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) |
|---|---|
| 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 |
|---|---|
| Country | Canada |
| City | Windsor |
| Period | 8/10/05 → 8/12/05 |
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ASJC Scopus subject areas
- Geometry and Topology
- Computational Mathematics
Cite this
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 conference › Paper
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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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