RE: the 6th chord and odd-limit theory

From: "Paul H. Erlich"

Graham, I am puzzled by your statement that these two four note chords
are complete chords of a nature which would seem to require them to be
six-note chords. Somehow, you hid two notes under your sleeve. Therefore
I will reserve judgment on your analysis until you clarify.

The reason these chords (12:15:18:20 and 14:18:21:24, call them 3:5:9:15
and 3:7:9:21 by octave equivalence)) are so interesting is that they are
9-limit chords but are not subsets of either the 9-otonality or the
9-utonality. I'm not sure if this finding directly contradicts anything
Partch said, but it seems to differ from the usual Partchian thinking
that for any odd limit, the corresponding otonal and utonal chords are
the basic harmonic units.

What are we to make of them? I think the best approach is to follow a
suggestion made by Dan Wolf some time ago, which is to construct a
lattice where 9 has its own axis and 3*3 will appear in a different
place than 9. At first I did not like this idea because I thought every
note should appear in only one place in the lattice, and although I saw
the case where 3*3 and 9 might be represented differently in a
temperament, such a temperament would not be consistent within the
9-limit so I didn't see the point of trying to analyze it with a 9- (or
higher) limit lattice. But these chords have convinced me that even in
JI, there is a good reason to have 9 distinct from 3*3.

If all the factors allowed in a consonant ratio are prime, then the
triangular lattice with prime axes works nicely. The smallest line
segments are the consonant intervals. For the 5-limit, the saturated
consonant chords (major and minor triads) are represented by the
smallest triangles, and are the supersets of all entities where every
note is direcly connected to every other note. In the 7-limit, the
saturated consonant chords (7-o and 7-u) are represented by the smallest
tetrahedra (and Wilson hexanies are represented by the smallest
octahedra . . .). If we want to continue to the 9-limit, 9-o and 9-u and
their subsets certainly appear to be the only structures where every
note is connected to every other note, but that ignores the equality of
3*3 and 9.

Taking this equality into account, the most obvious consequence is a
consonant interval which appears as two different unit segments in the
lattice: 3:1 appears as both 3:1 and 9:3. There are two ways of
adjoining these segments so that the intervals cancel each other out.
Therefore there are two structures in the lattice which appear to be
dissonant triads but are in fact consonant dyads. Let us choose one of
these structures, namely 9:3:3*3; looking at the other, 9:9*3:3*3, will
simply lead to inverses of whatever we find using the first.

Now to take advantage of the structure we need to find a note that is
connected (by a unit segment) to 9 but not to 3*3, and another note that
is connected to 3*3 but not to 9. In order for the resulting chord to be
consonant, the notes must be consonant with each other as well as with
3. This seems like a tall order, but either (5, 5*3) or (7, 7*3) are
clearly found using the lattice. Thus we get the the two chords above.
We promised to look at the inverses, but these chords are their own
inverses, so nothing new is gained. Since 5 is not connected to 7*3 and
3 is not connected to 7*5, nothing with more than 4 notes will be
directly connected each to all others and exploit 3*3

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 1 Jul 1997 14:03 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04783; Tue, 1 Jul 1997 14:03:34 +0200
Date: Tue, 1 Jul 1997 14:03:34 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA04784
Received: (qmail 9272 invoked from network); 1 Jul 1997 12:02:57 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 1 Jul 1997 12:02:57 -0000
Message-Id: <199707011213.IAA04860@ne02.northeast.net>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

>In my efforts to un-subscribe from THIS list, I have sent (without result)
>the following FOUR messages to listproc@eartha.mills.edu
>UNSUBSCRIBE
>UNSUBSCRIBE tuning
>UNSUBSCRIBE *
>UNSUBSCRIBE ALL

Interesting... You definitely have the address correct, which is
probably the most common mistake I'm told. The other common mistake
putting the command in the title line rather than the message-content.

Were I to guess though, I suspect that the problem is that the list
processor doesn't understand upper-case.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Wed, 2 Jul 1997 06:12 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA25560; Wed, 2 Jul 1997 06:12:29 +0200
Date: Wed, 2 Jul 1997 06:12:29 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA25529
Received: (qmail 15865 invoked from network); 2 Jul 1997 04:11:03 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 2 Jul 1997 04:11:03 -0000
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu