source file: mills2.txt Date: Wed, 12 Mar 1997 15:01:37 -0800 Subject: RE: Positive Linear Temperaments From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE John C., Perhaps you didn't see what I was trying to do with the optimal positive temperaments I proposed. The idea is that we are optimizing the _diatonic_ dominant seventh chord with respect to a tuning of 4:5:6:7 by minimizing the weighted or unweighted mean of the squared errors of all six intervals in that chord. No schismatic thirds, augmented sixths, etc. are used. There is a discussion currently about whether the 4:5:6:7 is the classical dominant seventh chord and whether it is too consonant. In most cases I would say that 4:5:6:7 should not be used for the dominant seventh and therefore a negative tuning such as meantone is best, as only the triads should be rendered as consonant as possible. But there are other styles for which diatonic dominant sevenths or minor 6th (that means minor with an added major 6th) chords are tonics, and should be tuned accordingly. My results indicate that mildly positive tunings are better for such styles. 12-tET is a compromise that leans slightly toward the latter; in fact I think it unlikely that mixolydian and dorian would have achieved their current popularity had a negative system become standard in Western music. -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 00:03 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04955; Thu, 13 Mar 1997 00:03:56 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA04952 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id PAA15168; Wed, 12 Mar 1997 15:02:17 -0800 Date: Wed, 12 Mar 1997 15:02:17 -0800 Message-Id: <009B12C606EBAEF3.6A55@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu