Mathematics and Music

Exam #2

Wednesday, April 12, 11:00 - 12:00 pm

The second exam covers homework assignments 5 - 7, Chapters 1, 6 and 7 and all the material covered in class from Friday, March 3 up to and including Friday, April 7. 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. Different 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. Change Ringing: general theory, 6 rules for an extent, verifying the rules, permutations, factorial, S_n, rounds, plain hunting, using permutations to describe an extent, Plain Bob Minimus

  3. Group Theory: Definition of a group (know and understand the 4 properties), examples of groups and nongroups (eg. the integers are a group under addition but not multiplication), order of a group, subgroup, S_n as a group, first lead of an extent is often a subgroup of S_n, symmetries of the square (dihedral group of degree 4)

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

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

  6. Mathematical Concepts: multiplication to find the frequency of a given note, ratios, irrational versus rational numbers (know the proof 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 Monday's class, April 10th. Please come prepared with specific questions.