Mathematics and Music

Homework Assignment #1

Due Wednesday, January 24, 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. Answer the following questions, writing neatly in COMPLETE SENTENCES (typing is not necessary, but legibility is.) Please cite any references, Internet sources, etc. that you used to help with the assignment.

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

Mathematics and music, the most sharply contrasted fields of intellectual activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind ...
H. Von Helmholtz, 1884

Read each of the following items and then answer the questions below.

• Music and mathematics: an overview, Susan Wollenberg, (introduction in the course text, pp. 1 - 9)
• The two most original creations of the human spirit, Elmer Mode, Mathematics Magazine, Vol. 35, No. 1 (Jan. 1962), 13 - 20. Handout from class: PDF file available here .
• Math and music: the deeper links, Edward Rothstein, The New York Times, (Aug. 29, 1982), Arts and Leisure Desk, Section 2, Page 1. Handout from class: PDF file available here .
Questions:
1. In class we discussed the famous technique of the 7 year-old Gauss adding all the whole numbers between 1 and 100. Use this technique to add together all the whole numbers between 1 and one million. Generalize to find a formula for the sum of all the whole numbers between 1 and n.
Note: Your second answer will depend on n. You may assume that n is a whole number greater than or equal to 2. You might try checking your formula with some small values of n. The list of sums obtained as n varies are called the triangular numbers. Can you explain why?

2. Give three concrete or structural traits mathematics and music share. Explain briefly.

3. Name four composers who have intentionally used mathematical concepts in their works. Provide names and examples.

4. Name four mathematicians/scientists who were/are also accomplished musicians or composers. (This question isn't just limited to examples from the readings. Feel free to explore using the Internet as well.)

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. Is Music a "science" or an "art"? Explain using specific examples and arguments from the readings (two to three paragraphs is sufficient.) How do you think each of the three writers would respond to this question?