Topics in Mathematics: Math and Music
SOLUTIONS to Exam 1 Review Questions

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

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

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.

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.

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.

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

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

Merengue (see CD #1).

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

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

Whole tone scale starting on D.

F# major scale.

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

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

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

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

Perfect fifth.

Tritone.

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.

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

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

Transposition to D♭ and E major.

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.