Mathematics and Music

Exam 2 Review Questions

Below are some types of questions you can expect to find on the exam. This is certainly not an exhaustive list and is meant to give a general overview. For more detailed topics, see the Exam 2 Review sheet.

General Questions:

1. Find the frequency of G above middle C in each of the three tuning systems: Pythagorean Tuning, Just Intonation and Equal Temperament.

2. In a short essay, compare the Pythagorean Scale with the scale given using Equal Temperament. What are the advantages and weaknesses of each system? Which came first and why?

3. Suppose you are writing a piece of music for the classical guitar and you plan to only use one key (say C major) and mostly the notes in the major scale, particularly the first, third, fifth and octave. Which of the three tuning systems would you use to tune your guitar? Explain.

4. Starting in the bass clef on the bottom line G, write the first ten notes corresponding to the frequencies in the overtone series of this G. Also notate the intervals between successive notes. Which notes in the series do not fit well with the usual notes of the piano?

5. What number do you multiply the fundamental frequency by to raise the pitch a perfect fourth using Equal Temperament? Prove that this number is irrational.

6. What are the three mandatory rules in change ringing necessary to qualify as a legitimate extent? What does it mean mathematically speaking for an extent to satisfy the "palindrome property"?

7. Write out a full extent on n = 3 bells. How many different extents on 3 bells are there? Explain.

8. Write out the first 8 changes of Plain Bob Minimus.

9. How many different ways are there to ring 12 bells if the odd numbered bells never move? In other words, Bell 1 stays as Bell 1, Bell 3 stays as Bell 3, etc.

10. Given the permutations a = (2 3 5 4 1) and b = (4 2 3 1 5), compute a*b, b*a, b^3 and a^(-1).

11. Explain why the integers do not form a group under multiplication.

12. Suppose that A is a subgroup of the group G, where G has order 30. List all the possible orders for A.

13. According to the text in Chapter 6, if a musical motif has two of the three symmetries ph, pv, or p2 (H, V, R_180 in our notation), then it automatically has the third remaining symmetry. Explain why this is true using the symmetries of the square and composition. Hint: If you have H and V as symmetries, for example, then you also have H*V. But H*V is equivalent to ???

14. Give an example of a group that is non-commutative.

15. Suppose that a and b are elements of a group (neither of which is the identity element) and that a^3 = e, b^2 = e and a b = b a. What is the inverse of a? What is the inverse of b? Simplify the quantities b a b, a^(2006) and (a b)^2.

16. Take the first 4 bars of Twinkle Twinkle Little Star (see class notes) in the key of C major and write the retrograde and retrograde-inversion of this motif (music would be provided to you on the exam.)

17. Give three musical examples where the mathematical symmetry of a vertical reflection is used. What is the musical name of this symmetry?