Math/Music: Structure and Form

Partial Solutions for Final Exam Review General Questions

  1. In 5 - 4 time you need 20 sixteenth notes to fill a measure (5 times 4).
    In 6 - 8 time you need 12 sixteenth notes to fill a measure (6 times 2).

  2. In 4 - 4 time, a triple-dotted eighth note gets 15/16 beats (1/2 + 1/4 + 1/8 + 1/16).
    In 3 - 8 time, a triple-dotted eighth note gets 15/8 beats (1+ 1/2 + 1/4 + 1/8).

  3. 5 - 5/4 + 5/16 - 5/64 + - ... . The sum of the infinite series is 5/(1 - -1/4) = 4.

  4. 500 x 1001 = 500,500. According to popular legend, Gauss showed the method of repeatedly grouping first and last terms to his grade school teacher.

  5. The lcm(a,b) = ab if and only if a and b are relatively prime (ie. the greatest common divisor of a and b is 1.)

  6. D flat major has five flats using the circle of fifths. Be sure to place the accidentals (flats) directly in front of the note on the same space or line as the notehead.

  7. This was question #4b on Exam 1. See solutions . The relative major of F# minor is A major.

  8. Discussed at the review session. Use 6 flats (in correct order) in the key signature and begin the piece on B flat.

  9. 52 white keys (88 total - 36 black = 52 white -- see Exam 1.)

  10. The two notes have frequency 330 Hz and 326 Hz. We "hear" their average, 328 Hz, with 4 beats per second (their difference.)

  11. The brain can distinguish between different instruments and sounds. It can also distinguish different frequencies of the overtone series. It can memorize sounds, filter out unwanted noises and recall particular sounds as needed.

  12. 10.5 Pi is 5.25 times around since 2Pi is equivalent to 1 time around. The sin(10.5 Pi) = 1.

  13. See class notes and the first chapter of the primary course text.

  14. The Pythagorean Comma is 3^(12)/2^(19). It comes from the fact that H^2 does not equal W or the fact that the circle of fifths does not close up (the spiral of fifths leading to B# does not equal C).

  15. Since (2^(7/12))^12 = 2^7, we have that 12 perfect fifths (going around the circle of fifths once) equals 7 octaves (this is the number of octaves in the circle of fifths going around once.)

  16. A minor 3rd. The ratio 30/25 = 6/5. In just intonation this is a minor third since a major triad starting on the tonic will be in the ratio 4:5:6 and the top half of this chord, the interval between 3 and 5 is a minor third. Alternatively, one could write out the overtone series and realize that the interval between 5f and 6f (ratio 6/5) is a minor third.

  17. Note that the A just below middle C will have frequency 220 Hz because this is half 440. The interval from A below middle C to F sharp above middle C is a Major 6th. Thus, in Pythagorean Tuning we have 220 times 27/16 = 371.25 Hz. For Just Intonation, a Major 6th is obtained by multiplying by 5/3 so we have 220 times 5/3 = 366.6666 Hz. Finally, for Equal Temperament we either have 440 divided by 2^(3/12) = 369.994 (down three half steps from A 440) or 220 times 2^(9/12) = 369.994 (up nine half steps from A 220.)

  18. See class notes or the first chapter of the primary course text.

  19. Done in the review session (or see class notes). The only note which does not fit well is the 7th overtone which is not in the scale of the tonic (in this case 7f gives C which is not in the D major scale.) This note falls between two notes on the piano.

  20. 2^(4/12) = 2^(1/3) (M3 is four half-steps.) To show this number is irrational, set 2^(1/3) = p/q, with p and q integers. Raise both sides to the 3rd power to obtain 2 = p^3/q^3. Cross multiplying gives 2 q^3 = p^3. The left-hand side of this equation is an integer which when factored into its prime factorization will have a number of 2's one greater than a multiple of 3 (ie. mod 3 the number of 2's is 1.) But the right-hand side has a prime factorization with a number of 2's that is exactly a multiple of 3 (ie. mod 3 the number of 2's is 0.) This violates the Fundamental Theorem of Arithmetic which states that a positive integer has a unique factorization into its prime factors.
    Note that the argument here differs from many of our preceding proofs because it is not about whether the number of 2's is even or odd, but how many there are upon division by 3.

  21. See class notes or the fourth chapter of the primary course text.

  22. A linear fractional transformation (LFT) is a function of the form y = (a + bx)/(c + dx) . Any projection from one line to a different one can be expressed in the form of an LFT. This is precisely what Strahle does in his guitar construction. The formula y = (17 - 5x)/(17 + 7x) , evaluated at x = 0, 1/12, 2/12, ..., n/12, ... gives the precise ratios used to raise the pitch by n half steps (in terms of string lengths).

  23. The continued fraction expansion for 31/11 is [2; 1, 4, 2]. The continued fraction expansion for the square root of 3 is [1; 1, 2, 1, 2, 1, 2, ...].