source file: mills2.txt Date: Fri, 14 Mar 1997 02:30:17 -0800 Subject: RE: Positive Linear Temperaments (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE Finally, optimizing the dominant ninth chord (10 intervals) of the diatonic scale with respect to a 4:5:6:7:9 tuning yields the fifth sizes (10+(2*log(3)+3*log(5)-3*log(7))/log(2))/20 octaves or 702.8187 cents (equal-weighted) (2642+(810*log(3)+631*log(5)-863*log(7))/log(2))/5060 octaves or 703.9231 cents (limit-weighted, 9:3 treated as 9-limit) (2714+(369*log(3)+631*log(5)-863*log(7))/log(2))/4988 octaves or 703.9515 cents (limit-weighted, 9:3 treated as 3-limit) So these are good tunings for diatonic music that uses the dominant ninth chord as a tonic. The subharmonic version of the diatonic dominant ninth chord is a diatonic dominant ninth chord, so making it approximate 4:5:6:7:9 automatically makes it approximate 1/9:1/7:1/6:1/5:1/4 just as well. I think it will tend to be heard as 4:5:6:7:9, though. Note that our optimized tunings have fifths spanning the range from just over 696 cents to just under 704 cents. It appears that for the diatonic scale, 12-equal is a nice central point from which other tunings can be regarded as small deviations. -Paul Erlich Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 14 Mar 1997 17:34 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17121; Fri, 14 Mar 1997 17:34:46 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17115 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id IAA11347; Fri, 14 Mar 1997 08:24:21 -0800 Date: Fri, 14 Mar 1997 08:24:21 -0800 Message-Id: <33298E56.46DE@dnvr.uswest.net> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu