The Distance Geometry of Deep Rhythms and Scales

We characterize which sets of k points chosen from n points spaced evenly around a circle have the property that, for each i = 1, 2, . . . , k−1, there is a nonzero distance along the circle that occurs as the distance between exactly i pairs from the set of k points. Such a set can be interpreted as the set of onsets in a rhythm of period n, or as the set of pitches in a scale of n tones, in which case the property states that, for each i = 1, 2, . . . , k − 1, there is a nonzero time [tone] interval that appears as the temporal [pitch] distance between exactly i pairs of onsets [pitches]. Rhythms with this property are called Erdős-deep. The problem is a discrete, one-dimensional (circular) analog to an unsolved problem posed by Erdős in the plane.