Math/Music: Structure and Form

Homework Assignment #1

Due Friday, Sept. 30, START of Class

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.

Music is the pleasure the human soul experiences from counting without being aware that it is counting.
- Gottfried Leibniz

1. Read Music and mathematics: an overview, Susan Wollenberg, introduction to the primary course text, edited by Fauvel, Flood and Wilson, pp. 1 - 9.

2. Read Chapter 2, "The Beat" in Math and Music: Harmonious Connections, Garland and Kahn (handout from class on 9/14.)

3. Listen to CD #1 Rhythm, available on Moodle and read the liner notes for the CD, 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.

4. What concrete or structural traits shared by mathematics and music are discussed by Wollenberg? Explain briefly.

5. What four subjects comprise the quadrivium, based on the teachings and writings of Plato and Pythagoras? What three subjects comprise the language-based trivium? Together, these seven fields formed the original "liberal arts."

6. How many quarter notes do you need to fill up a measure in 5 - 2 time? How many eighth notes?

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

8. In 7 - 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?

9. Suppose that while playing in 3 - 2 time, you stumble across a quarter 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.

10. Write out the first eight terms of the geometric sequence starting with a_0 = 1600 and ratio r = 1/4. 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? Give all your answers as fractions.

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

12. 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)? What is the lcm(12,20)?

13. 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.