Math, Music and Identity

Midterm Exam Review Solutions

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1. Listen, listen, listen. Be sure to read the CD liner notes as you listen.
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4. See the class handout on symmetry in music. A horizontal reflection is known as an inversion in music.
5. See the class handout on symmetry in music. A vertical reflection is known as a retrograde in music.
6. Inverses are not in the group. For example, 4^-1 = 1/4 but 1/4 is not an integer.
7. If a musical motif has symmetry H (inversion about the middle of the staff) and symmetry V (retrograde), then it will also have the symmetry obtained by composing these together. Since H*V=R₁₈₀, the motif will also be symmetric under a 180 degree rotation. If the inversion is not about the center of the staff, then the motif will be the same as its retrograde-inversion, but not identical to a 180 degree rotation.
8. Both D_n (the dihedral group of degree n) and S_n (the symmetric group on n symbols) are examples of non-commutative groups.
9. S is not closed under addition because the sum of two odd integers is an even integer. S is closed under multiplication, however, because the product of two odd integers is still odd. S does not form a group under multiplication because inverses, which are the reciprocals of the odd integers, are not contained in S. Thus, S fails to satisfy property 4 in the definition of a group.
10. The multiplication table for G is shown in Table 5.3 on p. 194 of the course text. The inverses are as follows: 0^-1 = 0, 1^-1 = 3 since 1 + 3 = 0 (mod 4), 2^-1 = 2 since 2 + 2 = 0 (mod 4), and 3^-1 = 1 since 3 + 1 = 0 (mod 4). G is identical (isomorphic) to the subgroup of D₄ given by the four rotations {e, R₉₀, R₁₈₀, R₂₇₀} (compare with the portion of the multiplication table for D₄ in the upper left block).
11. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. F_n = F_n-1 + F_n-2. The golden ratio is the ratio that subdivides a segment into two parts so that the ratio of the whole to the larger part is equivalent to the ratio of the larger part to the smaller part. The exact value of the golden ratio, usually symbolized by φ, is φ = (1 + sqrt{5})/2. The ratio of consecutive Fibonacci numbers approaches (limits on) the golden ratio as n gets larger and larger. Fibonacci numbers show up in nature in the number of opposing spirals on pine cones and sunflowers (typically consecutive Fibonacci numbers). They also occur as the number of petals in many flowers.
12. φ² = (3 + sqrt{5})/2, φ³ - 1 = 1 + sqrt{5}, φ⁴ - 3 φ = 2.
13. See the class handout on change ringing (or Section 6.1.2 of the course text).
14. (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 only two permissible moves: a = (1 2) or b = (2 3). Since a² = e and b² = e, we must alternate between a and b between changes. This results in either (ab)³ or (ba)³ as the only possible extents.
15. See the class handout on change ringing (or Section 6.1.5 of the course text). This isn't hard to learn if you recall our special dance class with Liz Lerman from last semester. The moves used to create Plain Bob Minimus are [(ab)³a c]³.
16. 6! = 720.
17. 88. Recall that the number of permissible moves on n bells is F_n+1 - 1. See Table 6.2 of the course text.
18. a*b = (2 3 5 1 4), b*a = (4 3 5 2 1), b³ = b, a^-1 = (5 1 2 4 3).
19. a^-1 = a³, b^-1 = b, bab = a implies ab = ba by multiplying by b on the left (or right) of both sides, a²⁰¹⁵ = a³, (ab)⁵ = ab (since ab * ab * ab * ab * ab can be regrouped as ab * a * (bab) * a * (bab) = ab * a⁴ = ab * e = ab).

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