source file: mills2.txt Date: Fri, 11 Jul 1997 12:20:02 +0200 Subject: RE: Meantone, Partch From: gbreed@cix.compulink.co.uk (Graham Breed) Paul Erlich wrote: >Topologically, meantone tunings wrap the 5-limit plane onto an infinite >cylinder. I like to see ETs as having as many dimensions as odd factors >with which they are consistent. ETs consistent with the 5-limit wrap >said plane onto a torus. A 2-D scale is linear with octave invariance, because each point will lie on a line drawn on Paul's cylinder, at the intersections with the original lattice. I generally don't like this idea of "linear temperaments" because it places too strong an emphasis on octave invariance. However, if you define a temperament as "a formula that produces a scale" that would make the difference. Applying octave invariance would wrap 3-D pitch space into a hypercylinder. Note that there would be two straight lines connecting any two points in this hypercylinder, so this space would also imply inversional invariance. A meantone would then wrap this into a toroidal hyperprism, and an ET a hypertorus. Quite something, given that an integer CET or TET would be a finite set of points in this hypertorus, having a Hausdorff dimension of zero. Anyway, back to Earth: >22tET can be considered 5-dimensional because >it consistently represents all ratios of 3, 5, 7, 9, and 11. This is a 1-D scale approximating 5-D harmony in my terminology, including 2 and implying 9. Call it an ET approximating the 11-limit if you prefer. However, I would certainly not say that the scale is 5 dimensional, rather that you are mapping it to a 5-dimensional space. The number of points in the full 5-D real pitch space depends linearly upon the number of octaves you look at, hence the scale has a Hausdorff dimension of 1. I wouldn't say that adding a direction for 9 alters the dimension. A definition, then: The dimension of a scale is the smallest number of fundamental intervals (FIs) such that every interval in the scale is a linear combination of integer multiples (phew!) of those FIs. This implies that none of the FIs can be a linear combination of integer multiples of the others. The FIs would be real numbers denoting the interval size in octaves, c*nts, or some such unit. Distinct prime numbers always constitute a set of FIs. Note the analogy reals->vectors, integers->scalars. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 11 Jul 1997 12:24 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00642; Fri, 11 Jul 1997 12:24:44 +0200 Date: Fri, 11 Jul 1997 12:24:44 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA00560 Received: (qmail 4660 invoked from network); 11 Jul 1997 10:24:34 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 11 Jul 1997 10:24:34 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu