source file: mills2.txt Date: Tue, 8 Jul 1997 18:24:15 +0200 Subject: RE: Meantone, Partch From: "Paul H. Erlich" John Chalmers wrote, >I agree that in general Meantone tone systems are 2-D, >but under the >common assumption of octave equivalence, the dimensionality may be reduced >to 1. In which case, both ET's and meantone tunings are generated by one >interval, the tempered fifth. Some ET's cannot be generated by their best fifth, e.g. 24 and 34. I like to see meantone systems as 2-D even if octave equivalence is assumed, since meantone systems are meant to approximate 5-limit JI. 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. 22tET can be considered 5-dimensional because it consistently represents all ratios of 3, 5, 7, 9, and 11. If we allow skipping of 13, we can add 15 and 17 and treat 22tET as a 7-dimensional tuning! Of course, one doesn't need a very large portion of the lattice to cover all 22 notes . . . Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Jul 1997 19:22 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17964; Tue, 8 Jul 1997 19:23:15 +0200 Date: Tue, 8 Jul 1997 19:23:15 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA19468 Received: (qmail 10441 invoked from network); 8 Jul 1997 17:23:05 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 8 Jul 1997 17:23:05 -0000 Message-Id: <199707081320_MC2-1A72-8171@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu