Below are the types of questions you can expect to find on the exam. This is certainly not an exhaustive list; it is intended to provide a general overview.

Questions:

1. Listening Questions: Upon listening, be able to identify, using both musical and mathematical terminology, what type(s) of symmetry is present in a piece (CD #3). You should also be able to identify the composer of the piece being played.


2. Give three reasons why the brain is so important to our hearing and understanding of sound.


3. Evaluate log₁₀(1000) + log₁₀(1/1000).


4. Evaluate log₂(log₂ 256).


5. By how many decibels should you raise the volume of a sound in order to increase the intensity by a factor of 50?


6. Suppose that a particular sound is increased by 40 decibels, by what factor has the intensity of the sound increased?


7. Graph two cycles of the function y = 4 sin( π ( t - 1/2)) . What are the amplitude, frequency, period, and phase shift of this function?


8. What angle (in radians) corresponds to going around the unit circle 11 times? What will the cosine of this angle be? What will the sine of this angle be?


9. Two notes are played together resulting in a sound wave of the form y = sin(400 Pi t) + sin(406 Pi t). What are the frequencies of each note? What is the frequency of the "note" we hear when they are combined? How many beats per second do you hear?


10. Suppose that partials with frequencies 330, 440, 550, 660, and 880 Hz are sounded together, each with approximately the same amplitude. What is the frequency of the residue pitch likely heard? Where is this note on the piano keyboard?


11. Write out the first 10 frequencies in the overtone series for a note whose fundamental is 150 Hz.


12. A string of length 40 cm and a string of length 80 cm are plucked simultaneously. What interval do you hear? What if the string lengths are 40 and 60 cm? Explain.


13. Using A440 Hz as your starting pitch, find the frequency of F♯ above middle C in each of the three tuning systems: Pythagorean tuning, just intonation and equal temperament (round to one decimal place).


14. In a short essay, compare and contrast the tuning systems of just intonation and equal temperament. What are the advantages and weaknesses of each system?


15. Starting in the bass clef on the bottom line G, write the first 12 notes corresponding to the frequencies in the overtone series of this G. Also notate the intervals between successive notes.


16. What is the ratio of the three notes in a major chord containing the 1st, 3rd, and 5th scale degrees using just intonation? Show that the notes in a minor chord using just intonation are in the ratio 10:12:15.


17. What numbers do you multiply the fundamental frequency by to raise the pitch a major seventh (M7) in each of the three tuning systems?


18. What number do you multiply the fundamental frequency by to raise the pitch a perfect fourth using Equal Temperament? Prove that this number is irrational.


19. How many cents are in a minor third using equal temperament? How does the answer change if you use just intonation? Approximately how many cents are there in the syntonic comma?


20. Take the first 4 bars of Twinkle Twinkle Little Star in the key of E major and write the inversion about G#, staying in the key of E major (use the treble clef). Also write the retrograde-inversion, inverting about G#, staying in the key of E major (use the treble clef).



21. Take the first 4 bars of Twinkle Twinkle Little Star in the key of E major and write the exact inversion about G# in the treble clef, using correct accidentals.


22. Give three musical examples (title and composer) where the mathematical symmetry of a horizontal reflection is used. What is the musical name of this symmetry?


23. Give three musical examples (title and composer) where the mathematical symmetry of a vertical reflection is used. What is the musical name of this symmetry?


24. Give an example of a group that is non-commutative, that is, there are elements a and b where a*b ≠ b*a.


25. Consider the set S of odd integers S = {±1, ±3, ±5, . . .}. Is the set S closed under addition? Is S closed under multiplication? Does S form a group if * = multiplication? Explain.


26. Consider the group G = {0, 1, 2, 3} with * = addition modulo 4 (e.g., 2 * 3 = 1 since 2 + 3 = 5 and 5 - 4 = 1.) Construct the multiplication table for G and state the inverse of each element.