The second exam covers Chapters 3 and 4, and Sections 5.1 and 5.3 of the course textbook From Music to Mathematics: Exploring the Connections. This includes homework assignments 4 – 6, CD #3, the Monochord Lab, and all the material covered in class since the first exam up to and including April 12. It is highly recommended that you review class handouts, homework problems, lecture notes, the CD liner notes, 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. Here are the solutions. The exam will be designed to take 50 minutes although you will have the full 75 minutes to complete the exam.

Exam Review: We will review for the exam during a portion of Tuesday's class on April 17th. Please come prepared with specific questions.

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 containing at least one octave will be provided on the exam for your use.

The following concepts are important material for the exam:

1. 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)


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


3. 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


4. The Three Major Tuning Systems: the Pythagorean scale, just intonation, equal temperament, strengths and weaknesses of each system, the spiral of fifths, know the frequency ratios or multipliers for each system, overtone series, rational versus irrational numbers, Pythagorean comma, syntonic comma, cents, know 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 only), piano keyboard, half steps and whole steps, major scale, circle of fifths, octave, intervals (2nd, 3rd, 4th, tritone, major, minor, perfect, etc.)


6. Group Theory: Definition of a group (know and understand the 4 properties), examples of groups and non-groups (e.g., the integers are a group under addition, but not multiplication), symmetries of the square (D₄, the dihedral group of degree 4), the musical subgroup of D₄ (M)


7. Musical Group Theory: transposition (translation), retrograde (vertical reflection), inversion (horizontal reflection), exact versus tonal inversions, retrograde-inversion (180 degree rotation), examples of each (see lecture notes and CD #3), be able to identify each in music, know how to apply each transformation to a given melody


8. Mathematical Concepts: logarithms, trigonometry (sine and cosine functions, finding basic values of, graphing, unit circle, radians, period, frequency, amplitude, phase shift, fundamental identity), greatest common divisor, multiplication or division to find the frequency of a given note, working with ratios, rational and irrational numbers (know the proof that the square root of 2 is irrational), group theory (see above), group multiplication tables, symmetry, modular arithmetic (e.g., addition mod 12)