for now
that there's this random division by 2 that sometimes turns up. For
now, it's algebraically simpler to drop the /2, and will make defining
generalized Weil norms easier later on, so I'm going to do that and
work with the unnormalized version for now.

Weil tuning is up next.

-Mike

On Thu, Aug 23, 2012 at 1:33 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> W(|a b c d e ... >) = |a| + |b| + |c| + |d| + |e| + ... + |a+b+c+d+e+...|
>
> Both of these functions have the structure of a norm on interval space
> which represents the Weil height of any interval.

Should this perhaps be called "Weil interval space" then?

-Mike