Homework 3 Answers
Math 109/Music 109A, Spring 2015
1.
47
1200
a. 21200 = √247 ≈ 1.03
1
b. 2−325/1200 = 2−13/48 = 48 13 ≈ 0.83
8/12
2/3
3
√2
c. 2
= 2 = √4 ≈ 1.59
1
d. 2−21/12 = 12 21 ≈ 0.30
√2
2.
a.
b.
c.
d.
≈ 261.626 Hz
≈ 77.782 Hz
≈ 987.767 Hz
≈ 207.652 Hz
a.
b.
c.
d.
≈ 430.539 Hz
≈ 645.080 Hz
= 32 Hz
≈ 101.594 Hz
3.
a. G3 ≈ 195.998 Hz
B3 ≈ 246.942 Hz
D4 ≈ 293.665 Hz
b. Ab4 ≈ 415.305 Hz
C5b ≈ 493.883 Hz
E5b ≈ 622.254 Hz
c. D3 ≈ 146.832 Hz
F3 ≈ 174.614 Hz
A3 = 220 Hz
C4 ≈ 261.626 Hz
d. B5b ≈ 932.328 Hz
Db6 ≈ 1108.731 Hz
F6b ≈ 1318.510 Hz
Abb
6 ≈ 1567.982 Hz
4.
Fret
1
2
3
4
5
6
7
8
9
10
11
12
Length of string (cm)
≈ 47.194
≈ 44.545
≈ 42.045
≈ 39.685
≈ 37.458
≈ 35.355
≈ 33.371
≈ 31.498
≈ 29.730
≈ 28.062
≈ 26.487
25
Distance (cm) from neck
≈ 2.806
≈ 5.455
≈ 7.955
≈ 10.315
≈ 12.542
≈ 14.645
≈ 16.629
≈ 18.502
≈ 20.270
≈ 21.938
≈ 23.513
25
5.
a. log10(0.0001) = log10(10-4) = -4log10(10) = -4
b. log3(243) = log3(35) = 5log3(3) = 5
2
2
7
c. log5( √25) = log5(5 ⁄7) = log5(5) = 2⁄7
7
d. log𝑏 ( 𝑛
1
√𝑏
𝑙
𝑙
𝑙
𝑛
𝑛
) = log𝑏 (𝑏−𝑛 ) = − log𝑏 (𝑏) = −
𝑙
e. log311 + log317 = log3(11*17) = log3187
f. log95 - 2log92 = log9(5⁄4)
log 21
log 21
g. log2 13 + log 4 21 = log 2 13 + 2 = log2 13 + 2
log2 4
2
1
= log2 13 + log2 21 = log 2 13 + log2 √21 = log2 13√21
2
log √𝑥
log √𝑥
1
2
h. 2log𝑐 𝑥 − 2log √𝑐 𝑥 = log𝑐 𝑥 4 − 𝑐 = log𝑐 𝑥 4 − 1 𝑐
4
= log𝑐 𝑥 − log𝑐 𝑥 = log𝑐 𝑥
3
log 𝑐 √𝑐
⁄2
6.
a.
f(x)
b.
g(x)
c.
r(x)
d.
f -1(x) = g(x)
r -1(x) = s(x)
f(x) = r(log5(2)x)
g(x) = log2(5)s(x)
s(x)
7. Let 𝛼 = log 𝑏 𝑥 and 𝛽 = log 𝑏 𝑦
So, 𝑥 = 𝑏𝛼 and 𝑦 = 𝑏𝛽
𝑥
𝑏𝛼
log 𝑏 = log 𝑏 𝛽 = log 𝑏 (𝑏𝛼−𝛽 ) = 𝛼 − 𝛽 = log 𝑏 𝑥 − log 𝑏𝑦
𝑦
𝑏
𝑥
∴ log 𝑏 = log 𝑏 𝑥 − log 𝑏 𝑦 ∎
𝑦
8.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
≈ 33.69 semitones
≈ 20.84 semitones
≈ 15.86 semitones
6 semitones up
≈ 17.31 semitones
≈ 386 cents down
≈ 316 cents up
≈ 884 cents down
≈ 1698 cents up
≈ 1982 cents up
up
down
up
up
9.
I
Answers may vary.
II
V7
I
10. There are a few possible interpretations. One possible interpretation:
Measure
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 1st ending
16 2nd ending
Chord 1 Chord 2
Ab
b
E
Ab
b
E
Fb
Eb
Fb
Eb
Abm
Abm
o7
B
Ab
b
F
Ab
Ab
Bo7
b
A
Fb
Ab
Ab
Ab
-