source file: m1452.txt
Date: Sat, 20 Jun 1998 07:19:45 -0500 (CDT)

Subject: Re: How to read lattice diagram

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

On Sat, 20 Jun 1998, Ed & Alita Morrison wrote:
> A long time ago Paul Hahn wrote,
> >    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
> 
>    I am not familiar with lattices of this type.  Would you please explain
> it for me?  The middle row is sequence of fifths but what is the meaning of
> the other rows and the fractions between them?

First, let me restore the brief explanation that originally accompanied
this diagram:

> >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.

If that doesn't do it for you, I'll put it a little less tersely.

The basic unit of this lattice is this tetrahedron which represents a
4:5:6:7 tetrad in three-dimensional space:

		    5/4
		    / \
		   /   \
		  / 7/4 \
		1/1 --- 3/2

(Imagine the 7/4 corner of the tetra sticking up out of the plane of the
screen towards you.)

The six edges of the tetra represent the six interval classes by which
the pitches of the tetrad are related.  Now, if you look again at the
original diagram, you'll see that every pitch in it is related to the
others around it by those same intervals, represented by the same
relative spatial position.

For example, the 8/7, 4/3, and 8/5 are each exactly opposite the 7/4,
3/2, and 5/4 relative to the 1/1, and together with it they form a
Utonal tetrad which is the inverse of the original Otonal tetrad.
Similarly, the interval between the 7/4 and the 5/4 is a 7/5, and that
same spatial relationship tells us where to place the 7/5 and 10/7 in
the diagram.

Does that help?

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
   /\        "Churchill? Can he run a hundred balls?"
  -\-\-- o                                                           
             NOTE: dehyphenate node to remove spamblock.          <*>

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End of TUNING Digest 1452
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