Archiving some stuff about the Weil norm off of Facebook

I posted this on Facebook to Paul about the Weil norm; I'm archiving
it here. This is shameless information vomit because I want to
preserve the information in case FB goes down. This might be useful
for those in the future who might be working on things related to
max(n,d) complexity and are scouring these archives for info on it.
This sums up the things I learned while working on it, but at this
point I'm done with it and am onto other things.

Perhaps unsurprisingly, the study the basic behavior of max(n,d) and
its logarithmic counterpart log(max(n,d)) ends up touching on a whole
bunch of random topics, including but not limited to

- Kees expressibility (the Weil norm is a generalization of it; Kees
expressibility can be derived from it as a seminorm)
- Truly Kees-optimal tunings that minimize the max odd-limit-weighted
error over all intervals
- HE and why the Farey series (really "Weil series") curve slopes down
- HE and where 1/sqrt(n*d) and 1/min(n,d) widths may come from
- Identifying Parncutt's model with HE seeded with min(n,d) instead of max(n,d)
- Building on Graham's work with "Farey limits" in composite.pdf,
which are the same as Weil limits, to figure out things minimizing
average Weil-weighted error rather than max
- Weil height doubling as a span-based complexity measure, since it's
secretly the average of (logarithmic) Tenney height and (logarithmic)
span.
- A "generalized musical norm" incorporating Tenney, Weil,
Tenney-Euclidean, Weil-Euclidean, etc norms, as well as a norm related
to Cangwu badness
- Unweighted Hahn distance is related to the W2 norm somehow (at least
in the 7-limit)
- The hexagonal lattice norm is the same as the W2 norm
- Augmented interval space organically appearing yet again, just like
when I was working with affine spaces

rearranged in order of how likely I think people are to care about any of them.

The above are all interesting points of departure for further
research, and what's below sums up what I know about them and how
they've come up.

BASIC GEOMETRIC THINGS

=What the Weil norm unit sphere looks like on a rectangular lattice=
1D: the endpoints of a line segment
2D: an irregular hexagon
3D: a cuboctahedron
4D+: "a higher dimensional hexagon"

=The relationship between the Weil norm and Kees expressibility=
Kees expressibility is really just a "seminorm" on interval space
which you can define by making all intervals octave-equivalent and
only getting the Weil norm of the simplest one. Some work here will
probably lead to a pure-octave Kees tuning which minimizes odd-limit
error over all intervals, but it requires knowing what the dual of a
seminorm is (is it a quasinorm?)

=What the dual norm of the Weil norm looks like=
Should just be the dual polygons to everything listed above; you can
apparently use something evil called the Legendre transformation to
work this out and I haven't done it yet

=What happens if you use a triangular lattice=
My thought is "Paul knows more about this than I do it's the same as
Kees expressibility basically"

ALGEBRAIC THINGS

=How the Weil norm is defined algebraically=
It can be shown that log(max(n,d)) = (log(n*d) + |log(n/d)|)/2. In
weighted monzo form, this means that the Weil height of |a b c> is
given by (|a| + |b| + |c| + |a+b+c|)/2, and this generalizes to
higher-dimensional monzos as you'd expect. However, we don't really
need to worry about this division by 2 at the end since it scales the
height of all monzos identically, so we can simply look at |a| + |b| +
|c| + |a+b+c| instead; this corresponds to log(max(n,d)^2) instead of
just log(max(n,d)). Keenan and I worked this out with him beating me
in the "race" to find the correct expression. Note all coordinates are
weighted.

Note that |a| + |b| + |c| + |a+b+c| is the same as the L1 height of
the vector |a b c; (a+b+c)>, where we simply add one "augmented"
coordinate at the end of the vector (with the semicolon serving as a
delimiter. This is hugely important, because we can come up with a
simple matrix mapping monzos of the form |a b c> to "augmented" monzos
of the form |a b c; (a+b+c)>; I will call this matrix W. Then if M is
a monzo, the Weil norm of M is simply given by ||W*M||_1!!

=What does this "augmented vector space" mean geometrically?=
So for the 3-limit, the Weil norm's unit sphere is a hexagon. A
hexagons, however, is a cross section of an octahedron. So what we're
really doing is, we're creating a larger 3D vector space where and
equipping it with the Tenney norm, which has an octahedron as its unit
sphere. Then, we're mapping all normal monzos to a 2D subspace of this
3D vector space, where the intersection of this special subspace and
the octahedron is exactly the Weil hexagon. So the unit sphere of this
larger, spurious vector space is an octahedron, but the unit sphere on
the subspace we care about is just a hexagon. The same applies to
higher-dimensional generalizations of this concept.

=Hey also, you mentioned augmented monzos before in your thing about
affine spaces, right? How the hell are those popping up again?=
It is indeed interesting that this space has now organically emerged
on more than one occasion, isn't it? There's a really deep connection
here I'm sort of still wrapping my head around.

=Hexagons are also cross sections of other things too; are there other
ways to define the Weil norm?=
Yes; there's also the Linf norm of the vector |a b c; (a+b) (a+c)
(a+b+c)> (which is magically related to Hahn distance somehow). Let's
call that the "alternate Weil norm", and note that this correspond to
exactly log(max(n,d)), not log(max(n,d)^2) or anything else.

PRACTICAL THINGS

=Why did this crash and burn?=
Minimax tunings are usually OK, and the Weil norm is usually OK, but
the two together seems to be bad.

=What if we used Graham's techniques in composite.pdf to work out RMS
Weil error or something?=
Graham Breed may have already worked some of this out, since his work
with "Farey limits" is exactly the same as what I'm now calling Weil
limits (at Gene's request). The matrices he worked out there have
off-diagonal terms, but I suspect that it'll work the same as the Weil
tuning in general, where you can map any normal monzo to an "augmented
monzo" in a special vector space where suddenly the off-diagonal terms
disappear.

=OK, so if we're going to come up with some composite Weil norm and
we're going to use Graham's composite.pdf techniques, we need to work
out an L2 version of the Weil norm, right?=
Yes.

L2 OPTIMIZATION GRAHAM BREED INFLUENCE TYPE THINGS

=How the W2 norm is defined=
The Weil norm is given by the following algorithm:
1) Map ordinary monzos to these special "augmented monzos"
2) Take the Tenney (T1) norm of the result

We thus say that the Weil norm is also called the W1 norm, and the L2
version is the W2 norm. To compute the W2 norm, just take the T2 norm
of the result in step 2 instead of the T1 norm. Easy, and the same
applies to any Lp generalization of the weil norm you want.

We can also call this the WE norm, for "Weil-Euclidean".

=Wait, you said that there was this other "alternate Weil norm" which
is an Linf norm, right? Why don't we take the L2 version of that
instead?=
So the alternate Weil norm is actually an Linf norm, so let's call it
the AWinf norm. So you know the W1 norm is log(max(n,d)^2), and the
AWinf norm is log(max(n,d)). So they're the same, up to a constant
scaling factor.

Miraculously, the W2 and AW2 norms are also the same exact thing up to
a constant scaling factor. The former is sqrt(|a|^2 + |b|^2 + |c|^2 +
|a+b+c|^2), and the latter is sqrt(|a|^2 + |b|^2 + |c|^2 + |a+b|^2 +
|a+c|^2 + |b+c|^2 + |a+b+c|^2). If you work it out algebraically, you
realize that one is just a multiple of the other. The latter is also
the L2 norm on a hexagonal lattice or something like that, Gene said.

Also miraculously, the W1 norm and the AWinf are the same exact thing,
up to a constant scaling factor.

So:
W1 ~= AWinf
W2 ~= AW2
Winf ~= AW1

is there a pattern beyond these three norms? Probably, but I haven't
figured it out. But we probably don't care about anything beyond W1
and Awinf anyway.

So that's nice.

=What the W2 norm looks like geometrically=
1D: the endpoints of a line segment
2D: an ellipse that's rotated to align with the hexagon
3D: an ellipsoid that's rotated to align with the cuboctahedron
4D+: higher-dimensional ellipsoids that are rotated to align with the
higher-dimensional hexagons

=Can we just immediately now use this to say that the W2 norm
minimizes RMS unweighted error over all Weil-bounded subsets of the
lattice?=
Not quite yet; another problem is that if we do the above, we now have
a bunch of "junk" monzos to worry about in the new space. I haven't
worked it out.

