source file: mills2.txt
Date: Sun, 6 Jul 1997 12:06:17 +0200

Subject: Re: the 6th chord and odd-limit theory (re-post)

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

As I don't seem to be the only person who missed Digest 1122,
I'm re-posting this with revisions.  Presumably, it will be
accompanied by the resurrected digest just to fill up all our
mail boxes.

All the numbers we'll be meeting today will be odd, so I won't say
so every time.  Numbers with more than one prime factor have a
special significance, so I'll call them "rhubarb numbers" for want
of knowing a better term.

For an n-limit chord, first choose a number m.  Either n or m, or
both, must be composite. m<n.  n and m  must not share a common
factor.  Providing neither n nor m are rhubarb, a saturated n-limit
chord can be constructed form all the divisors of n*m except n*m
and 1. This chord will be symmetrical: it's own utonal analog.

Examples are the two 9-limit symmetric chords already found.
Also, the 11-limit s-chord 33:11:9:3, the 25-limit s-chord
75:45:25:15:9:5:3 and the 35-limit s-chord
105:63:45:35:21:15:9:7:5:3.

When mj is rhubarb, we have a problem.  The intervals n*i/j and
n*j/i will be beyond the n-limit.  The same problem occurs when
nj is rhubarb unless i>m and j>m, so that i*m<n and j*m<n.
This problem complicates matters, but is not insoluble as I
shall demonstrate with the example n, m

[Or that's what I though when I orginally posted.  Squares of
primes are not a problem because n*i/i1.  However, with
mi*i, n*i*i/ii/1 will exceed the n-limit.  In practice,
treat 3*3*3', 3*3*3*3� and 5*5*55 as rhubarb.]

This gives us the chord 35:21:15:7:5:3.  The intervals 35/3 and
21/5 are >15-limit.  Therefore [21 or 5] and [35 or 3] must be
removed.  If 35 and 21 are removed, we have a 15-limit otonality.
Removing 5 and 3 gives its utonal analog.  The choice is then
removing [21 and 3] or [35 and 5].  The resulting chords 15-limit
s-chords 35:15:7:5 and 21:15:7:3 are then otonal/utonal analogs.

I think otonal/utonal pairs will arise whenever the rhubarb
problem occurs.  I haven't looked at any cases where n and m are
both rhubarb.  The first of these is n5, m3.  Doubly rhubarb
numbers can safely be ignored until someone starts writing 105-
limit music.

That should be all you need to know.

                         Graham

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