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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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 2
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.
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 3
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)?
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 4
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)
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 5
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 6
Canon Example
* Thanks to Anja Volk for helping to obtain this example.
0 8 16 18 26 34
4 12 20 22 30 38
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 7
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).
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 8
Canon Modulation
• R = fundamental rhythm
• S = set of displacements
• Change S to S + t.
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 9
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.
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 10
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.
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 11
Implementation
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 12
Implementation
S, Modulation 1
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 13
Example
• R = ( 0 1 4 5 12
25 29 36 42 58 59
63 )
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 14
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 15
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 16
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?
22Modulation 1
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 17
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?
22Modulation 1
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 18
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 19
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 20
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 21
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 22
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 23
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 24
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 25
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 26
Done!
• R = ( 0 1 4 5 12
25 29 36 42 58 59
63 )
• S = ( 0 22 38 40
54 56 )
Modulation 1
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 27
Implementation
S1, Modulation 2
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 28
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 29
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 30
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 31
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 32
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 33
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 34
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 35
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 36
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 37
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 38
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…..63 0.1
Modulation 2
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 39
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…..63 0.1
Modulation 2
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 40
Done!
• R = ( 0 1 4 5 12
25 29 36 42 58 59
63 )
• S1 = (16 38 54
56 70 0 )
Modulation 2
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 41
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 42
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 43
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] ) )
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 44
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*RR*S^
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 45
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^
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 46
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
ISE 599: Spring 2004: April 15 Andreatta et al: Geometric Groove 47
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…
