Math/Music: Structure and Form

Exam #2

Friday, December 2, In Class

The second exam covers homework assignments 3 and 4, Chapters 1 and 3 of Music and Mathematics: From Pythagoras to Fractals, the Monochord Lab and all the material discussed in class from Wednesday, October 26 up to and including Monday, November 21. 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 set of practice 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.

Exam Review: We will review for the exam during Wednesday's class on Nov. 30. Please come prepared with specific questions.

The following concepts are important material for the exam:

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

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

3. 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

4. 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)

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

6. Mathematical Concepts: 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 numbers (e.g., prove that the square root of 2 is irrational)