source file: mills3.txt Subject: RE: Basic dimensionality, and related issues From: PErlich@Acadian-Asset.com Dan Wolf wrote, >>(4) Using a TX81Z (with a resolution of 1.56 cent deviations from 12tet),= >> I >>perform >>a melody based upon random walks over a lattice with two dimensions: 3s a= >>nd >>9s. The >>best approximation of 9 does not coincide with the best approximation of >>3^2. For all practical purposes, 3 and 9, in this temperament, are >>relatively prime. Graham replied, >3*3=9 is a universal theorem. If you are using an interval that >is not 2*log(3), that interval is not log(9). Sorry, I agree with Dan here. What is true for rationals may not be true for their best approximations in a set of irrationals. 768tET is a poor example because you're likely to treat that many notes as a continuous pitch spectrum. 18tET (Busoni's third-tone system) is a much better example. By far the best and most consonant approximation of 9/4 is 21 steps. This can only be construed as 3/2*3/2 if two different values of 3/2 are used! In other words, the factorization of 9 into primes is at best irrelevant and at worst counterproductive in some cases. >Paul Erlich wrote: >>Also, if 9/5 and 9/7 are to be used as distinctive harmonic entities (i.e., >>consonances), and not only arising from 3*3/5 and 3*3/7, then it is useful to- >>include a 9-axis into the triangular lattice. >If the dissonance of a composite number is independent of its >factors, why use lattices at all? What an odd question! What could you possibly mean by it? >If you do, though, this >would introduce a concept of harmonic dimensionality different >from scale dimensionality. What I previously called "n-D >harmony" can be renamed "n-D JI" to avoid confusion here. Yes, there are these different and equally valid concepts of dimensionality. In my previous posts in which I talk about dimensions I am more concerned with harmonic dimensionality. $AdditionalHeaders: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C12564DA.00071273; Sun, 20 Jul 1997 03:17:14 +0200