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. Compute the least common multiple of 12 and 9, which is 36.

5. The lcm(a,b) = a if and only if the gcd(a,b) = b. A simple example is a = 4 and b = 2.

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. Remember that the notes are successive letter names (diatonic) and the interval pattern is W W H W W W H.

7. The notes are F#, G#, A, B, C#, D, E#, F#. The relative major of F# minor is A major, which has 3 sharps. We need the additional sharp on E to obtain the leading tone and we must use E# (as opposed to just F) to make the scale diatonic.

8. 52 white keys (88 total - 36 black = 52 white -- see Exam 1.)

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

10. 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.

11. 10.5 Pi is 5.25 times around since 2Pi is equivalent to 1 time around. The sin(10.5 Pi) = 1.

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

13. 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 or F# does not equal G flat).

14. 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.)

15. 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 the 3rd and 5th scale degrees 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.

16. 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 B below middle C is a Major 2nd or a whole step. Thus, in both Pythagorean Tuning and Just Intonation we have 220 times 9/8 = 247.5 Hz. For Equal Temperament we use the multiplier 2^(2/12) = 2^(1/6) to find 220 times 2^(1/6) = 246.94 Hz.

17. See the handout on the three musical tuning systems.

18. See class notes for examples and a list of the successive intervals. The first ten notes are D, D', A', D'', F#'', A'', C'', D''', E''', F#'''. 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 really falls between two notes on the piano.

19. 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 (i.e., 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 (i.e., 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.

20. See the handout on Strahle's guitar construction or the fourth chapter of the primary course text.

21. 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, ...].