Asymmetric Rhythms and Tiling
Canons
Dr. Rachel Hall
Saint Joseph’s University
rhall@sju.edu
Vassar College
May 8, 2007
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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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!
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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.
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Notation
Here are several ways to represent the same rhythm:
• Standard Western notation
or
• Drum tablature: x..x..x.
• Binary: 10010010
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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.
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Composition 001
• Choose a rhythm (not the same as mine!)
. . x . x .
• Write down all the patterns that are equivalent to your
rhythm.
...x.x
x...x.
etc.
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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.
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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?
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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.
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Fourteen rhythm cycles
of six beats

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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,
1
n
G
 x : gx  x
gG
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.

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Burnside’s lemma in action
The number of times a pattern
g : gx  x  x : gx  x
appears on this matrix equals the n G 
number of
fixing it.
gG
xX
Byrotations
inspection,
there are n = six
four-beat cycles


n = number of cycles
0º rotation



90º rotation
X = set of
(distinct)
patterns
G = group
of 90º
rotations
180º rotation
270º rotation
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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 = 24 = 16.
• Number of patterns fixed by 90° rotation = 21 = 2.
• Number of patterns fixed by 180° rotation = 22 = 4.
• Number of patterns fixed by 270° rotation = 21 = 2.
1
n
G
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
1
x : gx  x  16  2  4  2  6
4
gG
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General formula for number of
N-beat cycles
In general,
1
n
N
N

i1
2
gcd(i,N )
1

N
N d

(d)2

d|N
where  is the Euler phi-function:
(d) = number of positive integers less than or equal to
d that are relatively prime to d
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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.
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Examples
Asymmetric cycle
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.
.x.x...x
x.x.x...
.x.x.x..
..x.x.x.
...x.x.x
x...x.x.
.x...x.x
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DIY!
. . x . x .
• How can I fill in the rest of the beats to make a
pattern belonging to an asymmetric cycle?
• In general, there are 3N patterns of length 2N that are
members of asymmetric cycles.
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Counting asymmetric rhythm cycles
The number of asymmetric rhythm cycles of period 2N
is
1
2N

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N d

(d)

(d)3


d|2 N
d ev en
d|2 N
d odd


1 
N d 

 (2d)   (d) 3 

2N d|N
d|N


d odd




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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.
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Example
Schumann, “Kind im Einschlummern”
Voice 1:
x.xxxx..x.xxxx..
Voice 2: x.xxxx..x.xxxx..
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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.
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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?
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Asymmetric rhythms and
complementary canons
. . x . x .
. . x . x .
To make a rhythm asymmetric, you make the canon
complementary.
When will you get a tiling canon?
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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...:|
repeat sign
The methods of constructing n-voice canons, where the
voices are equally spaced from one another, are
similar to the asymmetric rhythm construction.
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A four-voice tiling canon
Voice
Voice
Voice
Voice
1: x.x.....|:x.x.....:|
2: x.x....|:.x.x....:|
3:
x.x.|:....x.x.:|
4:
x.x|:.....x.x:|
Entries: ee..ee..|:ee..ee..:|
inner rhythm = x.x.....
outer rhythm = ee..ee..
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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.
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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, …}
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Example (continued)
A = {0, 2}
B = {…, 0, 1, 4, 5, 8, 9, …}
Z AB
… 0 1 2 3 4 5 6 7 8 9 10 11 …

Every rhythmic tiling canon corresponds to an integer
tiling!
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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?
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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.
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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!
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