Unit 5:
Combinatorics and the mathematics
of drumming
November 26 
December 10 

Readings.
Book codes: AC=Africa Counts; MMC=Multicultural Math Classroom;
CP=Crest of the Peacock; UHN=Universal History of Numbers
Assignment #5. (due Wednesday, December
12th)

Write out the following rhythms in quarter and eighth notes. Practice
tapping out the rhythms. The time signature is 4/4.

1 2 3 + 4 +

1 + 2 + 3 + 4

1 + + 3 + 4

1 2 3 + +

1 + + + 4

Write the following rhythms using 1, 2, 3, 4, and +. Practice tapping
out the rhythms.





Find the total duration of the notes in the following patterns. (For
example,
= 1/4 + 1/8 + 1/4 + 1/4 = 7/8).



Complete the following chart of Fibonacci numbers. You'll need this
chart for the following exercises.
F_{1} 
F_{2} 
F_{3} 

F_{5} 
F_{6} 
F_{7} 

F_{9} 
F_{10} 
F_{11} 
F_{12} 
1 
1 

3 



21 





Given that F_{31} = 1,346,269 and F_{33} = 3,524,578,

find F_{32}.

find F_{34}.

The Irish jig is based around rhythm patterns in 6/8 time signature.
Write out all the rhythm patterns consisting of quarter and eighth notes
that have total duration 6/8. For example,
is one such pattern. Hint: the number of patterns is a Fibonacci
number.

How many rhythm patterns consisting of quarter and eighth notes that have
total duration 10/8? Of those, how many start with a quarter note?
How many start with an eighth note? Hint: all your answers
should be Fibonacci numbers.

Here are the first thirteen rows of Pascal's Triangle (note that the top
row is called Row 0). Complete the fourteenth row.

Refer to the handout on rhythm and Pascal's Triangle.

How many rhythm patterns formed of 4 notes are there, if each note is either
a quarter or an eighth note? (For example,
is a pattern of 4 notes.)

Of the patterns of 4 notes, which one has the longest duration, and what
is that duration? which has the shortest duration, and what is that
duration?

For each possible duration, list the number of patterns of 4 notes that
have that duration. Hint: the answer comes from a row of Pascal's
Triangleyou need not write out the patterns.

Write a formula for the sum of the numbers in the nth row, where
n
= 0, 1, 2, 3 ...

A prime number is an integer greater than 1 whose only divisors are 1 and
itself. For example, 2, 3, and 5 are prime but 6 is not. Study
the rows whose second entry is a prime number. What do all the numbers
in that row have in common?

Add the numbers between the red lines in Pascal's Triangle. For example,
the first few sums are 1, 1, 1+1=2, 1+2=3, and 1+3+1=5. What's the
pattern?

Extra Credit. Print out the template
for coloring Pascal's Triangle from the Math Forum. Instead of
writing in the numbers, color the hexagon if the corresponding number in
Pascal's Triangle is odd, and leave it white if the number is even.
You don't need to calculate the numbersjust remember how odd and even
numbers add!
Sample Test Questions.
You may bring a 5x7 index card with any notes you wish on it.
The test will be about 30 minutes long.

Several questions will be similar to homework questions. In particular,
you should be able to

Convert a rhythm pattern from eighth and quarter notes to the 1+2+3+4+
notation and vice versa.

Find the total duration of a given rhythm pattern.

List all the rhythm patterns of a given duration.

Predict how many rhythm patterns have a given duration, using the Fibonacci
sequence. Predict how many of those start with a quarter note and
how many start with an eighth note.

Find entries in the Fibonacci sequence.

Find entries in Pascal's Triangle.

Predict how many rhythm patterns can be formed with a specified number
of notes. Predict the range of possible durations and the number
of patterns of each duration, using Pascal's Triangle.

Investigate patterns in Pascal's Triangle and the Fibonacci numbers, as
in questions 1012.

There will be a few multiplechoice questions based on lectures and readings.
To help you focus, here are some key concepts we have discussed or will
discuss:

rhythm pattern

quarter note

eighth note

duration

time signature

Fibonacci sequence

Fibonacci number

Pascal's Triangle
You should be familiar with each concept and able to give an example for
each.
Multicultural Mathematics
home page.
Images are from the MacTutor
History of Mathematics Archive (used by permission).
Rachel W.
Hall / Department of Math and Computer Science / St. Joseph's University
/ rhall@sju.edu