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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. - PowerPoint PPT Presentation
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Asymmetric Rhythms and Tiling Canons
Dr. Rachel HallSaint Joseph’s University
Shippensburg University Student Math Conference
March 21st, 2007 Asymmetric rhythms and tiling canons 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.
March 21st, 2007 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!
March 21st, 2007 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.
March 21st, 2007 Asymmetric rhythms and tiling canons 5
Notation
Here are several ways to represent the same rhythm:
• Standard Western notation
• Drum tablature: x..x..x.
• Binary: 10010010
or
March 21st, 2007 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.
March 21st, 2007 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.
March 21st, 2007 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.
March 21st, 2007 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?
March 21st, 2007 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.
March 21st, 2007 Asymmetric rhythms and tiling canons 11
Fourteen rhythm cycles
March 21st, 2007 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.
March 21st, 2007 Asymmetric rhythms and tiling canons 13
Examples
Asymmetric cycle
x..x..x..x..x..xx.x..x...x.x..x...x.x..xx..x.x...x..x.x...x..x.xx..x..x.
Non-asymmetric cycle
x.x...x..x.x...xx.x.x....x.x.x....x.x.x....x.x.xx...x.x..x...x.xx.x...x.
March 21st, 2007 Asymmetric rhythms and tiling canons 14
DIY!
x x ....
How can I fill in the rest of the template to make an asymmetric cycle?
March 21st, 2007 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.
March 21st, 2007 Asymmetric rhythms and tiling canons 16
Example
Schumann, “Kind im Einschlummern”
Voice 1: x.xxxx..x.xxxx..Voice 2: x.xxxx..x.xxxx..
March 21st, 2007 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.
March 21st, 2007 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?
March 21st, 2007 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 ....
x x ....
March 21st, 2007 Asymmetric rhythms and tiling canons 20
Oh, those crazy canons!
A three-voice tiling canonx.....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
March 21st, 2007 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..
March 21st, 2007 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.
March 21st, 2007 Asymmetric rhythms and tiling canons 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, …}
March 21st, 2007 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!
€
Z = A⊕B
2 4 5310 6 7 8 9 1011 ……
March 21st, 2007 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?
March 21st, 2007 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.
March 21st, 2007 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!
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