source file: mills3.txt Date: Wed, 17 Dec 1997 00:08:18 +0100 Subject: Calculation of Modes From: Gregg Gibson Gregg Gibson said: > > > To give some idea of just how immense this universe is - and remember > > > these are _real_, usable resources, not purely theoretical, aurally > > > imperceptible variations - compare the number of modes (I here use the > > > term loosely merely to indicate a collection of seven notes used in a > > > melody) consisting of 12 notes taken 7 at a time: > > > > > > nCr where n = 12 & r = 7 > > > = 792 > > > with the number of modes consisting of 19 notes taken 7 at a time: > > > nCr where n = 19 & r = 7 > > > = 50,388 Graham Breed replied: > > Really, you should define modes relative to a tonic, so the numbers > > get smaller. With 12, you're choosing 6 from 11, which gives 462 > > modes. 19C6 is 12376. The contrast is still striking. Including > > all the transpositions, you get 462*12=5544 and 12376*19=235144. Gregg Gibson replied: > Thanks very much for your correction regarding the method of finding > numbers of 12-tone versus 19-tone modes. You are quite right that the > tonic must be omitted from the calculation. The corrected numbers would > be nCr (n=11, r=6) = 462 and nCr (n=18, r=6) = 18564 (not 12376, you > must have used n=17), or a factor of about 40 to 1. Since I am concerned > here with modes, not keys, the transpositions I neglect. I would like to thank Graham Breed again for pointing out to me that in the calculation of the numbers of modes in a given system, the tonic must be excluded from the algorithm. This is because, I take it, the tonic is by definition present in each mode, and so does not vary. This is one of those obvious points that it is easy to overlook. I have wondered off and on for years why my informal estimates of the numbers of 12-tone equal modes were in the vicinity of 400 - 500, not 792 as my formal calculations suggested. Now I know! SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: 22-tone equal & chromaticism PostedDate: 17-12-97 00:26:50 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: 17-12-97 00:24:47-17-12-97 00:24:48,17-12-97 00:24:26-17-12-97 00:24:27 DeliveredDate: 17-12-97 00:24:27 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 C125656F.00809A69; Wed, 17 Dec 1997 00:26:37 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17453; Wed, 17 Dec 1997 00:26:50 +0100 Date: Wed, 17 Dec 1997 00:26:50 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17442 Received: (qmail 6148 invoked from network); 16 Dec 1997 15:26:42 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 16 Dec 1997 15:26:42 -0800 Message-Id: <3497709E.2589@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu