source file: mills2.txt Date: Tue, 1 Jul 1997 12:19:33 +0200 Subject: RE: the 6th chord and odd-limit theory From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) 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