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The Distance Geometry of Deep Rhythms and Scales by E. Demaine, F. Gomez-Martin, H. Meijer, D. Rappaport, P. Taslakian, G. Toussaint, T. Winograd, D. Wood

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2 Rhythms and Scales A rhythm is a repeating pattern of beats that is a subset of equally spaced pulses.

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3 Rhythms and Scales A rhythm is a repeating pattern of beats that is a subset of equally spaced pulses. 0 8 4 12

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4 Rhythms and Scales A rhythm is a repeating pattern of beats that is a subset of equally spaced pulses. 0 8 4 12 clave Son

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5 Rhythms and Scales A scale is a collection of musical notes sorted by pitch. Diatonic scale

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6 Rhythms and Scales Pitch intervals in a scale are not necessarily the same Similar to a rhythm, a scale is cyclic Diatonic scale or Bembé C D E F G A B its geometric representation is similar to that of a rhythm

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7 Erdős Distance Problem (1989) Find n points in the plane s.t. for every i = 1,…, n-1, there exists a distance determined by these points that occurs exactly i times. Solved for 2 ≤ n ≤ 8 (0, 2) (1, 0) (–1, 0) (0, –1)

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8 Erdős Distance & Rhythms A rhythm that has the property asked by Erdős is called a Erdős- deep rhythm 0 4 274165 Multiplicity 5 9 10 14 15

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9 Erdős Distance & Rhythms A rhythm that has the property asked by Erdős is called a Erdős- deep rhythm 0 4 274165 Multiplicity 5 9 10 14 15 4

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10 Erdős Distance & Rhythms A rhythm that has the property asked by Erdős is called a Erdős- deep rhythm 0 4 274165 Multiplicity 5 9 10 14 15 4 6

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11 Erdős Distance & Rhythms A rhythm that has the property asked by Erdős is called an Erdős- deep rhythm 0 4 274165 Multiplicity 5 9 10 14 15 7 7

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12 Erdős Distance & Rhythms A rhythm that has the property asked by Erdős is called an Erdős- deep rhythm 0 4 274165 Multiplicity 5 9 10 14 15 4 4 4

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13 Winograd: Deep Scales The term deep was first introduced by Winograd in 1966 in an unpublished class term paper. He studied a restricted version of the Erdős property in musical scales He characterized the deep scales with n intervals and k pitches, with k = n/2 or k = n/2 + 1

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14 The Diatonic Scale is Deep C D E F G A B n = 12 k = 7 614325 Multiplicity

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15 Examples of Deep Rhythms Cuban Tresillo

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16 Examples of Deep Rhythms Cuban Tresillo Helena Paparizou Eurovision 2005 “My Number One”

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17 Examples of Deep Rhythms Cuban Tresillo Cuban Cinquillo

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18 Examples of Deep Rhythms Bossa–Nova

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19 Characterization Erdős-deep rhythms consist of : 1. D k,n,m = { i.m mod n | i = 0, …, k} 2. F = {0, 1, 2, 4} 6 - m and n are relatively prime - k ≤ n/2 + 1 n = 6 k = 4

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20 Characterization Erdős-deep rhythms consist of : 1. D k,n,m = { i.m mod n | i = 0, …, k} 2. F = {0, 1, 2, 4} 6 - m and n are relatively prime - k ≤ n/2 + 1 n = 6 k = 4

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21 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5

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22 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0

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23 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0 5

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24 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0 5 10

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25 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0 5 10 15

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26 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0 5 10 15 4

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27 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0 5 10 15 4 9

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28 Characterization: Example D 7,16,5 n = 16 k = 7 ≤ 9 m = 5 0 5 10 15 4 9 14

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29 Main Theorem A rhythm is Erdős-deep if and only if it is a rotation or scaling of F or the rhythm D k,n,m for some k, n, m with k ≤ n/2 + 1, 1 ≤ m ≤ n/2 and m and n are relatively prime.

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30 Deep Shellings A shelling of a Erdős-deep rhythm R is a sequence s 1, s 2, …, s k of onsets in R such that R – {s 1, s 2, …, s i } is a Erdős-deep rhythm for i = 0, …, k. 0 9 5 10 14 15 4

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31 Deep Shellings A shelling of a Erdős-deep rhythm R is a sequence s 1, s 2, …, s k of onsets in R such that R – {s 1, s 2, …, s i } is a Erdős-deep rhythm for i = 0, …, k. 0 9 5 10 14 15 4

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32 Deep Shellings A shelling of a Erdős-deep rhythm R is a sequence s 1, s 2, …, s k of onsets in R such that R – {s 1, s 2, …, s i } is a Erdős-deep rhythm for i = 0, …, k. 0 9 5 10 14 15 4

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33 Deep Shellings A shelling of a Erdős-deep rhythm R is a sequence s 1, s 2, …, s k of onsets in R such that R – {s 1, s 2, …, s i } is a Erdős-deep rhythm for i = 0, …, k. 0 9 5 10 14 15 4

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34 Deep Shellings A shelling of a Erdős-deep rhythm R is a sequence s 1, s 2, …, s k of onsets in R such that R – {s 1, s 2, …, s i } is a Erdős-deep rhythm for i = 0, …, k. 0 9 5 10 14 15 4 Corollary: Every Erdős-deep rhythm has a shelling

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35 Open Problem 1. Given the frequency of each distance, reconstruct the onsets of the deep rhythm (i.e. find m and n). This is a special case of the Beltway problem 0 9 10 14 15

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36 Thank you

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