source file: m1532.txt
Date: Tue, 22 Sep 1998 12:03:01 -0500 (CDT)

Subject: Re: scale derived by intersection of sets

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

On Mon, 21 Sep 1998, Robin Perry wrote:
> It involves generating two sets of ratios (as outlined below) then
> finding the intersection of the two.
> 
> In this case, the set to the right is exactly 3/2 of the set to the
> left.  I stopped where I have because I have not found any more common
> ratios beyond this point.

[snip lists of ratios separated from 3/2 and 1/1 by superparticulars]

There shouldn't be any more.  The first few superparticulars (those
larger than 8/7) don't bring you closer than a 16/15 to the central
pair, so once your list of superparticulars goes beyond 16/15 there
won't be any more overlap possible.

What a very interesting method of generating a scale, BTW.  3/2s and
superparticulars both seem to be very pleasing, and this method
guarantees a lot of them.

> The intersection of these two is:
> 
> 1/1  9/8  6/5  5/4  4/3  3/2  8/5  5/3  12/7  7/4  9/5  15/8  2/1

Here's lattice diagram for the scale.  Given the method of generation,
it's not suprising that the resulting shape is very symmetrical.

	     5/3 --- 5/4 ---15/8
	     / \     / \     / \
	    /   \   /   \   /   \
	   /     \ / 7/4 \ /     \
	 4/3 --- 1/1 --- 3/2 --- 9/8
	   \     / \12/7 / \     /
	    \   /   \   /   \   /
	     \ /     \ /     \ /
	     8/5 --- 6/5 --- 9/5

The scale can also be described as all the 5-limit pitches that form
(5-limit) consonances with either 1/1 and 3/2, plus the two 7-limit
pitches that form (7-limit) consonances with _both_ 1/1 and 3/2.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
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