source file: mills2.txt Date: Thu, 16 Jan 1997 12:24:06 -0800 Subject: Reply to PaulE From: John Chalmers Paul: I was referring to the Minkowski "Taxicab" "City Block," or "Manhattan" metric where the distance function is ds|dx| +|dy| +|dz|... rather than the square root of the sum of the squares of the Euclidean or relativistic metrics. Thus 6/5, 5/3, 16/15 and 15/8 would have the same distance from 1/1 as you point out. As for "punning," one may map the relevant region of the tonal lattice onto (or do I mean into?) some ET by using Fokker's application of the cross and/or box products after choosing the intervals (kommata) one wishes to set to zero. I think his papers on "Periodicity Blocks" are in the Bibliography. Whether one should use a pseudo-Cartesian (right angle) lattice or a polygonal/polyhedral one is for me mostly a matter of taste. At Paul E's earlier suggestion, I started plotting 7 and higher limit scales on oblique, polygonal lattices (the simpler ones are easily interpretable as polyhedrons in 3 space). I must admit that I got stunning graphics, though beyond about the 11-limit, I find the rectilinear mode easier to interpret. However, I got fascinated by the patterns, so, I wrote a series of Q&D Basic programs to plot scales (after factoring their ratios) on a series of polygonal lattices of the type Erv Wilson has used. I stopped at the centered-tridekagon and the 43-limit. I recommend this approach to those of you with faster computers, more memory, and higher resolution graphics than I have. I might mention the Carter Scholz's latest version of JiCalc plots scales on a 31-limit lattice (pseudo-Cartesian). It's downloadable from Mills. Also, it plays each lattice point when the note is clicked. -John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 17 Jan 1997 02:13 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA12385; Thu, 16 Jan 1997 11:01:41 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA12285 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id CAA22254; Thu, 16 Jan 1997 02:01:38 -0800 Date: Thu, 16 Jan 1997 02:01:38 -0800 Message-Id: <199701160500_MC2-F50-FF96@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu