Below are some types of questions you can expect to find on the exam. This is certainly not an exhaustive list and is meant to give a general overview. For more detailed topics, see the Midterm Exam Review sheet.

Questions:

1. Listening Questions: Upon listening, be able to identify, using both musical and mathematical terminology, what type(s) of symmetry and/or mathematics is present in a piece (CD #1 and #2).


2. Take the first 4 bars of Twinkle Twinkle Little Star in the key of E major and write the inversion about G#, staying in the key of E major (use the treble clef). Also write the retrograde-inversion, inverting about G#, staying in the key of E major (use the treble clef).



3. Take the first 4 bars of Twinkle Twinkle Little Star in the key of E major and write the exact inversion about G# in the treble clef, using correct accidentals.


4. Give three musical examples (title and composer) where the mathematical symmetry of a horizontal reflection is used. What is the musical name of this symmetry?


5. Give three musical examples (title and composer) where the mathematical symmetry of a vertical reflection is used. What is the musical name of this symmetry?


6. Explain why the integers do not form a group under multiplication.


7. According to the text in Chapter 6, if a musical motif has two of the three symmetries ph, pv, or p2 (H, V, R_180 in our notation), then it automatically has the third remaining symmetry. Explain why this is true using the symmetries of the square and composition. Hint: If you have H and V as symmetries, for example, then you also have H*V. But H*V is equivalent to ???


8. Give an example of a group that is non-commutative, that is, there are elements a and b where a*b does not equal b*a.


9. What are the three mandatory rules in change ringing necessary to qualify as a legitimate extent? What does it mean mathematically speaking for an extent to satisfy the "palindrome property"?


10. Write out a full extent on n = 3 bells. How many different extents on 3 bells are there? Explain.


11. Write out the first 8 changes of Plain Bob Minimus.


12. How many different ways are there to ring 12 bells if the odd numbered bells never move? In other words, Bell 1 stays as Bell 1, Bell 3 stays as Bell 3, etc.


13. Write out the full extent for Double Court Minimus [db(cb)^2d b]^3, where b = (2 3), c = (3 4) and d = (1 2). Is this a legitimate extent? Which of the 6 rules for an extent are satisfied?


14. How many allowable moves (permutations) are there with 10 bells? Recall that an allowable move is one which only interchanges adjacent bells.


15. Given the permutations a = (2 3 5 4 1) and b = (4 2 3 1 5), compute a*b, b*a, b^3 and a^(-1).


16. Given the permutation a = (2 4 1 5 3), without doing any calculations, how do we know there exists a positive integer k such that a^k = e? (This is a tricky one.)


17. Suppose that a and b are elements of a group (neither of which is the identity element) and that a^4 = e, b^2 = e and b a b = a. What is the inverse of a? What is the inverse of b? Show that a b = b a. Simplify the quantities a^(2011) and (a b)^5.


18. Given a tone row, in general how many different versions of the row are there for use in a piece of 12-tone music? What are the 4 general types of rows?


19. In 12-tone music, explain why going up a minor sixth is considered the ``same'' as going down a major third.


20. Given the following tone row P-0, write out in the treble clef, with correct accidentals, the tone rows P-5, R-5, I-0, I-3 and RI-3. What feature of this tone row is reminiscent of the one used by Schoenberg in his Piano Suite, Opus 25?






21. Explain why there are no 2 x 2 magic squares. How many "different" 3 x 3 magic squares exist?


22. What number do the rows, columns and main diagonals sum up to in an 8 x 8 magic square?


23. What composer used glissandi and ruled surfaces in one of his famous compositions? What was the name of the piece?