source file: m1587.txt
Date: Thu, 19 Nov 1998 15:05:53 -0800

Subject: MOS and Rothenberg

From: Carl Lumma <clumma@nni.com>

I've been pondering the relationship between the MOS and Propriety concepts
recently.  So I'm posting some questions to frighten and amuse.  I'm aware
that Clough formalized a lot of the MOS stuff, but I wouldn't know where
(or what) to get of his work.  Also, I'm not sure how best to get
Rothenberg's work...

>>has "stability" of 1.0 and "efficiency" of .7407.
>
>Shame on me, but I don't even know what these measure signify.  Help,
>please.

Or if the definitions of these terms appear in his 1969 paper "A Pattern
Recognition Model Applied to the Perception of Pitch".  Frog Peak or
somebody ought to distribute this stuff.  So the following is based only on
correspondence with Wilson, and Chalmer's excellent XH3 article "The
Application of Rothenberg's Pattern Recognition Model to the Structure of
Tetrachords and Tetrachordal scales" (and its version appearing in
Divisions of the Tetrachord)...

1. If A and B are the two unique interior intervals of a MOS, then Chalmers
has given that the scale is improper when A/B > 2 or < 1/2, proper when A/B
= 2 or 1/2, and strictly proper when 1/2 < A/B < 2.  But can anyone give a
formula that gives the propriety of a MOS using just its generator and
interval of equivalence, and show why it works?

2. Can a proper scale with one and only one ambiguous interval in each mode
exist?

3. What about a proper scale with one and only one ambiguous interval in
each interval (steps) class?

4. Can somebody show a method for finding which tunings will support a
given rank-order matrix?  Preferably one that works on both just and equal
step scales?

Carl