### 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 - 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 |

### ASJC Scopus subject areas

- Geometry and Topology
- Computational Mathematics

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## 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.