1. Asymmetric Rhythms and Tiling Canons
Dr. Rachel Hall
Saint Joseph’s University
EPADEL Fall 2006 Meeting
West Chester University

2. Asymmetric rhythms and tiling canons
Feel the beat
Classic 4/4 beat
Syncopated 4/4 beat
How are these rhythms different?
We will explore ways of describing rhythm mathematically.
October 28th, 2006
Asymmetric rhythms and tiling canons

3. Asymmetric rhythms and tiling canons
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 50 years before Fibonacci, and Pascal’s triangle 18 centuries before Pascal!
October 28th, 2006
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

5. Asymmetric rhythms and tiling canons
Notation
Here are several ways to represent the same rhythm:
Standard Western notation
Drum tablature: x..x..x.
Binary: or
October 28th, 2006
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

7. Asymmetric rhythms and tiling canons
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

8. Asymmetric rhythms and tiling canons
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

9. Asymmetric rhythms and tiling canons
Questions
How many different rhythm patterns with 6 beats are possible?
How many are in your rhythm cycle?
What are the possible answers to the previous question?
What does “6” have to do with it?
October 28th, 2006
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

11. Fourteen rhythm cycles
October 28th, 2006
Asymmetric rhythms and tiling canons

12. Asymmetric rhythms and tiling canons
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

13. Asymmetric rhythms and tiling canons
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
October 28th, 2006
Asymmetric rhythms and tiling canons

14. Asymmetric rhythms and tiling canons
DIY! x .
How can I fill in the rest of the template to make an asymmetric cycle?
October 28th, 2006
Asymmetric rhythms and tiling canons

15. Asymmetric rhythms and tiling canons
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

16. Asymmetric rhythms and tiling canons
Example
Schumann, “Kind im Einschlummern”
Voice 1: x.xxxx..x.xxxx..
Voice 2: x.xxxx..x.xxxx..
October 28th, 2006
Asymmetric rhythms and tiling canons

17. Asymmetric rhythms and tiling canons
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

18. Asymmetric rhythms and tiling canons
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?
October 28th, 2006
Asymmetric rhythms and tiling canons

19. Asymmetric rhythms and complementary canons
To make a rhythm asymmetric, you make the canon complementary.
When will you get a tiling canon? x . x .
October 28th, 2006
Asymmetric rhythms and tiling canons

20. Asymmetric rhythms and tiling canons
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
October 28th, 2006
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..
October 28th, 2006
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.
October 28th, 2006
Asymmetric rhythms and tiling canons

23. Asymmetric rhythms and tiling canons
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, …}
October 28th, 2006
Asymmetric rhythms and tiling canons

24. Asymmetric rhythms and tiling canons
Example (continued)
A = {0, 2}
B = {…, 0, 1, 4, 5, 8, 9, …}
Every rhythmic tiling canon corresponds to an integer tiling! … … 1 2 3 4 5 6 7 8 9 10 11
October 28th, 2006
Asymmetric rhythms and tiling canons

25. Asymmetric rhythms and tiling canons
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?
October 28th, 2006
Asymmetric rhythms and tiling canons

26. Asymmetric rhythms and tiling canons
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.
October 28th, 2006
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!
October 28th, 2006
Asymmetric rhythms and tiling canons