**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)

See also other papers by Erik Demaine.

Last updated January 4, 2022 by Erik Demaine.