source file: mills2.txt Date: Wed, 5 Mar 1997 15:26:16 -0800 Subject: Meaner tones From: John Chalmers Paul Erlich's and my last posts may have crossed on the Net, but I will try to answer his question in TD #1005. 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. Whether the subjective difference is proportional to the difference in the summed squares is something to be determined experimentally. After I did this (to put off doing my taxes, alas), I decided to look for a tuning that would equalize the errors in major/minor triads with fifths of 720 cents. I solved these two equations: X + Y 720 (X, Y are the Major and minor thirds) and X-386.3137 Y-315.6413 (errors in TM, tm). By substituting in, one gets 395.3362 for X, 324.6638 for Y, 9.0225 for the absolute differences and 488.43 for the total squared error as (9.0225^2 x 2 + 325.62), 407.03 without the tm. This is also the unweighted least squares minimum error solution as well. [(TM-X)^2 + (tm-720+X)^2, differentiate wrt X, set equal to zero, solve for X.] As both thirds are on the sharp side, this should be a pretty good tuning. (Essentially what I did was to take the error in the Fifth and split it between the two thirds.) For comparison, the corresponding values for 10-tet with its neutral third of 360 cents are 1018.03 and 2985.72. Ten fails by both tests again. Paul is correct that 15-tet is the more consonant of the two tunings he suggested and he is certainly correct that adding in the minor third (or Major Sixth) shows the differences dramatically. My point (in case someone forgot what we are discussing) is that in this case using just the fifth and major third is sufficient to demonstrate the difference in consonance, on the assumption that squared errors from JI is a good proxy measure consonance. In the meantone cases, I think it would be very difficult to hear the differences between the various tunings we've suggested, but Paul could be right. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 6 Mar 1997 01:05 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07210; Thu, 6 Mar 1997 01:05:32 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA07202 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id QAA14942; Wed, 5 Mar 1997 16:02:13 -0800 Date: Wed, 5 Mar 1997 16:02:13 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu