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Asymmetric Rhythms and Tiling Canons

Asymmetric Rhythms and Tiling Canons. Dr. Rachel Hall Saint Joseph’s University Shippensburg University Student Math Conference. Feel the beat. Classic 4/4 beat Syncopated 4/4 beat How are these rhythms different? We will explore ways of describing rhythm mathematically.

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Asymmetric Rhythms and Tiling Canons

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  1. Asymmetric Rhythms and Tiling Canons Dr. Rachel Hall Saint Joseph’s University Shippensburg University Student Math Conference

  2. Feel the beat • Classic 4/4 beat • Syncopated 4/4 beat • How are these rhythms different? • We will explore ways of describing rhythm mathematically. Asymmetric rhythms and tiling canons

  3. Math for drummers • The mathematical analysis of rhythm has a long history. • In fact, ancient Indian scholars discovered the Fibonacci numbers and Pascal’s triangle by counting rhythms in Sanskrit poetry. • They discovered the Fibonacci numbers fifty years before Fibonacci, and Pascal’s triangle 18 centuries before Pascal! Asymmetric rhythms and tiling canons

  4. 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. Asymmetric rhythms and tiling canons

  5. or Notation Here are several ways to represent the same rhythm: • Standard Western notation • Drum tablature: x..x..x. • Binary: 10010010 Asymmetric rhythms and tiling canons

  6. 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. Asymmetric rhythms and tiling canons

  7. Composition 001 • Choose a rhythm (not the same as mine!) • Write down all the patterns that are equivalent to your rhythm. . . . . x x ...x.x x...x. etc. Asymmetric rhythms and tiling canons

  8. 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. Asymmetric rhythms and tiling canons

  9. 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? Asymmetric rhythms and tiling canons

  10. 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. Asymmetric rhythms and tiling canons

  11. Fourteen rhythm cycles  Asymmetric rhythms and tiling canons

  12. 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. Asymmetric rhythms and tiling canons

  13. 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 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 x.x...x. Examples Asymmetric rhythms and tiling canons

  14. . . x . x . DIY! How can I fill in the rest of the template to make an asymmetric cycle? Asymmetric rhythms and tiling canons

  15. 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. Asymmetric rhythms and tiling canons

  16. Example Schumann, “Kind im Einschlummern” Voice 1: x.xxxx..x.xxxx.. Voice 2: x.xxxx..x.xxxx.. Asymmetric rhythms and tiling canons

  17. 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. Asymmetric rhythms and tiling canons

  18. 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? Asymmetric rhythms and tiling canons

  19. . . x . x . . . x . x . Asymmetric rhythms and complementary canons To make a rhythm asymmetric, you make the canon complementary. When will you get a tiling canon? Asymmetric rhythms and tiling canons

  20. Oh, those crazy canons! 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 Asymmetric rhythms and tiling canons

  21. 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.. Asymmetric rhythms and tiling canons

  22. 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. Asymmetric rhythms and tiling canons

  23. Tiling the integers A tiling of the integersis 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, …} Asymmetric rhythms and tiling canons

  24. Example (continued) A = {0, 2} B = {…, 0, 1, 4, 5, 8, 9, …} Every rhythmic tiling canon corresponds to an integer tiling! … … 0 1 2 3 4 5 6 7 8 9 10 11 Asymmetric rhythms and tiling canons

  25. 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? Asymmetric rhythms and tiling canons

  26. 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.). • In this case, there is an algorithm for constructing the translation set. • The answer is unknown for more than three primes. Asymmetric rhythms and tiling canons

  27. Inversion and monohedral tiling • Playing a rhythm backwards gives you its inversion. Tiling canons using a rhythm and its inversion are called monohedral. • 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! Asymmetric rhythms and tiling canons

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