Math/Music: Structure and Form

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

3. Write out the first four terms of an infinite geometric series that begins with 5 and has a ratio of -1/4. What is the sum of this infinite geometric series?

4. Suppose you come across a measure of music that asks you to play a polyrhythm of 12 against 9. How many pulses would you subdivide the measure into in order to see precisely where the pulse of each rhythmic pattern goes?

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

6. Without using key signatures, write out a D flat major scale in the bass clef using correct accidentals.

7. Without using key signatures, write out an F# harmonic minor scale (diatonic) in the treble clef using correct accidentals. What is the relative major of F# minor?

8. Using key signatures, transpose the first eight bars of Mary Had a Little Lamb to the key of F sharp major.

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

10. Two notes are played together resulting in a sound wave of the form
y = sin(660 Pi t) + sin(652 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?

11. Give two reasons why the brain is so important to our hearing and understanding of music.

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

13. In a short essay, compare the three tuning systems: the Pythagorean Scale, Just Intonation and Equal Temperament. What are the advantages and weaknesses of each system? In what order did they evolve and why?

14. Give the value of the Pythagorean comma and explain where it comes from.

15. Explain mathematically why the circle of fifths "closes" when using Equal Temperament.

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

17. 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.

18. For each scale degree (1 - 8), list the ratio used in Just Intonation to raise a note by that degree.

19. Starting in the bass clef on D, write the first ten notes corresponding to the frequencies in the overtone series of this D. Also notate the intervals between successive notes. Which note in the series does not fit well with the usual notes of the piano?

20. What number do you multiply the fundamental frequency by to raise the pitch a major third using Equal Temperament? Prove that this number is irrational.

21. Draw a diagram showing Strahle's construction of the placement of frets on a guitar.

22. Compute the continued fraction expansion for 31/11 and for the square root of 3.