source file: mills3.txt Date: Wed, 27 Aug 1997 19:41:19 +0200 Subject: Reply to Paul Erlich From: gbreed@cix.compulink.co.uk (Graham Breed) > [I] get the feeling that you're doing some interesting work, but you're > speaking a language too different from the majority of the tuning list > to engage many of us with your ideas. The last message in particular was This could well be the case. I worked out my ideas, including the basics of interval matrices, before I joined the tuning list and so before I realised there were other people interested in the same stuff as me. Naturally, I worked out my own terminology to explain things to myself. I'm trying to explain myself using the standard terms where possible, but it isn't easy because most of them don't even seem to have precise meanings, and some of my concepts also appear to be original. > The last message in particular was > very obscure. Can you try doing some hand-holding for us and explain > what you mean by "this works," "that doesn't work," a "0-comma scale," > and "this temperament"? Okay. By "this works" I meant that a 2-D scale could be constructed such that every 5-limit interval can be defined on that scale. I originally added the condition that a basis should be a tempered 2-3 plane. A 0-comma scale is a tempered scale with a just octave and another just interval, usually a perfect fifth. "This temperament" is the doubly positive temperament that Paul Erlich worked out. I think this can be called doubly positive temperament, as it covers all the ETs usually described as doubly positive. It is defined using the following matrix equation: (1 0) H' = (0 1)H' (5.5 -2) As this matrix involves a fraction, the tempered 2-3 plane is not a basis. The 0-comma scale involves a just fifth and a tritone equal to 6 steps in 12 tet. By my original criteria, it "doesn't work" but non-Pythagorean bases can be chosen. For example: (s) = (-0.5 1)H' (r) ( 3 -5) > Feel free to review you matrix/determinant stuff. A full description of interval matrices should arrive with this post. Even people with a non mathematical background should be able to follow most of it. Please ask if there's anything you don't understand. This post uses terminology from that one. I've brushed determinants under the carpet for the time being. > It may actually be valuable for understanding ancient Hindu > music. Then let's talk 7-limit. I'm more interested in making new computer music than describing old music, but I am nevertheless interested in what scales are used by different peoples at different times. Particularly when those scales can be defined economically by mathematics, ie JI and ETs. FWIW, I've now worked out 3 different 7-limit approximations to doubly positive temperament. SMTPOriginator: tuning@eartha.mills.edu From: gbreed@cix.compulink.co.uk Subject: Matrices for beginners PostedDate: 27-08-97 19:40:03 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $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: 28-08-97 10:21:45-28-08-97 10:21:46,28-08-97 10:19:19-28-08-97 10:19:19 DeliveredDate: 28-08-97 10:19:19 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C1256501.002DD4DC; Thu, 28 Aug 1997 10:20:36 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02473; Wed, 27 Aug 1997 19:40:03 +0200 Date: Wed, 27 Aug 1997 19:40:03 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA02471 Received: (qmail 6221 invoked from network); 27 Aug 1997 17:39:51 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 27 Aug 1997 17:39:51 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu