source file: mills3.txt Date: Tue, 16 Dec 1997 07:52:23 +0100 Subject: Reply to Graham Breed From: Gregg Gibson Unequal tunings waste the melodic substance no less surely than equal temperaments whose tuning degrees are either much narrower or much wider than 60 cents or so. A 19-tone unequal tuning would result in appreciably more intervallic confusion, given the fact that the 19-tone equal tuning degree is close to the limen. Modulation would also be made more difficult, and certain consonances would be too deteriorated for use. 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. 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. Gregg Gibson said: > > ... but the resulting modes conflate a number of > > aurally distinct modes into single, neutral modes, and lack much of the > > distinct melodic character of the 19-tone versions. Graham Breed replied: > > Which modes cease to be distinct? Of course, the melodic character > will be different in different tunings, but that is not to say one is > superior to the others. In good old 31 equal, you get the added bonus > of septimal chords, consonant or otherwise. I have enumerated all the possible diatonic and chromatic modes in 12-tone equal and 19-tone equal, using tree diagrams. I can assure you that many chromatic modes distinct in 19-tone equal are merged into a much smaller number of such modes in the 12-tone equal. The exact number of these modes depends on what criteria one uses, e.g. numbers and character of consonances, number of consonant chords, inclusion of septimals, etc. I will post these modes in due course - meaning when I have a solid day or two free for the purpose. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: Reply to John Starrett PostedDate: 16-12-97 08:47:12 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: 16-12-97 08:45:14-16-12-97 08:45:15,16-12-97 08:44:54-16-12-97 08:44:55 DeliveredDate: 16-12-97 08:44:55 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.002A93FB; Tue, 16 Dec 1997 08:47:00 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA16624; Tue, 16 Dec 1997 08:47:12 +0100 Date: Tue, 16 Dec 1997 08:47:12 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA16627 Received: (qmail 22966 invoked from network); 15 Dec 1997 23:47:08 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 15 Dec 1997 23:47:08 -0800 Message-Id: <34969448.3D68@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu