source file: mills3.txt Date: Tue, 26 Aug 1997 18:22:43 +0200 Subject: groups, matrices, etc. From: James Kukula Graham Breed comments on a possible approach to a general phonemics of scales: > I don't know the specifics of either group or category theory, but > this is definitely something like what I do with matrices. I'm no math whiz but matrices are a very classical way to represent groups and are also a very nice example of a category. With a group you'd want the matrices to all be square, that way any matrix can be multiplied by any other matrix, the product is again square, etc. With categories one can allow non-square matrices, and then you have to restrict multiplication so the rows of one matrix match the columns of the other - i.e. the natural structure of matrices lines up perfectly with the kind of restriction that the more abstract category theory supports. Just taking a finite subset of a group leaves you with, well, a finite subset of a group. I don't know a general way to handle this with any elegance. I'm just looking at this little book "Categories and Groupoids" by Philip Higgins from 1971. It's pretty serious math, but it's where I got the idea of just treating categories as algebraic objects in their own right. Usually they're used more as a way to talk about the structure of other mathematical objects. Anyway, since intervals are invertible, the natural phonemic structure of scales might be a groupoid. The best place to start is surely group theory. A fourth plus a tone is a fifth. A scale is first of all some rule about how intervals combine. Then one takes a subset of all the different possible combinations. The tricky part is, is there any structure to how the subset is picked. The whole groupoid tack is one blind grope toward some such structure. Jim SMTPOriginator: tuning@eartha.mills.edu From: touchles@sirius.com Subject: Silverman's JI dictionary PostedDate: 26-08-97 06:05:57 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 10:24:26-28-08-97 10:24:28,28-08-97 10:22:01-28-08-97 10:22:02 DeliveredDate: 28-08-97 10:22:02 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C1256501.002DFDA0; Thu, 28 Aug 1997 10:22:20 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00964; Tue, 26 Aug 1997 06:05:57 +0200 Date: Tue, 26 Aug 1997 06:05:57 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA00962 Received: (qmail 336 invoked from network); 26 Aug 1997 04:05:55 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 26 Aug 1997 04:05:55 -0000 Message-Id: <01BCB194.A1070800@ppp-astk20--205.sirius.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu