9-limit lattices

On which Paul Hahn writes:
> The tetrad on the left would be the 3D layer "below" the middle one in
> 4D space, and the one on the right the layer "above". All primary
> 9-limit intervals could then be seen as being one step away from the
> origin. If, as I said, we were 5-dimensional beings who could use
> 4-dimensional paper.

I've been reviewing saturated chords for my website. Although
Paul Erlich used these as an argument for giving 9 a separate
direction, the opposite seems to be the case to me. Take the
simplest example, 3:5:9:15 which looks like this:

5--15
\ / \
3---9

And you can see that all the intervals in there are also in the
9-limit otonality:

5
/ \
/ 7 \
1-----3-----9

Also that it's an Euler genus with the corners lopped off:

5--15-(45)
/ \ / \ /
(1)--3---9

As there are no 7s, we can use a tetrahedral lattice with 9 at
the apex. The chords then become:

5 5----15
/ \ \ /
/ 9 \ 9 \ /
1-----3 3

Which doesn't look as clear to me. The reason is that these
anomalous suspensions work because of the compositeness of 9.
A lattice that treats 9 as prime can't show that.

I can understand why 9 is a separate axis in Erv Wilson's
diagrams. The way the scales are generated, there's no confusion.
For general lattices, though, why? If you want the otonality
to look like a primary unit, how about a double linkage?

5
/ \
/ 7 \
1=====3-----9

I genuinely don't understand why even 5-dimensional beings
would want to give 9 its own direction. Now, a curved
structure in 5-dimensional space, that might work...

BTW, Paul Hahn's 9-limit construction is analogous to the
way I'm thinking about the 11-limit at the moment.