source file: mills3.txt Date: Tue, 23 Dec 1997 19:15:23 +0100 Subject: Re: 19, 22, 29, 41, 53 : 12 and multiples From: Paul Hahn On Tue, 23 Dec 1997, Steven Rezsutek wrote: > If you'll pardon my asking what is most likely an FAQ, could you > offer a quick explanation of what level-n consistancy is all > about? Level 1 consistency, or just consistency, at some limit x is what you have when all triads within the x-limit can be consistently represented, e.g. the tuning's closest approximation of the larger interval is the sum of the closest approximations of the two smaller intervals. Example: in 24TET, 5/4 is about 8 steps and 7/5 is about 12 steps, but 7/4 is closer to 19 steps than 20. Ergo, 24TET is not consistent at the 7-limit, because you can't tune a 4:5:7 triad so that all intervals are represented as accurately as the tuning allows. For higher consistency levels, consistent representations exist not only for all the x-limit intervals, but also combinations of up to n of them. Example: 12TET is level 3 consistent at the 5-limit because even a stack of three of the interval which is represented with the greatest error, the 6/5 (that is, 6/5 x 6/5 x 6/5 = 216/125) is represented consistently: the closest 12TET approximation to 216/125 is 9 steps, which equals 3 times the 3-step interval which best approximates the 6/5. You can find tables of consistency levels at http://library.wustl.edu/~manynote/consist.txt and http://library.wustl.edu/~manynote/consist2.txt The first is simply all the consistency levels at all the relevant limits up to 1200TET; the second only includes an ET if there is some limit at which it exhibits a higher consistency level than any smaller numbered ET, going up to 10000TET. (A period signifies level 1 consistency.) Incidentally, I think this may relate to why Haba isn't thought of much these days: as you can see by examining the tables cited above, simply subdividing the semitones of 12TET doesn't really yield you very good representations of harmonic intervals until you get up to 72TET which is level 2 consistent at the 11-limit, but that's a bit too big of a jump for some people; too many notes. ("--Herr Mozart." 8-)> ) Most of us find it quite daunting enough trying develop something that makes sense down in the 19/22/31 range. --pH http://library.wustl.edu/~manynote O /\ "Do you like to gamble, Eddie? -\-\-- o Gamble money on pool games?" NOTE: dehyphenate address to remove spamblock. <*> SMTPOriginator: tuning@eartha.mills.edu From: alves@orion.ac.hmc.edu (Bill Alves) Subject: Re: Danielou, Touma PostedDate: 23-12-97 19:48:51 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: 23-12-97 19:46:36-23-12-97 19:46:37,23-12-97 19:46:08-23-12-97 19:46:08 DeliveredDate: 23-12-97 19:46:08 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 C1256576.006723BE; Tue, 23 Dec 1997 19:48:29 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24604; Tue, 23 Dec 1997 19:48:51 +0100 Date: Tue, 23 Dec 1997 19:48:51 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA24623 Received: (qmail 23609 invoked from network); 23 Dec 1997 10:48:45 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Dec 1997 10:48:45 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu