Math, Music and Identity

Midterm Exam

Friday, March 27, In Class

The Midterm Exam covers homework assignments 1, 2, 3, and 4, Chapters 5 and 6 of the course text, and CD #1. It does not include 100 Years of Solitude or Musicophilia. It is highly recommended that you review homework problems, in-class worksheets, and your class notes. Many of the problems and questions we discussed in class are excellent examples of test questions. Note that most of the class lectures are available in PDF format from the course homepage.

A set of practice problems is available here. The solutions are now available. The exam will be designed to take 45 minutes although you will have a full hour 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. A partial copy of a piano keyboard will be provided on the exam for assistance with intervals.

Exam Review: We will review for the exam during Wednesday's class on March 25. Please come prepared with specific questions.

The following concepts are important material for the exam:

  1. 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), subgroup, Sn as a group, symmetries of the square (D4, the dihedral group of degree 4), the musical subgroup of D4

  2. Musical Group Theory: translation (transposition), vertical reflection (retrograde), horizontal reflection (inversion), 180 degree rotation (retrograde-inversion), examples of each, be able to identify each in music, know how to apply each transformation to a given melody

  3. Fibonacci Numbers and the Golden Ratio: Definition of, connection between Fibonacci and the golden ratio, appearance in nature, use by composers like Bartok and Debussy and why this is considered controversial, Lucas numbers

  4. Change Ringing: general theory, 6 rules for an extent, verifying the rules, allowable moves, permutations, factorial, Sn, rounds, plain hunting, "factoring" an extent into its moves (e.g., [(ab)3 a c]3), Plain Bob Minimus and its connection to D4

  5. Composers: you should know the examples we have discussed concerning how and where composers have utilized mathematical ideas in their works. Go over the liner notes to CD #1. A partial list of composers: Bach, Barber, Bartok, Beethoven, Gershwin, Handel, Haydn, Liszt, Machaut, Sousa

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

  7. Mathematical Concepts: permutations (multiplication of, inverse of), n factorial (n!), group theory (see above), group multiplication tables, working with identities, symmetry, modular arithmetic, recursive relations