The Interval Angle: A Similarity Measure for Pitch-Class Sets

Over the past Thirty Years, many measures have been proposed to gauge the similarity (or, mutatis mutandis, the difference) in sound of pitch-class sets. We propose here that a very simple measure be adopted: the angle between the interval-class vectors in six-dimensional space. We call this function Angle. The measure has essentially been proposed before: a variant called cosΤ was first mentioned in a seminar paper by David Rogers in 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 Castrén'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 λ2, is the same metric space from which come nearly all formulas from classical mechanics. Not only can Angle 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 would take into account the possibility that some pairs of intervals themselves sound more similar to each other than other pairs.