Damon Scott (Eastman School of Music, presenter) and
Eric J. Isaacson (Indiana University)
Over the years, many measures have been proposed to guage the difference in sound of pitch class-sets. We propose here that a very simple measure be adopted: take the angle between the inteval-class vectors in six-dimensional space as the measure of the difference in sound of the original pitch-class-sets. We call this function ANGLE. The measure has essentially been proposed before: a variant called cos-theta was first mentioned in a seminar paper of 1992. Apparently unnoticed, however, were the striking features of ANGLE which in our opinion indicate that it is a particularly useful similarity measure.
ANGLE looks and feels very much like a measure taken from physics; in fact, a measure identical in spirit is used throughout Astronomy to measure the dissimilarity of apparent positions of stars when they are viewed from earth. It also has a very high correlation with Castren's %Rel and RECREL. The function ANGLE is not very hard to calculate, and the metric space from which the formula for ANGLE comes, called L_2, is the same metric space from which come nearly all formulas from physics. ANGLE not only can measure the difference in sound of pitch-class-sets of different cardinality, but a readily obtainable generalization of it can measure the difference in sound of music taking into account octave doublings, note repetitions and chord inversions. Finally, it can be altered to produce variant measures which will conform to the fact that some pairs of intervals themselves sound more similar to each other than other pairs.