Math/Music: Structure and Form
Partial Solutions for Exam 2 Review Questions
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3 + -3 = 0.
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b = 1000.
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10 log_10 (50) which is approximately 17 dB.
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200 Hz and 203 Hz. The "note" we hear has a frequency of
201.5 Hz and the number of beats is 3 per second.
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See class notes or HW #4, problem #2.
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Amplitude is 4, period is 2, phase shift is 1/2. Graph will have roots at 1/2, 3/2, 5/2, 7/2, etc.
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22 Pi. cos(22 Pi) = 1, sin(22 Pi) = 0.
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cos(Pi/8) = (1/2)*sqrt[2 + sqrt(2)], sin(Pi/8) = (1/2)*sqrt[2 - sqrt(2)]
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Perfect 5th. Major Third.
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3^(12)/2^(19). Can't change keys very easily. B# not equal to C. Two half steps does not equal a whole step.
Circle of Fifths does not close up.
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150, 300, 450, 600, 750, 900, 1050, 1200 Hz
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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 octaves.
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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 #6.
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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 the I, IV and V chords, all
of which have the 4:5:6 ratio, the harmony will re-enforce the overtone series and sound the best.
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Pythagorean Tuning: 27/16
Just Intonation: 5/3
Equal Temperament: 2^(3/4)
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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 (5 + even or 0 = odd). But the right-hand side has a prime
factorization with an even or zero number of 2's. This violates the Fundamental Theorem of Arithmetic
which states that a positive integer has a unique factorization into its prime factors.
Thus, 2^(5/12) is irrational.
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There are 300 cents in a minor third in Equal Temperament (3 half steps). In Just Intonation there are
1200*log_2 (6/5) cents in a minor third, or approximately 315.6 cents. The syntonic comma, 81/80, is equivalent to
about 21.5 cents.