source file: mills2.txt Date: Tue, 14 Jan 1997 15:42:16 -0800 Subject: Comment on Kami's post From: John Chalmers Kami is correct about the 3)6 [ 1 1 1 3 5 7 ] 20-any being equivalent to the 3.5.7 Euler-Fokker Genus. I hadn't noticed this relation before and I don't know if Erv Wilson has either as he seldom used repeated factors. One can generalize these relations even further by considering the general 2)4 [x.y.x.w] hexany as the 6 vertices with 2 non-zero and 2 zero coordinates of the 16 vertices of the 4-D hypercube (tesseract). The 3)6 [x.y.z.w.v.u] 20-any is the 20 of the 64 vertices of the 6-D hypercube with 3 non-zero coordinates, and the 4)8 70-any is the analogous set of the 256 vertices of the 8-D "measure polytope," in Coxeter's terminology. One can write the vertices as N-dimensional vectors and embed them in a N-dimensional space where N is the number of prime factors in the tone space. Whenever 1 or 2 appears as a factor, the dimensionality may be lowered by the assumption of octave equivalence. Transposition thus becomes translation and other musical operations may be replaced by their corresponding vector operations. Different species of CPS (or any other kind of JI structure) may be obtained by rotating around other axes, planes, or hyperplanes. I have found such interpretations fascinating over the years. BTW, to measure distances between intervals use the Minkowski metric (Tenney's harmonic distance) rather than the Euclidean distance formula as the lattice is non-continuous. Enjoy your linear algebra class! --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 15 Jan 1997 08:22 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA09749; Wed, 15 Jan 1997 08:25:52 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA08894 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id XAA20222; Tue, 14 Jan 1997 23:25:42 -0800 Date: Tue, 14 Jan 1997 23:25:42 -0800 Message-Id: <32DC8757.4742@mail.pe.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu