source file: mills2.txt Date: Fri, 13 Sep 1996 11:15:21 -0700 Subject: Consistency generalized From: Paul Hahn I've been mulling over Paul Erlich's concept of consistency for a little while now. When he first described it it really caught my attention, because it dovetailed nicely with a concept I've used in my little theoretical musings for years. The tunings I am interested in are what you might call "level 2 consistent": not only are all intervals up to a given limit represented consistently, but all combinations of them as well. Thus, for example, at the 5-limit 7TET is consistent, but it is not level 2 consistent because 5/4 is represented by two steps, but 25/16 is closer to five steps than four. To generalize, an ET is level N consistent at the K-limit if all approximations of K-limit intervals vary from just by no more than 1/(2N) of a step. This means that all combinations of up to N K-limit intervals are represented consistently. While Paul E. explores ETs which are level 1 consistent, I tend to be more interested in level 2 consistency, which is a much more stringent criterion. How much more? Well, I wrote a program to figure out consistency levels at all limits up to 31 for all ETs up to 1200TET. (Overkill, I know, but CPU cycles are cheap.) The output is available at . ("." indicates level 1 consistency.) Perusing it, one finds that the first ET with level 2 consistency at the 7-limit is 31, at the 9-limit 41, at the 11-limit 72, at the 13-limit 270, and at the 15-limit 494. No ET below 1200 exhibits level 2 consistency at the 17-limit. Contrast this with 29 and 41 being level 1 consistent at the 15-limit, and 58 being level 1 consistent at the 17-limit. --pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote) O /\ "Churchill? Can he run a hundred balls?" -\-\-- o Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 13 Sep 1996 22:34 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA30898; Fri, 13 Sep 1996 22:35:44 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA28367 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA12813; Fri, 13 Sep 1996 13:35:42 -0700 Date: Fri, 13 Sep 1996 13:35:42 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu