Mathematics and Music

Exam #2

Friday, April 13, 10:00 - 11:00 am

The second exam covers homework assignments 5 - 7, CD #3, Chapters 1, 6 and 7 and all the material discussed in class from Friday, February 23rd up to and including Wednesday, April 4th. It is highly recommended that you review homework problems and your class notes. Many of the problems and questions we discussed in class are excellent examples of test questions.

A sample list of problems is available here along with solutions. The exam will be designed to take 45 minutes although you will have a full hour to take the exam.

Note: You will be given a scientific calculator for the exam which does NOT have graphing capabilities so be prepared to answer questions without your personal calculator.

The following concepts are important material for the exam:

1. The Three Tuning Systems: Pythagorean scale, Just Intonation, Equal Temperament, strengths and weaknesses of each system, overtone series, rational versus irrational numbers, Pythagorean comma, syntonic comma, why certain intervals sound "nice" together, how to find the frequency of a given note (eg. G above middle C)

2. Group Theory: Definition of a group (know and understand the 4 properties), examples of groups and non-groups (eg. the integers are a group under addition but not multiplication), subgroup, S_n as a group, symmetries of the square (dihedral group of degree 4)

3. Musical Group Theory: translations (transpositions), vertical reflection (retrograde), horizontal reflection (inversion), 180 degree rotations (retrograde-inversion), examples of each, be able to identify each in music, how to apply each transformation to a given melody

4. Change Ringing: general theory, 6 rules for an extent, verifying the rules, allowable moves, permutations, factorial, S_n, rounds, plain hunting, "factoring" an extent into its moves (eg. [(ab)^3 ac]^3), Plain Bob Minimus

5. General Music Theory: notation, writing and reading music in different clefs (treble and bass), piano keyboard, half steps and whole steps, major scale, circle of fifths, octave, intervals (2nd, 3rd, 4th, tritone, etc.)

6. Mathematical Concepts: multiplication to find the frequency of a given note, ratios, irrational versus rational numbers (prove that the square root of 2 is irrational), permutations (multiplication of, inverse of), n factorial, group theory (know the 4 properties), group multiplication tables, symmetry

Exam Review: We will review for the exam during Wednesday's class, April 11th. Please come prepared with specific questions.