Mathematics and Music
Partial Solutions for Exam 2 Review Questions
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Pythagorean Tuning: 391.1 Hz.
Just Intonation: 391.1 Hz.
Equal Temperament: 392 Hz.
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See class notes and Ch. 1 of the text. Also, see HW#5, problem 2.
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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 ratios of just intonation which re-enforce the overtone series
will sound the best.
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See HW#5, problem 3. The notes, in order, are G, G', D', G'', B'', D'', F'', G''', A''', B'''.
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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.
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See the handout from class on change ringing.
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(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.
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See the handout from class on change ringing.
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6! = 720.
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a*b = (2 3 5 1 4), b*a = (4 3 5 2 1), b^3 = b, a^(-1) = (5 1 2 4 3)
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Inverses are not in the group. For example, 4^(-1) = 1/4 but 1/4 is not
an integer.
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The order of a subgroup always divides evenly into the order of the larger group.
Thus, the only possibilities are 1, 2, 3, 5, 6, 10, 15, 30.
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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.
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Both D_4 and S_n are examples of non-commutative groups.
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a^(-1) = a^2, b^(-1) = b, bab = a, a^(2006) = a^2.
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See class notes from Friday, April 7th where we did an example.
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"Lean on Me," "I Got Rhythm" by George Gershwin, Bach's "Musical Offering", specifically his Crab Cannon,
Handel's "Messiah," Haydn's "Piano Sonata no. 41, minuet".
A vertical reflection is known musically as retrograde.