source file: mills2.txt Date: Mon, 7 Jul 1997 23:44:19 +0200 Subject: RE: Muddy Waters From: "Paul H. Erlich" I have not recieved TD 1122, so I write with some tentativeness . . . I wrote that in the rectangular lattice philosophy, >>i.e., a major seventh is just as dissonant/complex as a major >>sixth. Marion replied, >I must be missing something here. I thought a major seventh was >15:8, and a major sixth was 5:3. The last time I checked the LCM >of 5:3 was 15 and the LCM of 15:8 was 120. Note that I did state that I was assuming octave equivalence, by which I meant exact equivalence in all respects. (If we drop the assumption altogether, then I would argue that prime or odd limits will not do; every integer should be a distinct limit representing a certain degree of harmonic complexity.) Equating the LCM and ROHS ideas as Graham has done requires that we consider 2 to be a distinct factor. So I should really be comparing 5:3 with 15:1 directly if I am specifically addressing the LCM variety of the rectangular lattice philosophy. 15:1 is a very wide interval so the usual roughness we associate with dissonance does not come into play. But if we could somehow control for the width of the interval, I claim that 5:3 should be considered more consonant than 15:1. For example, I think 14:1 is slighlty more consonant that 15:1, once the unfamiliarity of the former interval is overcome. But 5:3 is more consonant than any 7-limit interval. I think a basic argument for the triangular as opposed to rectangular lattice representations can be founded upon traditional theory. Traditional theory deems 15:1 and all its octave equivalents to be dissonant, and 5:3 and all its octave equivalents to be consonant. Aside from second inversions, traditional theory deems major and minor triads to be consonant, while all inversions of the triads 1:3:15 and 1:5:15 are considered dissonant. A 2-d triangular lattice therefore seems a far better fit to traditional theory than a 2-d rectangular lattice. To address Graham's questions: One way of constructing a lattice with a useful metric is to make the length of each interval, according to a "city-block metric," a monotonic function of its (odd) limit. James Tenney and Graham have both proposed using log(n) as the length of each step along the axis representing the factor n. The problem I have with that is that m/n and m*n will both have a metric of log(m)+log(n). Following traditional theory, Partch, and my own ears, I think that for any odd number n, the ratios n/m or m/n, where m is an odd number less than n, and arbitrary powers of 2 are allowed, are more consonant that ratios involving odd numbers larger than n. The idea of a lattice and an associated metric is to make the distance associated with a consonant interval shorter than that associated with a dissonant interval. One way to make the lattice my beliefs is to use isosceles triangles; the base, representing 3:1, is given length log(3), while the sides, representing 5:1 and 5:3, are both given length log(5). One could also envision using scalene triangles and giving 5:3 a longer length than 5:1 if one felt that 5:1 was more consonant than 5:3. One could even preserve the appearance of the rectangular lattice and allow a "short-cut" diagonally through the block for 5:3, but not for 15:1. Then the length of 5:3 would be sqrt(log(3)^2 + log(5)^2), but 15:1 would still be log(3)+log(2). This sort of representation just barely works because sqrt(log(3)^2 + log(5)^2)9487 and the smallest compound interval length is log(9)1972. In the 5-limit case, the plane is filled by triangles. The right-side-up triangles are major triads, and the upside-down triangles are minor triads. In the 7-limit case, space is filled by tetrahedra and octahedra. The right-side-up tetrahedra are 7-o tetrads, the upside-down tetrahedra are 7-u tetrads, and the octahedra are Wilson hexanies. In general, a lattice of mutually prime axes will have the property that space is filled by all possible CPSs where the set of factors is the set of axes. One of each of these CPSs fit together like pieces of a puzzle to form an Euler genus that uses each factor exactly once. Since the Euler geni fill the space, so do the CPSs. When the axes are not mutually prime, as is the case when 9-limit ratios are considered consonant, things get a little stranger since the same note may appear in more than one place in the lattice. The easiest way around this is to use the prime lattice and introduce a lot of "shortcuts" corresponding to 9-limit and other odd ratios, but that destroys the CPS-filling-Euler-genera-filling-space picture. However, the usual otonal and utonal hypertriangles, the most compact structures in these lattices, will fail to be the only saturated consonant entities if the axes are not prime, so the CPS picture is not so useful in those cases anyway . . . Marion, I am using geometrical pictures to express my thoughts and your LCM idea, as Graham pointed out, is equivalent to a particular geometrical picture as well, a rectangular one where 2 is a distinct factor that has its own axis, and the log of the LCM is just the distance (according to the city-block metric) between the two most distant points in the chord. In other words, the LCM is a measure of the longest extent of the ROHS of a chord. Since I feel that a minor triad is more consonant than a major seventh chord, I don't like this method and would prefer instead to use the triangular lattice. What replaces the LCM in that case? If we use the isosceles triangles eluded to above, the answer is just the Partch limit! Much more to come . . . Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Jul 1997 04:16 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA28899; Tue, 8 Jul 1997 00:18:55 +0200 Date: Mon, 7 Jul 1997 18:39:02 +0200 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu