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.

Questions:

1. Using both musical and mathematical terminology, listen to the following piece and describe what type of symmetry is present.

2. Using A440 Hz, find the frequency of G above middle C in each of the three tuning systems: Pythagorean Tuning, Just Intonation and Equal Temperament.

3. 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?

4. 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.

5. 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.

6. What numbers do you multiply the fundamental frequency by to raise the pitch a major sixth (M6) in each of the three tuning systems?

7. 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.

8. 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"?

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

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

11. 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.

12. Write out the full extent for Double Court Minimus [db(cb)^2db]^3, where b = (2 3), c = (3 4) and d = (1 2). Is this a legitimate extent? Which of the 6 rules for an extent are satisfied?

13. 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).

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

15. 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 ???

16. Give an example of a group that is non-commutative, that is, there are elements a and b where a*b does not equal b*a.

17. 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^(2007) and (a b)^2.

18. Take the first 4 bars of Twinkle Twinkle Little Star in the key of C major (starting on middle C) and write the retrograde and retrograde-inversion of this motif where the inversion is done about the note G above middle C. Stay in the key of C major. (Music would be provided to you on the exam.)

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