source file: mills2.txt Date: Tue, 17 Sep 1996 17:29:44 -0700 Subject: RE: Consistency generalized From: PAULE Paul Hahn, Your generalization is quite correct and produces the desired result of a more stringent criterion. The musical meaning of integer levels of consistency higher than 1, namely, the importance of matching just intonation in combinations of n-limit intervals, I find dubious. So level 2 consistency seems like an arbitrary requirement, as level 2.1 consistency and level 1.9 consistency would lead to results which are slightly different but just as meaningful musically. One thing I looked at long ago was what we would now call level 3 consistency, since 12-tET is just barely level 3 consistent at the 5-limit. It seemed reasonable that an equal temperament could be "fuzzified" so that it covered one-third of frequency space, while two-thirds remained out-of-tune with respect to the equal temperament. The simplest qualifiers are: 3-limit: 5-tET 5-limit: 12-tET 7-limit: 31-tET 9-limit: 171-tET 11-limit: 342-tET (=2*171) My Commodore 64 couldn't do the higher-limit calculations within a reasonable period of time, but these days it's just a few seconds: 13-limit: 5585-tET (=5*1117) 15-limit: 5585-tET 17-limit: 16808-tET (=22*764) 19-limit: 20203-tET (=89*227) -Paul Erlich Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 18 Sep 1996 02:29 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08515; Wed, 18 Sep 1996 02:31:38 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA07845 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id RAA16328; Tue, 17 Sep 1996 17:31:33 -0700 Date: Tue, 17 Sep 1996 17:31:33 -0700 Message-Id: <960917190242_480834378@emout09.mail.aol.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu