Math/Music: Structure and Form

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 but is intended to provide a general overview.

Questions:

1. What is log(1000) + log(1/1000) if each logarithm is written to the base 10?

2. When you increase the amplitude of a sound by 40 decibels, by what factor do you increase the volume of sound?

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

4. 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?

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

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

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

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

12. Starting in the bass clef on the bottom line G, write the first ten notes corresponding to the frequencies in the overtone series of this G. Also notate the intervals between successive notes.

13. Using A440 Hz, find the frequency of G above middle C in each of the three tuning systems: Pythagorean Tuning, Just Intonation and Equal Temperament (one decimal place).

14. In a short essay, compare and contrast tuning a scale using Just Intonation versus using Equal Temperament. What are the advantages and weaknesses of each system?

15. 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 abundace of major I, IV and V chords. Which of the three tuning systems would you use to tune your guitar? Explain why.

16. What numbers do you multiply the fundamental frequency by to raise the pitch a major sixth (M6) in each of the three tuning systems?

17. What number do you multiply the fundamental frequency by to raise the pitch a perfect fourth using Equal Temperament? Prove that this number is irrational.

18. 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?