Mathematics and Music

Exam 1 Review Questions

Below are some types of questions you should expect to find on the exam. This is certainly not an exhaustive list and is meant to give a general overview.

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 in music: major, minor, Gregorian chant, early polyphony, atonal music, etc. (CD#2)

General Questions:

1. How many eighth notes do you need to fill up a measure in 7 - 4 time? in 5 - 8 time?

2. In 3 - 4 time, how many beats does a triple-dotted quarter note get? How many beats does it get in 5 - 2 time?

3. What is the sum of an infinite geometric series that begins with 3 and has a ratio of 2/3?

4. Give an example of two integers a and b such that the lcm(a,b) = b. What must the gcd(a,b) equal for this to be true?

5. Write out a whole tone scale, using correct accidentals, beginning on D, in both clefs.

6. Without using key signatures, write out an F# major scale in the treble clef using correct accidentals.

7. What key has 5 flats? Write the major scale for this key in the bass clef. List the flats for this key in correct order.

8. If you start at middle C, go up a perfect fifth, then down a major third, what note have you arrived at? Notate all three notes in the treble clef. Be able to find all three notes on the piano keyboard.

9. The first jump in the melody of Twinkle Twinkle Little Star is what interval?

10. How many half steps are there in a major sixth? How many whole steps are there in an octave? Name a song that begins with a major sixth.

11. Using key signatures, transpose Twinkle Twinkle Little Star from the key of C major to the key of E flat major (music would be provided to you on the exam.)

12. How many notes on the modern piano keyboard? Roughly how many octaves fit into the piano keyboard?

13. What is log(1000) + log(1/1000) if each logarithm is written to the base 10? When you increase the amplitude of a sound by 30 decibels, by what factor do you increase the volume of sound?

14. 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?

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

16. Graph the function y = 4 sin( Pi ( t - 1/2)) . What are the amplitude and period of this function?

17. What angle (in radians) corresponds to going around the unit circle 5 times? What will the cosine of this angle be? What will the sine of this angle be?

18. Set A = Pi/4 in the double angle formula cos(2A) = 2cos^2(A) - 1 to find cos(Pi/4). What is sin(Pi/4)?

19. A string of length 40 cm and a string of length 60 cm are plucked simultaneously. What interval do you hear?

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

21. Write out the first 8 frequencies in the overtone series for a note whose fundamental is 150 Hz.