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The Distance Geometry of Deep Rhythms and Scales

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

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The Distance Geometry of Deep Rhythms and Scales

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

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

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

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

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

  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. Erdős Distance & Rhythms • A rhythm that has the property asked by Erdős is called a Erdős- deep rhythm 0 15 14 Multiplicity 4 5 10 2 7 4 1 6 5 9

  9. Erdős Distance & Rhythms • A rhythm that has the property asked by Erdős is called a Erdős- deep rhythm 0 15 14 4 Multiplicity 4 5 10 2 7 4 1 6 5 9

  10. Erdős Distance & Rhythms • A rhythm that has the property asked by Erdős is called a Erdős- deep rhythm 0 15 14 4 6 Multiplicity 4 5 10 2 7 4 1 6 5 9

  11. Erdős Distance & Rhythms • A rhythm that has the property asked by Erdős is called an Erdős-deeprhythm 0 15 14 Multiplicity 4 7 7 5 10 2 7 4 1 6 5 9

  12. Erdős Distance & Rhythms • A rhythm that has the property asked by Erdős is called an Erdős-deeprhythm 0 15 14 4 4 Multiplicity 4 5 4 10 2 7 4 1 6 5 9

  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 Winograd: Deep Scales • The term deep was first introduced by Winograd in 1966 in an unpublished class term paper.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, withk = n/2ork = n/2 + 1

  14. The Diatonic Scale is Deep C B D A Multiplicity E G F 6 1 4 3 2 5 n = 12 k = 7

  15. Examples of Deep Rhythms Cuban Tresillo

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

  17. Examples of Deep Rhythms Cuban Cinquillo Cuban Tresillo

  18. Examples of Deep Rhythms Bossa–Nova

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

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

  21. Characterization: Example D7,16,5 n = 16 k = 7 ≤ 9 m = 5

  22. Characterization: Example D7,16,5 0 n = 16 k = 7 ≤ 9 m = 5

  23. Characterization: Example D7,16,5 0 n = 16 k = 7 ≤ 9 m = 5 5

  24. Characterization: Example D7,16,5 0 n = 16 k = 7 ≤ 9 m = 5 5 10

  25. Characterization: Example D7,16,5 0 15 n = 16 k = 7 ≤ 9 m = 5 5 10

  26. Characterization: Example D7,16,5 0 15 n = 16 k = 7 ≤ 9 m = 5 4 5 10

  27. Characterization: Example D7,16,5 0 15 n = 16 k = 7 ≤ 9 m = 5 4 5 10 9

  28. Characterization: Example D7,16,5 0 15 14 n = 16 k = 7 ≤ 9 m = 5 4 5 10 9

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

  30. 0 15 14 4 5 10 9 Deep Shellings • A shelling of a Erdős-deep rhythm R is a sequence s1, s2, …, sk of onsets in R such that R – {s1, s2, …, si} is a Erdős-deep rhythm for i = 0, …, k.

  31. Deep Shellings • A shelling of a Erdős-deep rhythm R is a sequence s1, s2, …, sk of onsets in R such that R – {s1, s2, …, si} is a Erdős-deep rhythm for i = 0, …, k. 0 15 14 4 5 10 9

  32. Deep Shellings • A shelling of a Erdős-deep rhythm R is a sequence s1, s2, …, sk of onsets in R such that R – {s1, s2, …, si} is a Erdős-deep rhythm for i = 0, …, k. 0 15 14 4 5 10 9

  33. Deep Shellings • A shelling of a Erdős-deep rhythm R is a sequence s1, s2, …, sk of onsets in R such that R – {s1, s2, …, si} is a Erdős-deep rhythm for i = 0, …, k. 0 15 14 4 5 10 9

  34. Deep Shellings • A shelling of a Erdős-deep rhythm R is a sequence s1, s2, …, sk of onsets in R such that R – {s1, s2, …, si} is a Erdős-deep rhythm for i = 0, …, k. 0 15 14 • Corollary: Every Erdős-deep rhythm has a shelling 4 5 10 9

  35. Thank you

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