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.
@conference{d3d21d8a30724cf1a99de3a439b97fa4,
title = "The distance geometry of deep rhythms and scales",
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.",
author = "Demaine, {Erik D.} and Francisco Gomez-Martin and Henk Meijer and David Rappaport and Perouz Taslakian and Godfried Toussaint and Terry Winograd and Wood, {David R.}",
year = "2005",
month = "1",
day = "1",
language = "English (US)",
pages = "163--166",
note = "17th Canadian Conference on Computational Geometry, CCCG 2005 ; Conference date: 10-08-2005 Through 12-08-2005",

}

TY - CONF

T1 - The distance geometry of deep rhythms and scales

AU - Demaine, Erik D.

AU - Gomez-Martin, Francisco

AU - Meijer, Henk

AU - Rappaport, David

AU - Taslakian, Perouz

AU - Toussaint, Godfried

AU - Winograd, Terry

AU - Wood, David R.

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34848897081&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34848897081&partnerID=8YFLogxK

M3 - Paper

AN - SCOPUS:34848897081

SP - 163

EP - 166

ER -