source file: mills2.txt Date: Thu, 6 Mar 1997 11:34:44 -0800 Subject: Meaner tones again From: John Chalmers Paul E: The weights I used were 3, 5, and 5. My error function was thus 3*(F-X)^2 + 5*(TM-4*X+2*OCT)^2 + 5*(MS-3*X+0CT)^2. (FĪnts(3/2), TMĪnts(5/4), MScents(5/3), OCT1200) Your weights are 9 25 25, the squares of these. I weighted the derivatives, you the squared errors themselves. I'm not sure which is more correct mathematically, but your unweighted gives a slightly lower sum of sq'ed errors. I thought our original question was optimising meantone-like tunings. When one goes beyond the meantone range, other functions must be used (8 fifths down for TM, 14 down for the Harmonic 7th, 8 d or 9up for the TM for doubly positive systems). Yes, I agree that JI triads are better than MT. I agree that the 15/8 is much less important than the 6/5. In fact if one sets the errors of the third and fifth equal, one gets the 1/3 comma meantone, if opposite in sign, the 1/5 comma. In the first case the 6/5 is just, the second, the 15/8. I have analysed the tunings in my post in terms of the abs values of the errors. In which case the errors of the 5/3 and 15/8 may or may not be equal in a given tuning. The sum of the F and T errors may or may not equal the errors in either the 5/3 or 15/8. I could take the algebraic sum of the errors rather than the abs value instead, I suppose. Somewhat to my surprise, using the ABS function did not generate any tunings different from the ones I got by the lst squares technique or simply setting the errors equal or opposite. The exact solution depended upon whether I minimized the abs(sum) or sum of ABS (errors) as expected, but the results were still tunings I already had. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 6 Mar 1997 22:26 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07942; Thu, 6 Mar 1997 22:26:42 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA08006 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id NAA04439; Thu, 6 Mar 1997 13:24:52 -0800 Date: Thu, 6 Mar 1997 13:24:52 -0800 Message-Id: <009B0E05094CC68A.6381@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu