Math, Music and Memory

Final Exam

Tuesday, Dec. 16, 8:00 - 10:30 am, Brooks 454

The final exam is CUMULATIVE, that is, it covers all the material from the first day of class onwards. This is most of the sections in Chapters 1--4 of A First Course in Math and Music. This does not include any of the common readings from the cluster. 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 (HW 1--7), your class notes, in-class handouts, the Monochord lab, CDs 1--2 (and the liner notes), the two midterm exams (solutions posted on course webpage), and the midterm exam 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 solutions. The exam will be designed to take 1.5 - 2 hours although you will have the full 2.5 hours to take the exam.

Note: You will be allowed a scientific calculator for the exam which does NOT have graphing capabilities. Please bring your own scientific calculator to the exam. Also, a portion of a blank piano keyboard will be provided on the exam for your use.

The following concepts are important material for the exam:

  1. General Music Theory: notation, writing and reading music in different clefs (treble, bass, alto, and tenor), rhythm, time signature (examples; CD #1), dotted notes and rests (duration), tied notes, polyrhythmic music (examples; CD #1), the piano keyboard, accidentals (sharps, flats, naturals, etc.; correct placement of on staff)

  2. Scales, Intervals and Other Important Items from Music Theory: half steps, whole steps, chromatic scale, whole tone scales, major scales, natural and harmonic minor scales, diatonic scale spelling, major and minor chords, dominant-tonic chord progression, 12-bar blues, circle of fifths, key signatures, octave, intervals (m2, M2, m3, M3, P4, tritone, etc.), polyphony and tonality (CD #2), relative major and minor keys

  3. Sound: sound as change in air pressure, attributes of sound (loudness, pitch, timbre and duration), the incredible ear-brain system, sound intensity and decibels (dB), frequency and hertz (Hz)

  4. Mathematics of Sound: logarithms, sine waves, basic trigonometry, trig identities, sketching sine waves, period and frequency, resonance

  5. Pitch, Frequency, and Length: residue pitch, how ratios relate to pitch (for example, taking 1/2 the length of a string, or doubling the frequency, raises the pitch an octave), Monochord Lab, overtone series

  6. The Three Major Tuning Systems: the Pythagorean scale, just intonation, equal temperament, strengths and weaknesses of each system, the spiral of fifths, frequency ratios or multipliers, overtone series, rational versus irrational numbers, Pythagorean comma, syntonic comma, cents, why certain intervals sound "nice" together, how to find the frequency of a given note using ratios or multipliers (e.g., G above middle C)

  7. Important Mathematical Concepts: geometric sequence and series, infinite geometric series, least common multiple, greatest common divisor, relatively prime numbers, logarithms, trigonometry (sine and cosine functions, graphing, unit circle, radians, period, frequency, amplitude, phase shift, identities), multiplication or division to find the frequency of a given note, working with ratios, irrational and rational numbers (e.g., prove that the square root of 2 is irrational)