source file: mills2.txt Date: Wed, 30 Oct 1996 09:02:10 -0800 Subject: gaRY: Diatonic Key Detection From: PAULE See the book, "Harmony and Tonality" edited by J. Sundberg and published by the Royal Swedish Academy of Music, Stockholm in 1987. Although there are articles about microtonality, the one in question is about 12-tet. I think Carol Krumhansl contributed to this one. Major and minor keys are characterized by a set of twelve "probe tone ratings," experimentally derived ratings of how well each of the twelve tones fits into a particular key. The minor and major "profiles" are transposed to assign to each of the twenty-four keys a point in twelve-dimensional space. Using multidimensional scaling, a useful statistical technique, the authors find that the relationships between the keys can be well represented on the surface of a torus, shown as a recangle where the top and bottom edges are assumed to connect and the left and right edges connect as well. The pattern shows the two circles of fifths (major and minor) as a series of diagonal lines (which connect through the edges to form an unbroken chain), with parallel and relative keys nearby to each other. A Bach piece is subject to the same multidimensional scaling procedure (I'm not sure how it is subdivided, or whether durations of notes count), and is shown to worm its way around a small region of the rectangle-torus. The results agreed very well with a traditional analysis of the piece. The probe-tone profiles are psychologically derived, but Georg Hajdu "shows" in Interface 22 that they may be related to the third or fourth octave of the harmonic series, at least for the major scale. He dismisses the seventh harmonic somehow; I think because it's too far from 12-tet to be important. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 30 Oct 1996 18:36 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02112; Wed, 30 Oct 1996 18:37:33 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA04670 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA18922; Wed, 30 Oct 1996 09:37:31 -0800 Date: Wed, 30 Oct 1996 09:37:31 -0800 Message-Id: <23961030173432/0005695065PK4EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu