Math, Music and Memory
Solutions for Final Exam Review General Questions
In 5 - 4 time you need 20 sixteenth notes to fill a measure (4 per beat times 5 beats in a measure).
In 6 - 8 time you need 12 sixteenth notes to fill a measure (2 per beat times 6 beats in a measure).
In 2 - 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 - 4 time (a rock waltz!)
12 - 3 + 3/4 - 3/16 + - ... . The sum of the infinite series is 12/(1 - -1/4) = 48/5 = 9.6.
Compute the least common multiple of 22 and 8, which is (22 * 8)/2 = 88.
The lcm(a, b) = a if and only if the gcd(a, b) = b. A simple example is a = 8 and b = 2.
The gcd(6, 8, 14) = 2. The lcm(6, 8, 14) is the same as lcm( lcm(6, 8), 14)
or lcm(24, 14), which equals 168.
D♭ 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 spelling) and the interval pattern for any major scale is W W H W W W H.
The relative major of C♯ minor is E major, which has 4 sharps.
We need an additional sharp on B to obtain the leading tone (a half step below the tonic).
We must use B♯ (as opposed to just C) to make the scale diatonic (consecutive letter names).
The interval pattern for any harmonic minor scale is W H W W H WH H.
The key of F♯ has 6 sharps. In order, they are F♯, C♯, G♯, D♯,
A♯, and E♯.
There are 52 white keys. There are 7 white keys in an octave and roughly 7 octaves on the modern 88-key piano. This gives 49 white keys. Three more come from the C, B, and A keys at the bottom of
the piano. Alternatively, there are 36 black keys (see Exam #1), so 88 - 36 = 52.
The note is the G above middle C. Going up a P5 from A gives an E; another P5 gives a B;
then down a M3 gets us to G. Since A220 is the A just below middle C (A440, an octave higher, is
the A above middle C), the G we have landed on is the first G above middle C. This note is a minor 7th above A220 (10 half steps). To find its frequency
in just intonation, we have 220 * 3/2 * 3/2 * 4/5 = 396 Hz. In equal temperament,
we have 220 * 2^(10/12) ≈ 392 Hz.
A minor second (half step).
The 5th bar of a 12-bar blues is usually the IV chord (sub-dominant). In the key of
B♭, this is the major chord starting on the 4th scale degree, or E♭.
The chord is the 4th, 6th, and 8th scale degrees, which corresponds to the notes
E♭, G, and B♭.
20 dB. We have 10 * log10(100) = 10 * 2 = 20 dB. If we increase the intensity
by a factor of 200, we have 10 * log10(200) ≈ 23 dB.
10.5π is 5.25 times around since 2π is equivalent to going once around. We have sin(10.5π) = 1, since the sine of an angle is given by the y-coordinate of the
corresponding point on the unit circle.
The period of this sine wave is 2π/660π = 1/330.
Thus, the frequency is 330 Hz, which corresponds to the E just above middle C on
the piano. This can be found by the fact that 440/330 = 4/3, so that the note
in question is a P4 below A440.
The 7th and 11th overtones are approximations; they are each flatter than
the corresponding note shown.
In order, the ratios for the Pythagorean scale are:
(1) 1:1, (2) 9:8,
(3) 81:64, (4) 4:3,
(5) 3:2, (6) 27:16,
(7) 243:128, (8) 2:1.
The syntonic comma is 81/80. It comes from the gap between the two different whole steps
in just intonation: (9/8)/(10/9) = 81/80. It is also the distance between the multipliers
that are different in the Pythagorean scale and just intonation, (e.g., the M3 is (81/64)/(5/4) = 81/80).
Since (2^(7/12))^12 = 2^7, we have that 12 equally-tempered perfect fifths (going around the circle of fifths once) equals 7 octaves (the number of octaves in the circle of fifths going around once), so the circle of fifths is a true circle.
However, in Pythagorean tuning or just intonation we have a spiral of fifths, since
(3/2)^12 ≠ 2^7, with the gap between these two numbers equal to the Pythagorean comma.
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, the interval between 5f and 6f (ratio 6/5)
in the overtone series is a minor third.
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 M2 or a whole step. Thus, in both Pythagorean tuning and just intonation
we have 220 * 9/8 = 247.5 Hz.
For equal temperament we use the multiplier 2^(2/12) = 2^(1/6) to obtain
220 * 2^(1/6) = 246.94 Hz.
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 that when factored into
its prime factorization will have a number of 2's one greater than a multiple of 3.
But the right-hand side has a prime factorization with a number of 2's that is exactly a multiple of 3 or 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 other 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.
False. 2^(1/2), the square root of 2, is an irrational number. However,
when it is multiplied by itself (the product of two irrational numbers), we obtain 2,
a rational number. Another example is 3^(1/2) * 1/3^(1/2) = 1.