source file: mills2.txt
Date: Tue, 8 Jul 1997 19:51:10 +0200

Subject: Re: Octave invariance

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

David Finnamore wrote:

>How so?  The way I larnt it, a simple C major triad in root position consists
>of a minor third stacked on top of a major third.  In first inversion, it
>consists of a perfect fourth stacked on top of the minor third.  In second
>inversion it's that perfect fourth with the major third stacked above it.
> From what perspective are those the same set of intervals?

Um, yes.  Technically there are six intervals in a major root position
triad: plus a major third, plus a minor third, plus a perfect fifth,
minus a major third, minus a minor third and minus a perfect fifth.
In octave equivalent terms, minus a fifth is the same as plus a fourth.
In these terms, then, all inversions of a major triad contain the same
intervals.  I had overlooked the obvious simplification of only taking
positive intervals and this does, indeed, distinguish simple inversions.
However, in the specific method David mentions the chords C-E-G-C, E-G-C-E
and G-C-E-G have identical dissonance.  I consider this to be a problem.
Taking all positive intervals, the first chord can be distinguished from
the last two (it's all in the sixths) which is less of a problem.

My mention of utonal/utonal degeneracy betrays some muddy thinking
on my part.  What I meant is that a major triad has the same
intervals as a minor triad.  This is a completely different issue
to inversions and octave equivalence, and I shouldn't have mentioned
it in that context.

What I mean by a "privileged root" is taking one note from the chord,
calling it the root, and treating it in a privileged way when you
analyse the chord.  In traditional harmony, the root of an x major
or minor chord is x.  It is considered that a chord is more
consonant if the root is the lowest note and the root is the most
common note in the chord.  The problem with privileged roots in
general is that you have to specify why one note in particular should
be the root, and why it should have that privilege conferred upon it.

The best way, IMHO, of having octave invariance without inversional
invariance is to take the lowest note in the chord to be the root,
and measuring only positive intervals that include it.  Hence,
C-E-G-C contains a major third, perfect fifth and perfect unison
(octave invariant octave).  This is a sensible simplification,
distinguishes all simple inversions of major and minor triads and
tetrads, and is in good agreement with traditional harmony.  A
fundamental theory, though, has to explain why this approximation
works.  Although I don't know of such a theory, it would have to be
octave specific.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
           with SMTP-OpenVMS via TCP/IP; Wed, 9 Jul 1997 01:11 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22258; Wed, 9 Jul 1997 01:11:38 +0200
Date: Wed, 9 Jul 1997 01:11:38 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA22294
Received: (qmail 8339 invoked from network); 8 Jul 1997 23:11:32 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
  by localhost with SMTP; 8 Jul 1997 23:11:32 -0000
Message-Id: <970708191023_-892188239@emout06.mail.aol.com>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu