source file: mills2.txt Date: Tue, 1 Jul 1997 12:14:47 +0200 Subject: RE: Partch Limit vs Prime Limit (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: "Paul H. Erlich" Let those of us who don't understand Partch's work not accuse him of muddy thinking!!! As for priority, I seriously doubt the claim that the prime-limit concept predated Partch's work; in fact, it is often merely the result of misunderstanding Partch. I think we have at least four schools of thought here; add in the question of dualism and there are eight philosophies about the mathematical characterization of chords. All these, by the way, leave aside the issues of temperament and octave equivalence, so so will I for now. Let us sidestep the issue of odd vs. prime limits for now, and consider another dichotomy within the context of 5-limit lattices. There are those who subscribe to a rectangular matrix/ROHS/LCM/Tenney harmonic distance idea. According to any of these philosophies, 15:1 is "the same" as 5:3, i.e., a major seventh is just as dissonant/complex as a major sixth. (Note that this conclusion is independent of the odd/prime question; it is important to keep that question from entering the present dichotomy.) In the rectangular matrix point of view, both intervals involve moving one unit along the 3-axis and one unit along the 5-axis. The relevant ROHS is this a 1X1 rectangle. The LCM of both intervals is 15. Since I think the major sixth is as basic a consonance as the major third (for many reasons including some recently discussed experiments on tuning major triads), I prefer a triangular lattice. Here 5:3 has an axis unto itself, so the lattice is filled with triangles, which represent major or minor triads depending on their orienetation. Now 5:3 is just one step, while 15:1 is two. Considering that actual occurences where a 15:1 is emphasized in music are typically in the form 15:5:1 or 15:3:1 (or 15:5:3:1), while 5:3 is happy all by itself, the triangular lattice seems more appropriate. In fact, the major seventh in equal temperament is 11 times closer to 17:9 than to 15:8; the reason that doesn't matter is that the individual steps in the triangular lattice are indeed well-represented in equal temperament, and 15:8 is merely a by-product of linking consonant intervals. 15:8 heard alone is indeed a dissonance and so its exact tuning is not crucial. Finally, note that the smallest triangles in the rectangular lattice are the major triad, the minor triad, the 15:5:1, and the 15:3:1. Are these equivalent in consonance? I think it is far easier to believe that the major and minor triads are more consonant that the other two, and the triangular lattice expresses this by the more compact geometric representation of the major and minor triads. Now, sticking to the 5-limit, we can address the odd vs. prime issue. Clearly, _any_ two points in the lattice, no matter how distant, form a 5-limit interval, if the prime definition is used. If the limit is supposed to be related to dissonance/complexity, that should mean that any two points on the lattice form a more consonant interval than an 8:7. But this is absurd. In fact, I would consider 9:8 slighlty more dissonant than 8:7, if heard in isolation. The Partch theory calls the former a 9-limit interval, correctly expressing its more complex nature. The claim that 27:16, heard in isolation, has a certain 3-limit quality to it, is nonsense to me, and is probably nothing more than a result of familiarity with the effects of Pythagorean tuning, and of listening to chords like 27:9:3:1. Relative primality is all that matters to the ear; absolute primality of the terms in a ratio can have no conceivable importance to the auditory phenomenon, and 27:16 differs materially from 23:16 or 29:16 only in the simpler ratios they approximate (i.e., we are essentially getting into issues of temperament). It's hard to try to discuss these issues seperately since they are all >so intertwined . . . Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 1 Jul 1997 12:16 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00391; Tue, 1 Jul 1997 12:16:22 +0200 Date: Tue, 1 Jul 1997 12:16:22 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA00389 Received: (qmail 4850 invoked from network); 1 Jul 1997 10:15:26 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 1 Jul 1997 10:15:26 -0000 Message-Id: <009B69A0B034BCFE.99DA@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu