source file: m1430.txt
Date: Thu, 28 May 98 18:21 BST-1

Subject: 9-limit lattices

From: gbreed@cix.compulink.co.uk (Graham Breed)

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.