Math, Music and Memory

Exam 2 Review Questions

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

Questions:

1. What is log10(1000) + log10(1/1000)?

2. What is log2(log2 256)?

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

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

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

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

7. 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?

8. Derive a formula for sin(3A) containing only sin(A) and powers of sin(A).

9. 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?

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

11. A string of length 40 cm and a string of length 60 cm are plucked simultaneously. What interval do you hear? What if the string lengths are 40 and 50 cm? Explain.

12. What is the Pythagorean comma and why is the Pythagorean scale generally not used in modern music?

13. 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 musical intervals between successive notes.

14. 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).

15. 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?

16. Suppose you are writing a piece of music for the classical guitar and you plan to use only one key (say C major), only the notes in the major scale, and an abundance of major I, IV and V chords. Which of the three major tuning systems would you use to tune your guitar? Explain why.

17. 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.

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

19. What number do you multiply the fundamental frequency by to raise the pitch a perfect fourth using equal temperament? Is this number rational or irrational?

20. 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?