Math, Music and Memory

Final Exam Review Questions

Below are some types of questions you can expect to find on the final exam. This is certainly not an exhaustive list and is meant to give a general overview. Keep in mind that one of the goals of a final is to test for an overall understanding (synthesis) of the material from the entire course. For a more detailed list of topics, see the final exam review sheet.

Listening Questions:

1. Be able to recognize the time signature or rhythmic features of a piece by listening (see CD #1).

2. Be able to hear the difference between a chromatic, whole tone, major, or minor scale.

3. Be able to recognize different types of tonality and/or musical styles: major, minor, Gregorian chant, early polyphony, atonal, modern music, etc. (see CD #2).

General Questions:

1. How many sixteenth notes do you need to fill up a measure in 5 - 4 time? in 6 - 8 time?

2. In 2 - 4 time, how many beats does a triple-dotted eighth note get? How many beats does it get in 3 - 8 time?

3. What is the time signature of Billy Joel's The Piano Man?

4. Write out the first four terms of an infinite geometric series that begins with 12 and has a ratio of -1/4. What is the sum of this infinite geometric series?

5. Suppose you come across a measure of music that asks you to play a polyrhythm of 22 against 8. How many pulses would you subdivide the measure into in order to see precisely where the pulse of each rhythmic pattern goes?

6. What property must two integers a and b have so that the lcm(a, b) = a? Give an example.

7. Find gcd(6, 8, 14) and lcm(6, 8, 14).

8. Without using key signatures, write out a D♭ major scale (one octave) in the bass clef using correct accidentals. Be able to find the corresponding notes on the piano keyboard.

9. Without using key signatures, write out a C♯ harmonic minor scale (one octave, diatonic spelling) in the treble clef using correct accidentals. What is the relative major of C♯ minor?

10. What key has 6 sharps? List the sharps for this key in correct order.

11. How many white keys are there on the modern piano keyboard? Explain how you arrived at your answer.

12. Beginning on A220, go up by two perfect fifths and then down by a major third. What note have you arrived at? Find the frequency for this note in just intonation and equal temperament.

13. What musical interval is used so effectively by composer John Williams in his famous shark theme of Jaws?

14. What chord is typically found in measure 5 of a 12-bar blues? If the blues is in the key of B♭, what three notes make up this chord?

15. By how many decibels should you raise the volume of a sound in order to increase the intensity by a factor of 100? by a factor of 200?

16. What angle (in radians) corresponds to going around the unit circle 5 1/4 times? What will the sine of this angle be?

17. What is the frequency of the sound wave y = sin(660π t)? What note does this correspond to and where can you find it on the piano?

18. Starting at the bottom of the staff in the bass clef, write the first twelve notes corresponding to the frequencies in the overtone series built on F. Also notate the intervals between successive notes. Which notes in the series do not fit well with the usual notes of the piano?

19. For each scale degree (1 - 8), list the ratio used in Pythagorean tuning to raise the tonic by that degree.

20. Give the value of the syntonic comma and explain where it comes from.

21. Explain mathematically why the circle of fifths "closes" up when using equal temperament, but does not close up in either the Pythagorean scale or just intonation.

22. A string of length 25 cm and a string of length 30 cm are plucked simultaneously. What interval do you hear?

23. Assuming that A above middle C has a frequency of 440 Hz, find the frequency of the B just below middle C in each of the three tuning systems: Pythagorean, just intonation and equal temperament.

24. What number, when multiplied by the fundamental frequency, raises the pitch a major third in equal temperament? Prove that this number is irrational.

25. True or False: The product of two irrational numbers is always irrational.