Math/Music: Aesthetic Links

Midterm Exam Review Solutions

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.


  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. Ans. Listen, listen, listen. Be sure to read the CD liner notes as you listen.

  3. 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).

  4. Ans.

  5. 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.

  6. Ans.

  7. 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?

  8. Ans. Bach - nearly any fugue he wrote did this; I Got Rhythm by George Gershwin; Mikrokosmos No. 141 by Bartok; The Thunderer by John Philip Sousa. A horizontal reflection is known as an inversion in music.

  9. 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?

  10. Ans. Lean on Me by Bill Withers; The Musical Offering by Johann Sebastian Bach; the ``Hallelujah Chorus'' from the Messiah by George Frideric Handel; Piano Sonata No. 41, ``Minuet and Trio'' in A major by Joseph Haydn. A vertical reflection is known as a retrograde in music.

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

  12. Ans. Inverses are not in the group. For example, 4^(-1) = 1/4 but 1/4 is not an integer.

  13. 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 ???

  14. Ans. If a musical motif has symmetry H (ph) and symmetry V (pv), then it will have the symmetry obtained by composing these together. But this is H*V=R_180 (p2) so R_180 is guaranteed. Similar arguments work for the other two cases since H*R_180=V and V*R_180=H.

  15. 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.

  16. Ans. Both D_4 and S_n are examples of non-commutative groups.

  17. 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"?

  18. Ans. See the handout from class on Change Ringing .

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

  20. Ans. (1 2 3), (2 1 3), (2 3 1), (3 2 1), (3 1 2), (1 3 2), (1 2 3). There are only two legitimate extents because at any given stage there are two possible moves: a = (1 2) or b = (2 3). Since a^2 = e and b^2 = e, we must alternate between a and b between changes. This results in either (ab)^3 or (ba)^3 as the only possible extents.

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

  22. Ans. See the handout from class on Change Ringing . This isn't hard to learn if you recall the moves used to create Plain Bob Minimus, [(ab)^3 ac]^3.

  23. 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.

  24. Ans. 6! = 720.

  25. 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?

  26. Ans. (1 2 3 4), (2 1 3 4), (2 3 1 4), (2 3 4 1), (2 4 3 1), (2 4 1 3), (2 1 4 3), (1 2 4 3) is the first lead of Double Court Minimus. The second lead is (1 4 2 3), (4 1 2 3), (4 2 1 3), (4 2 3 1), (4 3 2 1), (4 3 1 2), (4 1 3 2), (1 4 3 2) and the third lead is (1 3 4 2), (3 1 4 2), (3 4 1 2), (3 4 2 1), (3 2 4 1), (3 2 1 4), (3 1 2 4), (1 3 2 4), (1 2 3 4). This is a legitimate extent because the first three rules are satisfied. However, rule #4 is not satisfied since Bells 2, 3 and 4 rest in the same position for several changes in a row. Rules 5 and 6 are satisfied.

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

  28. Ans. 88. See homework #2 (make a table).

  29. 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).

  30. Ans. a*b = (2 3 5 1 4), b*a = (4 3 5 2 1), b^3 = b, a^(-1) = (5 1 2 4 3)

  31. 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.)

  32. Ans. Start looking at higher and higher powers of a. Call the set A = {a, a^2, a^3, a^4, ...}. This set is finite because A is a subset of S_5 which has 5! = 120 elements so eventually we must repeat. Either we keep multiplying until we find a k value where a^k = e, or we will exhaust all 120 elements and a^120 = e.

  33. 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.

  34. Ans. a^(-1) = a^3, b^(-1) = b, bab = a implies ab = ba by multiplying by b on the left (or right) of both sides, a^(2011) = a^3 = a^(-1), (ab)^5 = ab (since ab * ab * ab * ab * ab = ab * ba * ab * ba * ab = a^4 * ab = ab).

  35. 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?

  36. Ans. There are typically 48 different versions of a prime row (including the row itself.) Although this is not the case for every tone row, as some may only generate a total of 24 (e.g., the chromatic scale or two whole tone scales adjoined). The 4 general types of rows (12 of each) are P-n (transposition of P-0 up n half steps), R-n (retrograde of P-n), I-n (inversion of P-n about the starting note or transposition of I-0 up n half steps), RI-n (retrograde inversion of P-n = retrograde of I-n).

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

  38. Ans. Going up by a minor sixth is going up by 8 half steps, while descending by a major third is 4 half steps. If we think of these numbers as 8 and -4 (minus because we went down), then they are 12 half steps apart, or an octave. But notes that differ by an octave are considered the ''same'' in 12-tone music. If you start on any note and go up by a minor 6th, you will arrive at the same note (in name) as if you had gone down by major 3rd. The intervals m6 and M3 are called inversions of one and other.

  39. 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?

  40. Ans. The first and last notes of the row are a tritone apart, just as the primary row used by Schoenberg.

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

  42. Ans. By symmetry, we may assume that the upper-left corner is 1. By running through the 6 possible cases, one can see that it is never possible to get all of the rows, columns and main diagonals to sum up to 5, the magic constant for n = 2. There is only 1 (up to rotation and reflection) 3 x 3 magic square.

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

  44. Ans. The magic constant M_8 = 8*(8^2 + 1)/2 = 4*65 = 260.

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

  46. Ans. Iannis Xenakis in Metastasis.