=How does the W2 tuning relate to the T2 tuning?=
In practice, UNLIKE the T1 (TOP) tuning vs the W1 (Weil) tuning, the
T2 (TE) and W2 (WE) tunings agree almost all the time. Or, at least
the POTE and POWE ones agreed a lot (again this is all empirical for
now).

So how about them apples! This is, again, probably related to what
Graham proved in composite.pdf.

REALLY COOL THINGS I NEVER EVEN GOT TO

=Are there norms that interpolate between Tenney and Weil norms?=
Yes, there's a class of norms corresponding to log(n*d) + k*|log(n/d)|
for some free parameter k. So if n>d, this means it's the same as
(n*d) * (n/d)^k if you get rid of logs. k=1 is n*n, k=0 is n*d, and
k=-1 is d*d, so in general k=1 is max(n,d)^2, k=0 is n*d, and k=-1 is
min(n,d)^2.

A nice "geometric" version of this is given by throwing that scaling
factor from before back in, where we divide by 2. This is the magic
bullet:

(log(n*d) + k*|log(n/d)|)/(1+k)

If you look at the unit sphere for this: for k=0 you get the usual
Tenney norm, as k goes to 1 the sides of the Tenney norm diamond get
"pulled out" to become the Weil norm, and as k goes to -1 the sides
get "pinched" all the way in.

I have pictures of this but it's all disorganized now.

=So the most general norm possible would have two free parameters; one
interpolating between Weil and Tenney, and the other interpolating
between L1 and L2 and beyond, right?=

Yes, and that first parameter magically corresponds to a weighting
factor controlling how much you care about the logarithmic span of the
interval.

=Whoa!=
I know!

=Does Cangwu badness fit into that somehow?=
Still not sure.

HARMONIC ENTROPY THINGS

=OK, so the thing we're calling "Farey series" HE is really "Weil
limit HE", right?=

Yes. We were calling it a Farey series at first, but really we're
looking at all positive rationals under a certain Weil height, given
by big N.

=And the basic gist of it is, Weil height is something like the
average of (logarithmic) Tenney height and span, right?=

Exactly. For some rational n/d, it's exactly (TH(n/d) + span(n/d))/2,
which is (log(n*d) + |log(n/d)|)/2. By controlling the weight on the
second term, e.g. (log(n*d) + k*|log(n/d)|)/2, you can transform
between max(n,d), sqrt(n*d), and min(n,d). If you drop the /2 at the
end, you get max(n,d)^2, n*d, and min(n,d)^2.

=Wait, could this have anything to do with the slope in HE? For
instance, the "Farey series" approach, which is really just the Weil
height approach, has a downward slope as span increases.=

That would appear to be the case, though I don't have it all worked
out algebraically. For Weil height, as intervals increase in span,
they also increase in Weil height, so there are less of them
competing. This (almost certainly) leads to the characteristic
downward slope of the Farey series HE curve, which should probably be
a "Weil series curve" or something.

=What about the 1/d widths?=
Those widths are really 1/min(n,d). There must also be some beautiful
relationship here but I'm not sure what it is yet.

=What if we use min(n,d) to seed the HE curve rather than max(n,d)?=

Well, note that min(n,d) isn't a height, so there are an infinite
number of intervals for any min(n,d) = N bound that you want to set.
For instance, saying that min(n,d) <= 1 gives you 1/1, 2/1, 3/1, 4/1,
5/1, ... etc. But hey, saying that min(n,d) <= 7 is basically what
Parncutt and Terhardt were doing, right?

=Gadzooks, that gives us Parncutt HE!=

Egads even! So this basically predicts that Parncutt HE will look
exactly like normal HE, but with an upward rather than a downward
slope.

=So since Weil height is secretly the average of span and Tenney
Height, could we also interpret it as a consonance measure that
penalizes for span as well as Tenney height?=

Yes, it works out really nicely like that. If you don't care about
span, this identity will make no sense at all to you, but you can
still use the other properties of the Weil norm. If you do care about
span, then this is yet another added bonus.

=So basically, the takehome point is, for HE, the "Farey series"
approach is basically a bound on intervals by Weil height, which
secretly has Tenney height built in at its core as well as taking into
account span, right?=

Yes, exactly, and this is reflected visually by the shape of the
curve. Tenney Height is what's really powering the whole thing.

=============================================================

That's it for now. Phew, moving on!

-Mike