The distance geometry of deep rhythms and scales

Erik D. Demaine, Francisco Gomez-Martin, Henk Meijer, David Rappaport, Perouz Taslakian, Godfried Toussaint, Terry Winograd, David R. Wood

    Research output: Contribution to conferencePaper

    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 Erdos-deep. The problem is a discrete, one-dimensional (circular) analog to an unsolved problem posed by Erdos in the plane.

    Original languageEnglish (US)
    Pages163-166
    Number of pages4
    StatePublished - Jan 1 2005
    Event17th Canadian Conference on Computational Geometry, CCCG 2005 - Windsor, Canada
    Duration: Aug 10 2005Aug 12 2005

    Conference

    Conference17th Canadian Conference on Computational Geometry, CCCG 2005
    CountryCanada
    CityWindsor
    Period8/10/058/12/05

    Fingerprint

    Distance Geometry
    Geometry
    Erdös
    Circle
    Analogue
    Interval

    ASJC Scopus subject areas

    • Geometry and Topology
    • Computational Mathematics

    Cite this

    Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G., ... Wood, D. R. (2005). The distance geometry of deep rhythms and scales. 163-166. Paper presented at 17th Canadian Conference on Computational Geometry, CCCG 2005, Windsor, Canada.

    The distance geometry of deep rhythms and scales. / Demaine, Erik D.; Gomez-Martin, Francisco; Meijer, Henk; Rappaport, David; Taslakian, Perouz; Toussaint, Godfried; Winograd, Terry; Wood, David R.

    2005. 163-166 Paper presented at 17th Canadian Conference on Computational Geometry, CCCG 2005, Windsor, Canada.

    Research output: Contribution to conferencePaper

    Demaine, ED, Gomez-Martin, F, Meijer, H, Rappaport, D, Taslakian, P, Toussaint, G, Winograd, T & Wood, DR 2005, 'The distance geometry of deep rhythms and scales' Paper presented at 17th Canadian Conference on Computational Geometry, CCCG 2005, Windsor, Canada, 8/10/05 - 8/12/05, pp. 163-166.
    Demaine ED, Gomez-Martin F, Meijer H, Rappaport D, Taslakian P, Toussaint G et al. The distance geometry of deep rhythms and scales. 2005. Paper presented at 17th Canadian Conference on Computational Geometry, CCCG 2005, Windsor, Canada.
    Demaine, Erik D. ; Gomez-Martin, Francisco ; Meijer, Henk ; Rappaport, David ; Taslakian, Perouz ; Toussaint, Godfried ; Winograd, Terry ; Wood, David R. / The distance geometry of deep rhythms and scales. Paper presented at 17th Canadian Conference on Computational Geometry, CCCG 2005, Windsor, Canada.4 p.
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