Math/Music: Structure and Form

Final Exam

Wednesday, Dec. 14, 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 does not include Collapse. 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, the Monochord lab, 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 1.5 - 2 hours although you will have the full 2.5 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 Monday, Dec. 12, 3:00 - 4:30 pm in Brooks 454. 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 treble and bass clefs, rhythm, time signature (CD #1), dotted notes, tied notes and rests (duration), polyrhythmic music, piano keyboard, accidentals (sharps, flats, naturals, etc.)

  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), transposition, relative major of a minor key

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

  4. Mathematics of Sound: logarithms, sine waves, basic trigonometry, trig identities, sketching sine waves, pitch as frequency, resonance, beats (general rule of)

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

  6. The Three Tuning Systems: Pythagorean Tuning, Just Intonation, Equal Temperament, strengths and weaknesses of each system, 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. Strahle's Guitar Construction: general set up in order to place the frets, continued fractions, basics behind why it works

  8. Other 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, etc.), multiplication or division to find the frequency of a given note, ratios, irrational and rational numbers (e.g., prove that the square root of 2 is irrational), continued fractions