Mathematics and Music

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 more detailed topics, see the final exam review sheet.

Listening Questions:

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

2. Be able to recognize different types of tonality and/or musical styles: major, minor, Gregorian chant, early polyphony, 12-tone, atonal and modern music (CD#2 and #3)
Essay Questions:
1. 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?

2. Give four examples of composers who used mathematical concepts in their compositions. Be sure to give names and titles of pieces. Be specific about the mathematical connection in the works (eg. the composer uses retrograde in the main melody which is equivalent to a vertical reflection)

3. Bach, Schoenberg and composing change ringers all used ideas from musical group theory in their compositions. Using specific examples, explore the similarities and contrast the differences in their approaches.

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. What property must two integers a and b have such that the lcm(a,b) = ab?

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

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

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

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

9. 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?

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

11. What angle (in radians) corresponds to going around the unit circle 3 1/2 times? What will the cosine of this angle be?

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

13. Assuming that A above middle C has a frequency of 440 Hz, find the frequency of F sharp above middle C in each of the three tuning systems: Pythagorean Tuning, Just Intonation and Equal Temperament.

14. For each scale degree (1 - 8), list the ratio used in the Pythagorean Scale to raise a note by that degree.

15. 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 notes in the series do not fit well with the usual notes of the piano?

16. 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.

17. Write out a full extent on n = 4 bells, including a factorization using permutations. How many changes must it have?

18. How many allowable moves (permutations) are there with 10 bells? Recall that an allowable move is one which only interchanges adjacent bells.

19. Given the permutations a = (2 4 1 5 3) and b = (1 4 3 2 5), compute a*b, b*a, a^3 and b^(-1).

20. Given the permutation a = (2 4 1 5 3), without doing any calculations, how do we know there exists a positive integer k such that a^k = e?

21. Suppose that A is a subgroup of the group G, where G has order 50. List all the possible orders for A.

22. Suppose that a and b are elements of a group (neither of which is the identity element) and that a^5 = e, b^2 = e and b a b = a. What is the inverse of a? What is the inverse of b? Show that a b = b a. Simplify the quantities a^(2000) and (a b)^5.

23. Take the first 4 bars of Twinkle Twinkle Little Star (see class notes) in the key of C major and transpose them to E major. Write the retrograde-inversion of the motif in E major, taking the inversion about E and staying in the key of E major.

24. If the first 4 bars of Twinkle Twinkle Little Star were inverted strictly (not staying in the same key) about C, what key would the resulting motif be in?

25. Give three musical examples where the mathematical symmetry of a horizontal reflection is used. What is the musical name of this symmetry?

26. Given a tone row, how many different versions of the row are there for use in a piece of 12-tone music? What are the possible choices?

27. Are 3 and 11 equivalent mod 4? List three numbers equivalent to 3 mod 4.