Mathematics and Music

Final Exam

Friday, May 11, 8:30 - 11:30 am, Brooks 452

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, handouts, CD liner 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 on Wednesday, May 9, 12:00 - 1:30 pm in Brooks 452. 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, tone row

  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 (m2, M2, m3, M3, P4, 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: logarithms, sine waves, basic trigonometry, trig identities, sketching sine waves, the harmonic oscillator, pitch as frequency, resonance, beats (general rule of)

  5. The Three 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, allowable moves, permutations, factorial, S_n, rounds, plain hunting, "factoring" an extent into its moves (eg. [(ab)^3 ac]^3), Plain Bob Minimus

  7. 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)

  8. Musical Group Theory: translations (transpositions), vertical reflection (retrograde), horizontal reflection (inversion), 180 degree rotations (retrograde-inversion), know some examples of each, be able to identify each in music (CD #3), 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 or on the CD's (dates, style, personal history, etc.) A partial list: Sousa, Barber, Beethoven, Handel, Haydn, Mozart, Bach, Hindemith, Gershwin, Tchaikovsky, Chopin, Liszt, Stravinsky, Machaut, Schoenberg, Xenakis, Davies (CD #4 for the last three listed)

  10. Other Mathematical Concepts: geometric sequence and series, infinite geometric series, least common multiple, greatest common divisor, logarithms, trigonometry (sine function, graphing, unit circle, radians, attributes of a sine wave, period, frequency, amplitude, phase shift, 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, magic squares