source file: mills2.txt Date: Mon, 10 Mar 1997 09:57:56 -0800 Subject: RE: Meaner Tones (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE John Chalmers has derived some meantone tunings using the augmented sixth as the approximation to the harmonic seventh. But what if, more relevant to modern practice, the minor seventh must approximate the harmonic seventh, the minor third must approximate the 7:6, and the diminished fifth must approximate the 7:5? Add terms for these to the usual 5-limit terms, and the following "optimal" tunings are found: Equal-weighted: Fifth3+(log(3)+13*log(5)-11*log(7))/log(2))/75 octaves, or 702.2260 cents Limit-weighted: Fifth075+(81*log(3)+469*log(5)-539*log(7))/log(2))/3035 octaves, or 703.4736 cents So if one wants a diatonic system with optimal dominant seventh (and half-diminished seventh, if you subscribe to some form of duality) chords, one must use fifths slightly _larger_ than pure. -Paul Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 10 Mar 1997 19:29 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00782; Mon, 10 Mar 1997 19:29:46 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA00780 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id KAA04727; Mon, 10 Mar 1997 10:27:52 -0800 Date: Mon, 10 Mar 1997 10:27:52 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu