Topics in Mathematics: Math and Music

SOLUTIONS to Exam 1 Review Questions

1. 22 eighth notes are needed to fill up a measure in 11 - 4 time, while 7 eighth notes are needed in 7 - 8 time.

2. In 5 - 4 time, a triple-dotted quarter note gets 1 7/8 = 15/8 beats, while it gets 15/16 beats in 3 - 2 time.

3. The sum of an infinite geometric series that begins with 9 and has a ratio of -2/3 is 9/(1 - -2/3) = 9/(5/3) = 27/5. The first five terms are 9, -6, 4, -8/3, 16/9.

4. To subdivide a polyrhythm of 10 against 6 you need at least lcm(10,6) = 30 subdivisions to see the precise location of each note in the polyrhythm.

5. In order for the lcm(a,b) = 2a, we must have gcd(a,b) = b/2. Two examples satisfying this requirement are a = 4, b = 8 or a = 9, b = 6.

6. In order for the lcm(m,n) = n2, we must have gcd(m,n) = m/n. Two examples satisfying this requirement are m = 4, n = 2 or m = 9, n = 3.

7. lcm(2,5,10) = 10, lcm(3,5,10) = 30, and lcm(4,5,10) = 20.

8. Merengue (see CD #1).

9. Rite of Spring by Stravinsky (see CD #1).

10. Two notes enharmonically equivalent to A double sharp are B or C♭. Two notes enharmonically equivalent to F double flat are E♭ or D#.

11. Whole tone scale starting on D.

12. F# major scale.

13. G harmonic minor scale. The relative major of G minor is B♭ major.

14. G# natural minor scale. The relative major of G# minor is B major.

15. D♭ major scale. The key of D♭ has 5 flats. In order, they are B♭, E♭, A♭, D♭, and G♭.

16. E♭. The three notes, C, E♭, and G form a C minor chord.

17. Perfect fifth.

18. Tritone.

19. There are 9 half steps in a major sixth. There are 6 whole steps in an octave. Somewhere Over the Rainbow begins with an octave jump.

20. B major and minor chords. B major consists of B, D#, and F#, while B minor contains the notes B, D, and F#.

21. E and F# major are musically the closest keys to B major, while F major is the furthest from B major.

22. Transposition to D♭ and E major.

23. There are precisely 2 different whole tone scales when considering scales with the same total set of notes to be equivalent. For instance, a whole tone scale that starts on C has the same set of notes as the one that starts on D (C, D, E, F#, G#, A#). The other whole tone scale is (C#, D#, F, G, A, B). No matter which note we start on, any whole tone scale has to contain the same set of notes as one of these two choices. This is due to the repeating W W W W W W pattern.
In contrast, there are precisely 12 different major scales, one for each note of the chromatic scale. Here we are assuming enharmonic equivalence, so F# and G♭ major are identical scales, as are B and C♭ major, and C# and D♭ major.