Mathematics and Music

Final Exam

Saturday, May 6, 8:30 - 11:30 am, Brooks 132

The final exam is CUMULATIVE, that is, it covers all the material from the first day of class onwards. The goal is for you to synthesize knowledge from the entire course, bringing together common themes and subject material. It is highly recommended that you review homework problems, your class notes, the midterm exams and midterm review sheets. 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 partial solutions. The exam will be designed to take 2 hours although you will have the full three hours to take the exam. 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.

Exam Review: We will review for the exam Thursday, May 4, 3:00 - 4:30 pm in Brooks 132. Please come prepared with specific questions.

The following concepts are important material for the exam:

  1. General Music Theory: notation, writing and reading music in different clefs (treble and bass), rhythm, time signature (CD #1), dotted notes (duration), polyrhythmic music, piano keyboard

  2. Scales and Intervals: half steps (semi-tone), whole steps (whole-tone), chromatic scale, whole tone scale, major scale, natural and harmonic minor scales, circle of fifths, key signatures, octave, intervals (2nd, 3rd, 4th, tritone, etc.), polyphony and tonality (CD #2)

  3. Sound: sound as change in air pressure, attributes of sound (amplitude, frequency, timbre and duration), the incredible ear-brain system, oscillograph plots, hertz, decibels

  4. Mathematics of Sound: sine waves, basic trigonometry, sketching sine waves, the harmonic oscillator, resonance, beats (general rule of), the wave equation

  5. Different Tuning Systems: Pythagorean scale, Just Intonation, Equal Temperament, strengths and weaknesses of each system, the 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 for a given tuning system (eg. G above middle C), relationship between length of a string and pitch produced (eg. 1/2 the length of a string raises the pitch an octave)

  6. 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

  7. 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)

  8. 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

  9. Composers: be sure to know several examples of how and where composers have utilized mathematical ideas in their works. You should also know some basic facts about composers we have discussed in class (dates, style, etc.) A partial list: Sousa, Barber, Beethoven, Handel, Haydn, Mozart, Bach, Hindemith, Gershwin, Tchaikovsky, Chopin, Stravinsky, Machaut, Schoenberg, Xenakis

  10. Other Mathematical Concepts: geometric sequence and series, infinite geometric series, least common multiple, trigonometry (sine function, graphing, attributes of a sine wave, period, frequency, identities, etc.), using 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 (see above), group multiplication tables, symmetry, modular arithmetic