### 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 language | English (US) |
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Pages | 163-166 |

Number of pages | 4 |

State | Published - Jan 1 2005 |

Event | 17th Canadian Conference on Computational Geometry, CCCG 2005 - Windsor, Canada Duration: Aug 10 2005 → Aug 12 2005 |

### Conference

Conference | 17th Canadian Conference on Computational Geometry, CCCG 2005 |
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Country | Canada |

City | Windsor |

Period | 8/10/05 → 8/12/05 |

### Fingerprint

### ASJC Scopus subject areas

- Geometry and Topology
- Computational Mathematics

### Cite this

*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.

Research output: Contribution to conference › Paper

}

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.

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M3 - Paper

SP - 163

EP - 166

ER -