source file: mills2.txt Date: Thu, 6 Mar 1997 13:24:52 -0800 Subject: RE: Meaner Tones (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE >He suggested a tuning with Major Thirds (TM) of 372 cents, >Fifths (F) of 720 and minor thirds (tm) of 348 and asks that I >compare this to 15-tet. I can't tune up the triads easily today, but >I did calculate the following: > Fifth Major T minor t (in cents) >PE's tuning 720 372 348 >15-tet 720 400 320 >720-F 18.05 (720-F)^2 325.62 >T-372 14.3137 (T-372)^2 204.88 >400-T 13.683 (400-T)^2 187.31 >348-tm 32.36 (348-tm)^2 1047.09 >320-tm 4.36 (320-tm)^2 19.0 >Now let us sum the squared errors of the fifth and the Major >Third for both tunings: >15-tet 325.62 + 187.31 512.93 >PE's 325.62 + 204.88 530.5 >15-tet is the more consonant tuning by this test, though not by much. >Now let us add the squared error of the minor third to each. >15-tet 512.93 +19531.93 >PE's 530.5 + 1047.09 77.59 >I will stipulate that this is a very large difference, but we already >knew that 15 tet was the more consonant. Grr! OK, PE's is now redefined (for the third time) to have a perfect fifth of 720 cents and a major third whose error is exactly equal in magnitude and opposite in sign to that of 15-equal. There. Now tell me that this isn't a glaring counterexample to judging triadic consonance by looking at perfect fifths and major thirds only! By the way, if we want to take these sum squared errors and interpret them as measures of consonance, we should divide by the number of terms in the sum to obtain a mean squared error, then divide by twice the squared tolerance of the ear, and exponentiate the negative of that result (in case you didn't get it, that's a normal distribution model, normalized to give a value of one for just intonation). The tolerance of the ear is about 1% so the squared tolerance is about 300 square cents. >Another question is whether the sum of the squared or even absolute >errors >is the proper measure of out-of-tuneness. Errors in the flat direction >may be more serious than sharp ones, but both our techniques conflate >the two. For the case I'm trying to make with the "PE's" tuning, it doesn't matter whether you use squared or absolute error. As for preferring sharp intervals, presumably this would apply to all intervals as well as their inversions, and since we are considering all intervals along with their inversions, all such asymmetry will cancel out. What does it mean to prefer a sharp unison over a flat unison? (that's just a joke, but there is a serious (hehe) debate on this topic over on rec.music.compose) Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 7 Mar 1997 03:04 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08226; Fri, 7 Mar 1997 03:04:25 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA08258 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id SAA12588; Thu, 6 Mar 1997 18:02:44 -0800 Date: Thu, 6 Mar 1997 18:02:44 -0800 Message-Id: <199703061841_MC2-1239-3640@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu