Math/Music: Structure and Form

Partial Solutions for Exam 2 Review Questions

  1. 3 + -3 = 0.

  2. b = 1000.

  3. 10 log_10 (50) which is approximately 17 dB.

  4. 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.

  5. See class notes or HW #4, problem #2.

  6. 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.

  7. 22 Pi. cos(22 Pi) = 1, sin(22 Pi) = 0.

  8. cos(Pi/8) = (1/2)*sqrt[2 + sqrt(2)], sin(Pi/8) = (1/2)*sqrt[2 - sqrt(2)]

  9. Perfect 5th. Major Third.

  10. 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.

  11. 150, 300, 450, 600, 750, 900, 1050, 1200 Hz

  12. 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.

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

  14. See class notes and Ch. 1 of the text. Also, see HW #5, problem #6.

  15. 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.

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

  17. 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.

  18. 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.