Mathematics and Music

Homework Assignment #2

Due Friday, February 2, 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. Please cite any references (web based or text) that you may have 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 Chapter 2, "The Beat" in Math and Music: Harmonious Connections, Garland and Kahn (handout from class on 1/24.)

  2. Listen to the CD Math and Music I: Rhythm, distributed in class on 1/24. 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. 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.

  6. 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?

  7. 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 on 1/26 contained a polyrhythm of 12 against 22. What is the lcm(12,22)?

  8. 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.

  9. On music staff paper, write out an ascending chromatic scale (one octave) using sharps in both the treble clef and the bass clef beginning on the note F. Looking at the resulting set of notes, which two letter names are only used once? Why?

  10. Write out an ascending whole-tone scale (one octave) using flats in both the treble clef and the bass clef beginning on the note A flat.

  11. Write out a B major scale (ascending and descending one octave) in both the treble clef and the bass clef. Include accidentals (in this case, sharps) as needed alongside the correct notes. In other words, do not use a key signature.

  12. Write out the first eight measures of Mary Had a Little Lamb in the key of B flat major and in common time. Include accidentals (in this case, flats) as needed alongside the correct notes and list the words below the correct notes. In case you've forgotten, the words are:

    Mary had a little lamb
    Little lamb, little lamb
    Mary had a little lamb
    It's fleece was white as snow

    Hint: The song starts on the third of the scale.

  13. Take the music from the previous question (no words) and add another eight measures in retrograde (ie. a vertical reflection.) Your total "piece" will be sixteen measures (still in B flat major). How does it sound?