Mathematics and Music

Partial Solutions for Exam 2 Review Questions

1. Listen and read the liner notes for CD #3.

2. Pythagorean Tuning: 391.1 Hz.
Just Intonation: 391.1 Hz.
Equal Temperament: 392 Hz.

3. See class notes and Ch. 1 of the text. Also, see HW#5, problem 4.

4. Just Intonation. Since we are staying in one key, we don't need to worry about how to sharp or flat notes to move into a new key. Since we are playing 1, 3, 5 and 8 of the scale, the simple, low-numbered ratios of just intonation which re-enforce the overtone series will sound the best.

5. See HW#5, problem 5. The notes, in ascending order, are G, G', D', G'', B'', D'', F'', G''', A''', B''' where the '' signify different ocatves.

6. Pythagorean Tuning: 27/16
Just Intonation: 5/3
Equal Temperament: 2^(3/4)

7. 2^(5/12) (P4 is five half-steps.) To show this number is irrational, set 2^(5/12) = p/q, with p and q integers. Raise both sides to the 12th power to obtain 2^5 = p^12/q^12. Cross multiplying gives 2^5 q^12 = p^12. The left-hand side of this equation is an integer which when factored into its prime factorization will have an odd number of 2's. But the right-hand side has a prime factorization with an even number of two's. This violates the Fundamental Theorem of Arithmetic which states that a positive integer has a unique factorization into its prime factors.

8. See the handout from class on Change Ringing .

9. (1 2 3), (2 1 3), (2 3 1), (3 2 1), (3 1 2), (1 3 2), (1 2 3). There are only two legitimate extents because at any given stage there are two possible moves: a = (1 2) or b = (2 3). Since a^2 = e and b^2 = e, we must alternate between a and b between changes. This results in either (ab)^3 or (ba)^3 as the only possible extents.

10. See the handout from class on Change Ringing . This isn't hard to learn if you recall the moves used to create Plain Bob Minimus, [(ab)^3 ac]^3.

11. 6! = 720.

12. (1 2 3 4), (2 1 3 4), (2 3 1 4), (2 3 4 1), (2 4 3 1), (2 4 1 3), (2 1 4 3), (1 2 4 3) is the first lead of Double Court Minimus. The second lead is (1 4 2 3), (4 1 2 3), (4 2 1 3), (4 2 3 1), (4 3 2 1), (4 3 1 2), (4 1 3 2), (1 4 3 2) and the third lead is (1 3 4 2), (3 1 4 2), (3 4 1 2), (3 4 2 1), (3 2 4 1), (3 2 1 4), (3 1 2 4), (1 3 2 4), (1 2 3 4). This is a legitimate extent because the first three rules are satisfied. However, rule #4 is not satisfied since Bells 2, 3 and 4 rest in the same position for several changes in a row. Rules 5 and 6 are satisfied.

13. a*b = (2 3 5 1 4), b*a = (4 3 5 2 1), b^3 = b, a^(-1) = (5 1 2 4 3)

14. Inverses are not in the group. For example, 4^(-1) = 1/4 but 1/4 is not an integer.

15. If a musical motif has symmetry H (ph) and symmetry V (pv), then it will have the symmetry obtained by composing these together. But this is H*V=R_180 (p2) so R_180 is guaranteed. Similar arguments work for the other two cases since H*R_180=V and V*R_180=H.

16. Both D_4 and S_n are examples of non-commutative groups.

17. a^(-1) = a^2, b^(-1) = b, bab = a, a^(2007) = e, (ab)^2 = a^2

18. See class notes from Monday, March 19th where we did an example.

19. "Lean on Me" by Bill Withers, "I Got Rhythm" by George Gershwin, Bach's "Musical Offering", specifically his Crab Cannon, the Hallelujah chorus from Handel's "Messiah," Haydn's "Piano Sonata no. 41, Minuet and Trio". A vertical reflection is known musically as a retrograde.