Asymmetric Rhythms and Tiling Canons

1
Asymmetric Rhythms and Tiling Canons

• Dr. Rachel Hall
• Saint Josephs 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.

3
Math for drummers

• The mathematical analysis of rhythm has a long
  history.
• In fact, ancient Indian scholars discovered the
  Fibonacci numbers and Pascals triangle by
  counting rhythms in Sanskrit poetry.
• They discovered the Fibonacci numbers fifty years
  before Fibonacci, and Pascals triangle 18
  centuries before Pascal!

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.

5
Notation

• Here are several ways to represent the same
  rhythm
• Standard Western notation
• Drum tablature x..x..x.
• Binary 10010010

6
Periodic rhythms

• If a rhythm is played repeatedly, its 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
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.
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.

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?

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.
• Burnsides lemma is used to count these cycles.

11
Fourteen rhythm cycles
?
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.

13
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
• 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.

14
DIY!
How can I fill in the rest of the template to
make an asymmetric cycle?
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.

16
Example

• Schumann, Kind im Einschlummern
• Voice 1 x.xxxx..x.xxxx..
• Voice 2 x.xxxx..x.xxxx..

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.

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?

19
Asymmetric rhythms and complementary canons

• To make a rhythm asymmetric, you make the canon
  complementary.
• When will you get a tiling canon?

20
Oh, those crazy canons!

• A three-voice tiling canon
• x.....x..x.xx.....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
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..

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.
• Heres one of period 72. Youll hear the whistle
  sound the outer rhythm about halfway through.

23
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,

24
Example (continued)

• A 0, 2
• B , 0, 1, 4, 5, 8, 9,
• Every rhythmic tiling canon corresponds to an
  integer tiling!

2
4
5
3
1
0
6
7
8
9
10
11
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?

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.

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!