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

 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.

• 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 10-12.
• There will be a few multiple-choice 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.

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