Mathematics and Music

Homework Assignment #2

Due Friday, February 3, 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 and complete exercises 1 - 11 in Chapter 1 of The Music Kit, 2nd ed., Workbook, Tom Manoff (handout from class on 1/25.)

  2. Read Chapter 2, "The Beat" in Math and Music: Harmonious Connections, Garland and Kahn (handout from class on 1/25.)

  3. Listen to the CD Math and Music I: Rhythm, available on reserve in the music library. Liner notes for the CD are available here. You may be tested on some of this music so be sure to read the notes and listen carefully.

  4. How many eighth notes do you need to fill up a measure in 3 - 4 time? How many sixteenth notes?

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

  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)? What is the lcm(8,12)?

  8. True or False: For any two positive integers a and b, lcm(a,b) = ab . If true, explain why. If false, what conditions on a and b are needed to make the statement true?

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

  10. Write out the first eight measures of Mary Had a Little Lamb in the key of F major and in common time. Include 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.

  11. Take the music from the previous question and add another eight measures in retrograde (ie. a vertical reflection.) Your total "piece" will be sixteen measures (still in F major). How does it sound?

  12. Take the eight measures of Mary Had a Little Lamb and write the melody in inversion (ie. horizontal reflection about the third of the scale) starting on the same note but staying in the key of F major. Your total "piece" will be eight measures. How does this one sound? Which reflection seems to work better musically?