1. The Geometric. rhythmic canons between theory, implementation and musical experiment by Moreno Andreatta, Thomas Noll, Carlos Agon and Gerard Assayag presentation by Elaine Chew ISE 599: Spring 2004

2. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove2 Algebraic Definitions R = rhythm, represented by a locally finite set of rational numbers marking onset times dR = period of rhythm, smallest rational number s.t. R = R + dR pR = pulsation, GCD of IOI in R V = voice, a displacement of R by s = R + s S = set of all translations p = pulsation of canon, GCD(pR,pS), largest possible “tick size” for counting time in R+S.

3. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove3 Transformation to Z/Zn Convert everything element in R to integral counts of the canon’s “tick size”, i.e. r = r/p. Now R is a subset of the set of integers, Z. n = period of the new R = dR/p. By definition, if there is an onset at r  R, then there is another at r + n, r + 2n, r + 3n … Hence, we only need to store the non-repeating parts of R, i.e., a subset of Z, Z/nZ (the cyclic group, Z mod n). Note: should n be LCM(nR, nS)?

4. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove4 Generating Voices R  Z/Zn = inner rhythm S  Z/Zn = outer rhythm Vs = R + s, where s  S Example n = 16 R = (0 3 4 6 8) S = (0 1) V0 = (0 3 4 6 8), V1 = (1 4 5 7 9)

5. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove5 Types of Canons Rhythmic Tiling Canon I: Union of all Vs covers Z/Zn II: The voices are pairwise disjoint Regular Complementary Canons of Maximal Category, RCMC-Canons (VuZA, 1995) III: pR = pS

6. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove6 Canon Example * Thanks to Anja Volk for helping to obtain this example. 0 8 16 18 26 34 4 12 20 22 30 38

7. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove7 RCMC Canons Constitute a “factorization” of Z/Zn into two non-periodic subsets, R and S. Smallest such canon has period 72 and 6 voices (no cyclic group smaller than Z/72Z can be factorized into two non- periodic subsets).

8. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove8 Canon Modulation R = fundamental rhythm S = set of displacements Change S to S + t.

9. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove9 Rhythmic Re-interpretation R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) n = 72 New set of displacements if t = 16: S1 = ( 16 38 54 56 70 0 ) Note: additions are mod 72.

10. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove10 Rhythmic Re-interpretation R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) n = 72 New set of displacements if t = 16: S1 = ( 16 38 54 56 70 0 ) Note: additions are mod 72.

11. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove11 Implementation

12. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove12 Implementation  S, Modulation 1

13. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove13 Example R = ( 0 1 4 5 12 25 29 36 42 58 59 63 )

14. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove14 R = V0 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5………..12……………………..25……29……….... 36……….42……….58.59…..63 0 Modulation 1

15. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove15 R = V0 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5………..12……………………..25……29……….... 36……….42……….58.59…..63 0 Modulation 1

16. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove16 V22 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5………..12… …………………..25……29………... 36……….42……….58.59 ……..63? 22 Modulation 1

17. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove17 V22 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5………..12… …………………..25……29………... 36……….42……….58.59 ……..63? 22 Modulation 1

18. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove18 V38 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5……….12….………………….25……29……. ….36……….42………58.59..…63 38 Modulation 1

19. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove19 V38 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5……….12….………………….25……29……. ….36……….42………58.59..…63 38 Modulation 1

20. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove20 V40 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5……….12….………………….25……29…. ……..36……….42………58.59…..63 40 Modulation 1

21. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove21 V40 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5……….12….………………….25……29…. ……..36……….42………58.59…..63 40 Modulation 1

22. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove22 V54 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) ….………..25……29….……...36……….42……….58.59…..63 54 0.1…….4..5………..12……….. Modulation 1

23. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove23 V54 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) ….………..25……29….……...36……….42……….58.59…..63 54 0.1…….4..5………..12……….. Modulation 1

24. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove24 V56 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5………..12……. ….……..……..25…...29….……...36………..42……….58.59…. 56..63 Modulation 1

25. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove25 V56 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) 0.1…….4..5………..12……. ….……..……..25…...29….……...36………..42……….58.59…. 56..63 Modulation 1

26. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove26 Done! R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S = ( 0 22 38 40 54 56 ) Modulation 1

27. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove27 Implementation  S1, Modulation 2

28. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove28 V0 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 0 0.1…….4..5………..12……………………..25……29……….... 36……….42……….58.59…..63 Modulation 2

29. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove29 V0 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 0 0.1…….4..5………..12……………………..25……29……….... 36……….42……….58.59…..63 Modulation 2

30. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove30 V16 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 16 0.1…….4..5………..12…..…………..…..25……29………... 36……….42……….58.59…….63 Modulation 2

31. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove31 V16 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 16 0.1…….4..5………..12…..…………..…..25……29………... 36……….42……….58.59…….63 Modulation 2

32. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove32 V38 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 38 0.1…….4..5……….12….………………….25……29……. ….36……….42………58.59..…63 Modulation 2

33. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove33 V38 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 38 0.1…….4..5……….12….………………….25……29……. ….36……….42………58.59..…63 Modulation 2

34. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove34 V54 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) ….………..25……29….……...36……….42……….58.59…..63 54 0.1…….4..5………..12……….. Modulation 2

35. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove35 V54 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) ….………..25……29….……...36……….42……….58.59…..63 54 0.1…….4..5………..12……….. Modulation 2

36. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove36 V56 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 0.1…….4..5………..12……. ….……..……..25…...29….……...36………..42……….58.59…. 56..63 Modulation 2

37. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove37 V56 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 0.1…….4..5………..12……. ….……..……..25…...29….……...36………..42……….58.59…. 56..63 Modulation 2

38. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove38 V70 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 70 …….4.5.………..12……………………..25……29……….…36.. ……….42…..….58.59…..630.1 Modulation 2

39. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove39 V70 R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) 70 …….4.5.………..12……………………..25……29……….…36.. ……….42…..….58.59…..630.1 Modulation 2

40. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove40 Done! R = ( 0 1 4 5 12 25 29 36 42 58 59 63 ) S1 = (16 38 54 56 70 0 ) Modulation 2

41. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove41 Augmented Canons An affine transformation –[a,t](x) = (ax + t mod)(m*n) –where a is augmentation factor, t is summand –where R has m elements and S has n A(R), A(S) sets of augmentation factors T(R), T(S) corresponding translations R*S is all pairs of consecutive transformations | R*S | = set of all elements in R*S

42. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove42 Example R = ( [11,0] [5,1] [5,3] [11 10] ) S = ( [11,0] [5,1] [5,5] ) [a,t](0) = t mod 12 R(0) = ( 0 1 3 10 ) S(R(0)) = [ ( 0 11 9 2 ) ( 1 6 4 3 ) ( 5 10 8 7 ) ] S*R

43. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove43 R*S R = ( [11,0] [5,1] [5,3] [11 10] ) S = ( [11,0] [5,1] [5,5] ) R*S = (( [1,0] [7,1] [7,5] ) ( [7,1] [1,6] [1,2] ) ( [7,3] [1,8] [1,4] ) ( [1,10] [7,9] [7,5] ) )

44. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove44 S*R^ R = ( [11,0] [5,1] [5,3] [11 10] ) S = ( [11,0] [5,1] [5,5] ) S*R^ = (( [1,0] [7,1] [7,5] ) ( [7,11] [1,6] [1,10] ) ( [7,9] [1,4] [1,8] ) ( [1,2] [7,3] [7,7] ) ) Note that | S*R | = | R*S^ | but S*R  R*S^

45. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove45 Properties In non-augmenting case, –A(R) = (1 1 … 1) and A(S) = (1 1 … 1) –Canon duality: S*R = (R*S)^ When both (R,S) and (S,R) generate canons, –| R*S | = | S*R^ | –R*S = S*R^

46. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove46 Augmented and Dual Canons S*R(0) = (( 0 11 9 2 ) ( 1 6 4 3 ) ( 5 10 8 7 ) ) R*S^(0) = (( 0 11 7 ) ( 1 6 2 ) ( 3 8 4 ) ( 10 9 5 ) ) augmented canon dual canon

47. ISE 599: Spring 2004: April 15Andreatta et al: Geometric Groove47 Example R = ( 0 1 3 10 ) S = ([11,0] [5,1] [5,5]) V1 = s1(R) V2 = s2(R) V3 = s3(R) 0 11 33 110… 1 6 16 51 6166 76 111… 5 10 20 55 65 70 80 115…