gaRY: Diatonic Key Detection

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.


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