Math/Music: Aesthetic Links

Midterm Exam

Monday, March 28, In Class

The Midterm Exam covers homework assignments 1, 2 and 3, Chapters 6, 7 and 8 of Music and Mathematics: From Pythagoras to Fractals, CD's 1 and 2, and all the material discussed in class from Monday, February 7 up to and including Wednesday, March 23. 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. Note that many of the class lectures are available in PDF format from the course homepage.

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: Calculators are not allowed for this exam. All calculations needed will be doable by hand. A partial copy of a piano keyboard will be provided on the exam.

Exam Review: We will review for the exam during Friday'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, S_n as a group, symmetries of the square (D_4 dihedral group of degree 4)

  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. Change Ringing: general theory, 6 rules for an extent, verifying the rules, allowable moves, permutations, factorial, S_n, rounds, plain hunting, "factoring" an extent into its moves (e.g., [(ab)^3 ac]^3), Plain Bob Minimus and its connection to D_4

  4. Twelve-Tone Music: general theory, four types of rows, notation, know how to convert the prime row P-0 into other rows, identifying rows in music

  5. Composers: be sure to know several examples of how and where composers have utilized mathematical ideas in their works. You should also know some basic facts about composers who we have discussed in class or who appear on one of the CD's (dates, style, personal history, etc.) A partial list: Bach, Beethoven, Handel, Haydn, Mozart, Hindemith, Gershwin, Liszt, Bartok, Schoenberg, Davies, Xenakis, Reich

  6. General Music Theory: notation, writing and reading music in different clefs (treble and bass), piano keyboard, half steps and whole steps, major scale, 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, group theory (see above), group multiplication tables, working with identities, symmetry, modular arithmetic, magic squares, magic constant