source file: mills2.txt
Date: Thu, 26 Jun 1997 19:35:57 +0200

Subject: Re: the 6th chord and odd-limit theory

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

On Tue, 17 Jun 1997, Paul Erlich wrote:
> I wonder if there are any 7-limit analogues to these chords (i.e., chords in 
> which each interval is within the 7-limit but the chord as a whole is in a 
> higher limit). I think the answer is no. Anyone care to come up with a 
> counterexample?

Now that I understand the question, it's easy to find a counterexample, or
prove a slightly stronger version of the statement.

To demonstrate this I will use ratio space axes.  As I still don't know of a
standard notation, I'll have to explain and use mine.

As an example, here's a chord:

  (0  0  0 1)                 (0 0 0 0)                   (0 1 1 1)
  (0 -1  0 1)                 (0 1 0 0)                   (0 0 1 1)
  (0  0 -1 1)H (0 0 0 1)H - (0 0 1 0)H (0 -1 -1 0)H + (0 1 0 1)H
  (0  0  0 0)                 (0 0 0 1)                   (0 1 1 0)

In each matrix, each row is a note and each column corresponds to a prime
factor.  The matrix on the right is the otonal version.  In ratio space
terms, this is defined so that the lowest number in each column is zero.
This chord is then revealed to be 15:21:35:105.

This is a 105-limit otonality, but a 7-limit utonality (the middle
representation).  The intervals are all 7-limit or below.  You can tell this
because each interval involves adding a factor in one column and subtracting
one from another.  The intervals are, in fact, 5/7, 5/3, 3, 7/3, 5 and 7.

The following assumes octave equivalence -- i.e. you ignore the first
column.  It also assumes there are only four columns in total (true for
all 7-limit intervals).

If you take the lower of the otonal and utonal limits, though, this can't
be done.  In a 7-limit interval, one column can go up by one, and another
column can go down by one.  If any note in the otonal representation has a
number in any column that is not 0 or 1, there must be a >7-limit interval.
If any note has more than one 1 in it, and another has more than one 0 in it,
the interval between these notes must be >7-limit. Hence a chord with only
<limit intervals can only have up to four distinct notes and such a chord
must be a subset of either the chord above or its otonal analog (1:3:5:7).

This only is true because 3*3>7.  It doesn't apply, therefore, to higher
prime limits where the otonal matrix needn't be binary.

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