source file: mills3.txt Date: Thu, 29 Jan 1998 18:30:34 +0100 Subject: Re: 60 vs 45 From: Paul Hahn On Thu, 29 Jan 1998, Carl Lumma wrote: > While I agree that equilateral triangles work better than squares for > modeling triads, I still feel much better with right angles in 3D > (7-Limit). For one thing, while both triangles and squares tile a plane, > tetrahedrons aren't space-filling like cubes. You don't need the tetrahedra to fill space. The lattice fills space perfectly fine (if you look at the Voronoi cells of the vertices, they're rhombic dodecahedra, which _do_ fill space); but instead of dividing space up into cubes you get alternating tetrahedra and octahedra. The octahedra are actually kind of interesting as chords, almost like the Petrushka chord. > 5/4-------15/8 > / \ . / > / . \ / > / \ / > 1/1-------3/2 > > ..The dots (7/4 and 21/16) are a 3/2 apart, but are not represented by the > same length or direction as the other 3/2's in the diagram. What to do? This isn't right--21/16 should be plotted, not where you have it, but like this: 5/4-------15/8 / \ / / . \ / . / \ / 1/1-------3/2 in which case the spatial/visual relationship between pitches 3/2 apart is preserved. --pH http://library.wustl.edu/~manynote O /\ "Do you like to gamble, Eddie? -\-\-- o Gamble money on pool games?" NOTE: dehyphenate address to remove spamblock. <*> SMTPOriginator: tuning@eartha.mills.edu From: Paul Hahn Subject: Re: 60 vs 45 PostedDate: 29-01-98 18:43:07 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 29-01-98 18:42:14-29-01-98 18:42:15,29-01-98 18:40:59-29-01-98 18:40:59 DeliveredDate: 29-01-98 18:40:59 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C125659B.00613DA2; Thu, 29 Jan 1998 18:42:07 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01774; Thu, 29 Jan 1998 18:43:07 +0100 Date: Thu, 29 Jan 1998 18:43:07 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA01772 Received: (qmail 23136 invoked from network); 29 Jan 1998 09:42:20 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 29 Jan 1998 09:42:20 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu