source file: m1428.txt
Date: Tue, 26 May 1998 09:38:43 -0500 (CDT)

Subject: Small miscellany

From: Paul Hahn <Paul-Hahn@library.wustl.edu>

Apologies, my tuning friends: I've left several responses to my messages
(both on the list and privately) hanging; somehow I didn't seem to get
any less busy when the summer began, but am just as busy with different
things.  But here are a couple of drive-by comments:

I've generated a couple more consistency tables, and the best candidates
I've spotted so far for synthesizers are probably 1224 (2^3 * 3^2 * 17)
and 1848 (2^3 * 3 * 7 * 11).  You can see their consistency levels at
various limits at <http://library.wustl.edu/~manynote/consist4.txt>.

It's diameter, not radius, please, and yes, an ET having a diameter less
than or equal to its consistency level is a desirable property in my
book.  I think of it being "quasi-JI-like", in a way.

It's kind of hard to represent on a 2D screen with ASCII characters what
I would do with a 4D 9-limit lattice, but here's a go:

    10/9 --- 5/3 --- 5/4
     / \     / \10/7 / \
    /   \   /   \   /   \
   /14/9 \ / 7/6 \ / 7/4 \
16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8
           \ 8/7 / \12/7 / \ 9/7 /
            \   /   \   /   \   /
             \ / 7/5 \ /     \ /
             8/5 --- 6/5 --- 9/5

This is a projection of part of the 3D 7-limit lattice into a plane.
The triangles are in the 3-5 plane, and the ratios inside the triads are
either one layer above or below depending on whether they're inside
major or minor triads.  This shows all pitches separated from the 1/1 by
intervals I consider consonant or primary within the 9-limit; however,
without drawing lines which intersect and become confusing I can't
draw lines for all those intervals, even when I'm not restricted to
ASCII art.  Several of the 9-limit intervals look like secondaries
instead of primaries when represented this way.  In 4 dimensions it
would look like this:

                     5/3 --- 5/4
                     / \10/7 / \
    10/9            /   \   /   \         3/2 --- 9/8
     / \           / 7/6 \ / 7/4 \          \ 9/7 /
    /   \        4/3 --- 1/1 --- 3/2         \   /
   /14/9 \         \ 8/7 / \12/7 /            \ /
16/9 --- 4/3        \   /   \   /             9/5
                     \ / 7/5 \ /
                     8/5 --- 6/5

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.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
   /\        "Churchill? Can he run a hundred balls?"
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