Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Francisco Gomez-Martin, Henk Meijer, David Rappaport, Perouz Taslakian, Godfried T. Toussaint, Terry Winograd, and David R. Wood, “The Distance Geometry of Deep Rhythms and Scales”, in Proceedings of the 17th Canadian Conference on Computational Geometry (CCCG 2005), Windsor, Ontario, Canada, August 10–12, 2005, pages 163–166.

Abstract:

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.

Length:

The paper is 4 pages.

Availability:

The paper is available in PostScript (324k), gzipped PostScript (148k), and PDF (184k).

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Related papers:

DeepRhythms_CGTA (The Distance Geometry of Music)