Homework should be turned in at the BEGINNING OF CLASS. You are certainly encouraged to work on assignments with other classmates, although the solutions you turn in should be YOUR OWN WORK AND WRITING. Be sure to cite any references, websites, Internet sources, etc. that you used for assistance with the assignment.

Note: Please list the names of any students or faculty you worked with on the assignment.

1. Read Chapter 2, "The Beat" in Math and Music: Harmonious Connections, Garland and Kahn (handout from class on 9/17.)


2. Listen to CD #1 Rhythm, available on Moodle. Liner notes for the CD are available here. You may be tested on some of this music so be sure to read the liner notes and listen carefully. One approach to effectively absorbing the music is to make brief notes about each piece, listing significant details as you listen.


3. How many eighth notes do you need to fill up a measure in 3 - 2 time? How many sixteenth notes?


4. In 2 - 4 time, how many beats does a double-dotted eighth note get? How many beats does it get in 6 - 8 time?


5. In 5 - 8 time, a measure begins with a dotted quarter rest. How many beats remain in the measure? What if the measure began with a double-dotted quarter rest?


6. Suppose that while you are playing in 4 - 4 time, you stumble across a half note that is dotted infinitely often. (The composer has come up with some clever notation for this, so you are not actually looking at an infinite number of dots!) How many beats should you hold your note? Explain your reasoning.


7. Write out the first eight terms of the geometric sequence starting with a_0 = 200 and ratio r = 1/5. If these terms are added together, what is the sum of the resulting geometric series? If there were an infinite number of terms, what would the sum of the infinite geometric series be?


8. Find the sum of the infinite geometric series 9 - 6 + 4 - 8/3 + 16/9 - + ... .


9. What is the least common multiple of 7 and 11, that is, what is the lcm(7,11)? What is the lcm(7,11,15)? One of the Chopin excerpts discussed in class contained a polyrhythm of 12 against 22. What is the lcm(12,22)?


10. Find a formula for the least common multiple lcm(a,b) of two positive integers a and b. Note that the answer is NOT simply ab. Your formula will need to involve the greatest common divisor gcd(a,b). For example, the gcd(12,18) = 6 since 6 divides each number evenly and it is the largest such number to do so. While 2 and 3 are common divisors of 12 and 18, they are not the greatest ones. Try playing around with some simple examples as you search for the formula.