source file: mills2.txt Date: Wed, 15 Jan 1997 15:45:50 -0800 Subject: RE: Comment on Kami's post From: PAULE John Chalmers wrote, >Whenever 1 or 2 appears as a factor, the dimensionality may be >lowered by the assumption of octave equivalence. I was under the impression that having 1 (or 2) as a factor was common, based on some material John sent me, which led me to the discovery I mentioned in my comment on Kami's post. It certainly is not a trivial factor and seems to aid in the geometrical interpretation of these things. Without a 1 (or 2), the CPS is just a "border" between these tiled hyperpolyhedra. Whether or not you assume octave equivalence doesn't alter the dimensionality of a given CPS, as far as I can see, since no octave-equivalent pairs of tones are present. An exception would have to be made in the case of 2 appearing as a factor along with the repetition of other factors, but John said, >Erv Wilson [. . .] seldom used repeated factors. On another point: >to measure distances between intervals use the Minkowski metric >(Tenney's harmonic distance) I would dispute this, since it takes the axes as orthogonal, contrary to my view of Euler geni (?) as oblique. Tenney would view 6/5 as spanning the same harmonic distance as 16/15, that distance being log(3)+log(5)g(15), while I would argue that the former (and its octave equivalence class) is simpler and so should be represented by a shorter distance. Here's how I would measure distance. I take each axis as representing a new prime number, starting with 3. In two dimensions (5-limit), the solution is to make major triads or 1)3 [1.3.5] CPS, and minor triads or 2)3 [1.3.5] CPS, isosceles triangles (which fill the plane) where the three edges 3/1, 5/1, and 5/3 have length log(3), log(5), and log(5), respectively (i.e., the length is the log of the limit). By contrast, Tenney would take these lengths as log(3), log(5), and log(15), respectively, so that the consonant triads are no more compact in space than a chord like 12:15:16. In three dimensions (7-limit or 9-limit), one additionally represents 7/1, 7/3, and 7/5 as edges of length log(7), and space is littered with transpositions of 1)4 [1.3.5.7] and 3)4 [1.3.5.7] tetrads (otonal and utonal, in Partch language), which have one of each of the six edge types (are these called isosceles tetrahedra?), and the "holes" are filled with transpositions of the 2)4 [1.3.5.7] hexany (isosceles octahedra?), which have two of each of the six edge types. 9-limit constructs are not as pretty geometrically, but the math works out: assuming you travel along edges only (is this what is meant by the Minkowski metric? My relativity teacher didn't think so!), an interval is consonant with respect to the n-limit if the shortest distance needed to traverse the interval is less than or equal to log(n), since log(3)+log(3)g(9), etc. The practical problem with these metrics is that they don't take "punning" into account. Eventually a really distant point will sound like a really close one. A simple, accurate, and consistent temperament can remedy this situation, but only with more (euclidean) dimensions around which to "wrap" the lattices. For example, the set of meantone temperaments makes three steps of 3/1 the same as one step of 5/3. So the plane becomes a tube. This tube may still be infinitely long, though, and so will have punning problems. But meantone equal temperaments like 12 and 19 cut it off and attach the two ends together, making a torus. The geometric distance between any two tones is then a good indication of how consonant the interval between them is. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 16 Jan 1997 01:19 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18226; Thu, 16 Jan 1997 01:22:38 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA18244 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id QAA14704; Wed, 15 Jan 1997 16:22:35 -0800 Date: Wed, 15 Jan 1997 16:22:35 -0800 Message-Id: <199701160021.LAA24279@anugpo.anu.edu.au> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu