Unit 5:
Combinatorics and the mathematics of drumming

November 26 -
December 10


Readings.
 

Topic Book Pages
Introduction to rhythm web site
web site
Fibonacci numbers handout
web site
Pascal's triangle handout
web site

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)

  1. Write out the following rhythms in quarter and eighth notes.  Practice tapping out the rhythms.  The time signature is 4/4.
    1. 1  2  3 + 4 +
    2. 1 + 2 + 3 + 4
    3. 1 +   + 3 + 4
    4. 1  2  3 +  +
    5. 1 +  +  + 4
  2. Write the following rhythms using 1, 2, 3, 4, and +.  Practice tapping out the rhythms.
    1.    
    2.    
    3.     
    4.     
  3. Find the total duration of the notes in the following patterns.  (For example,      =  1/4 + 1/8 + 1/4 + 1/4 = 7/8).
    1.   
    2.     
  4. Complete the following chart of Fibonacci numbers.  You'll need this chart for the following exercises.

  5.  
    F1 F2 F3    F5 F6 F7    F9 F10 F11 F12
    1 1    3          21            

     
  6. Given that F31 = 1,346,269 and F33 = 3,524,578,
    1. find F32.
    2. find F34.
  7. 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.
  8. 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.
  9. Here are the first thirteen rows of Pascal's Triangle (note that the top row is called Row 0).   Complete the fourteenth row.

  10. Refer to the handout on rhythm and Pascal's Triangle.
    1. 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.)
    2. 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?
    3. 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 Triangle--you need not write out the patterns.
  11. Write a formula for the sum of the numbers in the nth row, where n = 0, 1, 2, 3 ...
  12. 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?
  13. 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?

  14. 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 numbers--just 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.


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