source file: mills2.txt Date: Thu, 19 Sep 1996 15:16:19 -0700 Subject: RE: RE: Consistency generalized From: PAULE >the average error of random intervals is >1/4 step [given] level 2 consistency . . . Paul H., surely the average error is less than 1/4 step, since that is the maximum error. The mean error is actually zero, since errors can be positive or negative, but the standard deviation (square root of mean of squared errors) is a better measure of the overall size of the errors. Graphing the standard deviation with respect to all the JI intervals up to a given limit for all ETs starting with 1-tET gives a bumpy graph (with dips at famous ETs) which for the most part closely follows an overall curve. This curve is given by 1/sqrt(12) times the step size of the ET. In other words, without assuming any consistency at all, the standard error of random intervals is .2887 step. Of course, the mean absolute error of random intervals, without assuming any consistency at all, is .25 step, since all values from 0 to .5 step are taken on with equal probability. I have not yet figured out how to calculate the expected errors given consistency, but a good measure of how "special" an ET is is given by assuming "in-tune-ness" is given by a normal curve centered around the just interval with some appropriately psychoacoustically researched standard deviation (in this context, standard deviation refers to the "width" of the curve). Then the difference between the "in-tune-ness" of the standard deviation of the ET and the "in-tune-ness" of the nominal value of 1/sqrt(12) times the step size gives you the "specialness" of the ET. My first post to this digest was a table of the most "special" tunings, for various harmonic limits and various reasonable values of the standard deviation, and I later reposted it with all the inconsistent tunings deleted. Needless to say, most everyone's favorite ET was to be found somewhere on the table -- they're all the most special! -Paul E. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 20 Sep 1996 05:29 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA10972; Fri, 20 Sep 1996 05:31:30 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA09507 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id UAA19745; Thu, 19 Sep 1996 20:31:28 -0700 Date: Thu, 19 Sep 1996 20:31:28 -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