>As for your more complex psychoacoustic model, what meantone-like >tuning(s) does it generate?
If you are referring to the normal curve model I proposed, the answer is, the same ones! It's a monotonic decreasing function of the sum of the squared errors, so minimizing the latter is the same as maximizing the former. If nothing else, the function is nice because is gives all tunings a rating between zero and one.
>As for "expressive unisons," they can be very poignant. I believe >Wyschnegradsky made considerable use of them.
What is this referring to?
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 12 Mar 1997 19:18 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03417; Wed, 12 Mar 1997 19:18:02 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA03415 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id KAA20139; Wed, 12 Mar 1997 10:16:33 -0800 Date: Wed, 12 Mar 1997 10:16:33 -0800 Message-Id: <009B12A1F682E230.6D2A@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu
>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.
Uh, abs(sum) seems like a bad idea, since pos. and neg. errors cancel out in the sum. As for sum of abs(errors), you certinaly get different results from least squares. Using the three intervals in the 5-limit triad, the best meantone tunings according to absolute error have fifths of 696.57842466 cents (unweighted) and 696.57841680 cents (limit-weighted). I normally don't use so many decimal places but you need to to see the difference between these two! Maybe you only thought you got tunings you already had!
-Paul E.
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 13 Mar 1997 06:00 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06689; Thu, 13 Mar 1997 06:00:26 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA06685 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id UAA10749; Wed, 12 Mar 1997 20:58:57 -0800 Date: Wed, 12 Mar 1997 20:58:57 -0800 Message-Id: <33279CD4.1381@dnvr.uswest.net> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu