1. Asymmetric Rhythms and Tiling Canons Dr. Rachel Hall Saint Joseph’s University rhall@sju.edu Vassar College May 8, 2007

2. May 8th, 2007Asymmetric rhythms and tiling canons2 Feel the beat Classic 4/4 beat Syncopated 4/4 beat How are these rhythms different? We will explore ways of describing rhythm mathematically.

3. May 8th, 2007Asymmetric rhythms and tiling canons3 Math for drummers The mathematical analysis of rhythm has a long history. In fact, ancient Indian scholars discovered the Fibonacci numbers, Pascal’s triangle, and the rudiments of the binary number system by enumerating rhythms in Sanskrit poetry. They discovered the Fibonacci numbers fifty years before Fibonacci, and Pascal’s triangle 18 centuries before Pascal!

4. May 8th, 2007Asymmetric rhythms and tiling canons4 Beats, rhythms, and notes In music, the beat is the basic unit of time. A rhythm is a sequence of attacks (drum hits) or note onsets. A note is the interval between successive attacks. We will assume that every note begins on some beat.

5. May 8th, 2007Asymmetric rhythms and tiling canons5 Notation Here are several ways to represent the same rhythm: Standard Western notation Drum tablature: x..x..x. Binary: 10010010 or

6. May 8th, 2007Asymmetric rhythms and tiling canons6 Periodic rhythms If a rhythm is played repeatedly, it’s hard to tell where it starts. Two periodic rhythms are equivalent if one of them is the same as the other delayed by some number of beats. For example,.x.x..x. is equivalent to x..x..x. The set of all rhythms that are equivalent to a given pattern is called a rhythm cycle.

7. May 8th, 2007Asymmetric rhythms and tiling canons7 Composition 001 Choose a rhythm (not the same as mine!) Write down all the patterns that are equivalent to your rhythm. x... x....x.xx...x. etc.

8. May 8th, 2007Asymmetric rhythms and tiling canons8 Binary necklaces You can represent your rhythm as a necklace of black and white beads, called a binary necklace. The necklace can be rotated (giving you all the equivalent patterns) but not turned over.

9. May 8th, 2007Asymmetric rhythms and tiling canons9 Questions How many different rhythm patterns with six beats are possible? How many are in your rhythm cycle? What are the possible answers to the previous question? What does “six” have to do with it?

10. May 8th, 2007Asymmetric rhythms and tiling canons10 Counting rhythm cycles There are 64 rhythm patterns with six beats. Counting rhythm cycles is much more difficult. (Can you explain why?) It turns out that there are only 14 rhythm cycles with six beats. Burnside’s lemma is used to count these cycles.

11. May 8th, 2007Asymmetric rhythms and tiling canons11 Fourteen rhythm cycles of six beats

12. May 8th, 2007Asymmetric rhythms and tiling canons12 Burnside’s lemma Suppose a group G acts on a set X. Then the number n of equivalence classes in X under the action of G is the average, over G, of the number of elements of X that each element of G fixes. That is, In this case, X is a set of rhythm patterns, G is a group of cyclic permutations, and n is the number of rhythm cycles.

13. May 8th, 2007Asymmetric rhythms and tiling canons13 Burnside’s lemma in action n = number of cycles 0º rotation 90º rotation 180º rotation 270º rotation G = group of 90º rotations The number of times a pattern appears on this matrix equals the number of rotations fixing it. X = set of (distinct) patterns By inspection, there are n = six four-beat cycles

14. May 8th, 2007Asymmetric rhythms and tiling canons14 Using the formula to find n Suppose we didn’t know that n = 6. Then G = group of 90° rotations; |G| = 4. Number of patterns fixed by 0° rotation = 2 4 = 16. Number of patterns fixed by 90° rotation = 2 1 = 2. Number of patterns fixed by 180° rotation = 2 2 = 4. Number of patterns fixed by 270° rotation = 2 1 = 2.

15. May 8th, 2007Asymmetric rhythms and tiling canons15 General formula for number of N -beat cycles In general, where  is the Euler phi-function:  (d) = number of positive integers less than or equal to d that are relatively prime to d

16. May 8th, 2007Asymmetric rhythms and tiling canons16 Asymmetric rhythms A rhythm is syncopated if it avoids a beat that is normally accented (the first and middle beats of the measure). Can a rhythm cycle be syncopated? A rhythm cycle is asymmetric if all its component rhythm patterns are syncopated.

17. May 8th, 2007Asymmetric rhythms and tiling canons17 Examples Asymmetric cycle x..x..x..x..x..x x.x..x...x.x..x...x.x..x x..x.x...x..x.x...x..x.x Non-asymmetric cycle x.x...x..x.x...x x.x.x....x.x.x....x.x.x....x.x.x x...x.x..x...x.x

18. May 8th, 2007Asymmetric rhythms and tiling canons18 xx.... DIY! How can I fill in the rest of the beats to make a pattern belonging to an asymmetric cycle? In general, there are 3 N patterns of length 2 N that are members of asymmetric cycles.

19. May 8th, 2007Asymmetric rhythms and tiling canons19 Counting asymmetric rhythm cycles The number of asymmetric rhythm cycles of period 2 N is

20. May 8th, 2007Asymmetric rhythms and tiling canons20 Rhythmic canons A canon, or round, occurs when two or more voices sing the same tune, starting at different times. A rhythmic canon occurs when two or more voices play the same rhythm, starting at different times.

21. May 8th, 2007Asymmetric rhythms and tiling canons21 Example Schumann, “Kind im Einschlummern” Voice 1: x.xxxx..x.xxxx.. Voice 2: x.xxxx..x.xxxx..

22. May 8th, 2007Asymmetric rhythms and tiling canons22 More on canons Messaien, Harawi, “Adieu” Voice 1: x..x....x.......x....x..x...x..x......x..x...x.x.x..x....x.. Voice 2: x..x....x.......x....x..x...x..x......x..x...x.x.x..x....x.. Voice 3: x..x....x.......x....x..x...x..x......x..x...x.x.x..x....x.. A canon is complementary if no more than one voice sounds on every beat. If exactly one voice sounds on each beat, the canon is a tiling canon.

23. May 8th, 2007Asymmetric rhythms and tiling canons23 Make your own canon Fill in the template in your worksheet to make your rhythm into a canon. Is your canon complementary? If so, is it a tiling canon? What is the relationship to asymmetry?

24. May 8th, 2007Asymmetric rhythms and tiling canons24 Asymmetric rhythms and complementary canons To make a rhythm asymmetric, you make the canon complementary. When will you get a tiling canon? xx.... xx....

25. May 8th, 2007Asymmetric rhythms and tiling canons25 Canons with more than 2 voices A three-voice tiling canon x.....x..x.x|:x.....x..x.x:| x.....x.|:.x.xx.....x.:| x...|:..x..x.xx...:| The methods of constructing n-voice canons, where the voices are equally spaced from one another, are similar to the asymmetric rhythm construction. repeat sign

26. May 8th, 2007Asymmetric rhythms and tiling canons26 A four-voice tiling canon Voice 1: x.x.....|:x.x.....:| Voice 2: x.x....|:.x.x....:| Voice 3: x.x.|:....x.x.:| Voice 4: x.x|:.....x.x:| Entries: ee..ee..|:ee..ee..:| inner rhythm = x.x..... outer rhythm = ee..ee..

27. May 8th, 2007Asymmetric rhythms and tiling canons27 Tiling canons of maximal category A tiling canon has maximal category if the inner and outer rhythms have the same (primitive) period. None exist for periods less than 72 beats. Here’s one of period 72. You’ll hear the whistle sound the outer rhythm about halfway through.

28. May 8th, 2007Asymmetric rhythms and tiling canons28 Tiling the integers A tiling of the integers is a finite set A of integers (the tile) together with a set of translations B such that every integer may be written in a unique way as an element of A plus an element of B. Example: A = {0, 2} B = {…, 0, 1, 4, 5, 8, 9, …}

29. May 8th, 2007Asymmetric rhythms and tiling canons29 Example (continued) A = {0, 2} B = {…, 0, 1, 4, 5, 8, 9, …} Every rhythmic tiling canon corresponds to an integer tiling! 24531067891011 ……

30. May 8th, 2007Asymmetric rhythms and tiling canons30 Results and questions Theorem (Newman, 1977): All tilings of the integers are periodic. Can a given set A tile the integers? If so, what are the possible translation sets?

31. May 8th, 2007Asymmetric rhythms and tiling canons31 Partial answers Only the case where the size of the tile is divisible by less than four primes has been solved (Coven, Meyerowitz,Granville et al.). The proof uses results about the factorization of polynomials over the field of integers modulo N. In this case, there is an algorithm for constructing the translation set. The answer is unknown for more than three primes.

32. May 8th, 2007Asymmetric rhythms and tiling canons32 Inversion and monohedral tiling Playing a rhythm backwards gives you its inversion. Tiling canons using a rhythm and its inversion are called monohedral. Monohedral tiling canons can be aperiodic. Beethoven (Op. 59, no. 2) uses x..x.x and.xx.x. to form a monohedral tiling canon. Not much is known about monohedral tiling. Maybe you will make some discoveries!