1. 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

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

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

4. 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

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

6. 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

7. 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)

8. 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

9. 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

10. 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

11. 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

12. 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

13. 13 Winograd: Deep Scales  The term deep was first introduced by Winograd in 1966 in an unpublished class term paper.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

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

15. 15 Examples of Deep Rhythms Cuban Tresillo

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

17. 17 Examples of Deep Rhythms Cuban Tresillo Cuban Cinquillo

18. 18 Examples of Deep Rhythms Bossa–Nova

19. 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

20. 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

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

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

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

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

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

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

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

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

29. 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.

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 36 Thank you