source file: mills3.txt Date: Mon, 2 Feb 1998 20:17:03 +0100 Subject: RE: Decatonic tuning and 7-limit lattices From: "Paul H. Erlich" >}My point about tuning tritones isn't quite covered in Paul's paper. >}To quote: "Since any perfect 6th10 in the scale is likely to >}represent 7:5 and 10:7 equally often, the interval achieves its >}greatest accuracy at exactly 600 cents." No, at least 1 of the 4 >}tritones in a pentachordal scale has only one interpretation. My >}point is that, as at least two have to be ambiguous, these worst >}intervals in 22-equal can't be improved by any tempering. A well >}temperament may improve the average tuning, though, and I haven't >}looked at this properly yet. > >You are correct, but with every new key one modulates to, my comments become >more and more correct. > >}In abstract terms, the point of fcc is that, along with hcp, it >}has the highest possible packing fraction for spheres. > >Correction: the highest _known_ packing fraction for spheres. The >sphere-packing "theorem" is as yet unproved. > >}That's why >}it's so common in crystals. > >Naah, crystal structures have little if nothing to do with the principle of >close packing. > >} >} >} 5/3-------5/4 >} / \ / \ >} / \10/7 / \ >} O O / 7/6 \ / 7/4 \ >} | / \ / \ >} \___/ 4/3-------1/1-------3/2 >} \ / \ / >} \ 8/7 / \12/7 / >} \ / 7/5 \ / >} \ / \ / >} 8/5-------6/5 > >}And there's probably a name for that structure, maybe 7-limit >}tonality diamond. > >Yup, but it's the Wilson picture rather than the Partch picture of the >7-limit tonality diamond. > >}In case people are getting deluged with all this theory, I'll >}highlight the point that these lattices are very useful for >}constructing 7-limit scales, and taking chords from them. > >I second that! SMTPOriginator: tuning@eartha.mills.edu From: alves@orion.ac.hmc.edu (Bill Alves) Subject: Re: Da Capo Press e-mail PostedDate: 02-02-98 20:21:44 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: 02-02-98 20:20:44-02-02-98 20:20:45,02-02-98 20:19:24-02-02-98 20:19:25 DeliveredDate: 02-02-98 20:19:25 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 C125659F.006A420F; Mon, 2 Feb 1998 20:20:37 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05867; Mon, 2 Feb 1998 20:21:44 +0100 Date: Mon, 2 Feb 1998 20:21:44 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA05864 Received: (qmail 24805 invoked from network); 2 Feb 1998 11:17:22 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 2 Feb 1998 11:17:22 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu