Physics 198
Homework 15
Due October 22
1. At what point should we divide a 65 cm guitar string (as Pythagoras did) so that the two
segments sound pitches sound octave apart?
Solution:
If two sound pitches are one octave apart, the ratio of the frequencies is 2:1. Since the frequency
is inversely proportional to the length, the ratio of the length of two segments should be 1:2.
L1  13  65cm  21.7cm
L1  23  65cm  43.3cm
2. What is a linear system?
Solution:
A linear system is one in which doubling the driving force doubles the response. If two driving
forces are applied to a linear system simultaneously, the response will be the sum of the
responses to the driving forces individually. That is, the response of the system to one driving
force is not affected by the presence of the second one.
3. What is the maximum number of beats between two sounds that can be heard?
Solution:
As long as the frequency difference Δf is less than about 10Hz, the beats are easily perceived.
When Δf exceeds 15Hz, the beat sensation disappears. The maximum number of beats between
two sounds that can be heard is about 12-15Hz.
4. What are the two most consonant musical intervals?
Solution:
The two most consonant musical intervals are 2:1 octave (C/C) and 3:2 perfect fifth (G/C).
5. What are second-order beats?
Solution:
Second-order beats are the beats between two tones whose frequencies are nearly but not quite in
a simple ratio. They are also called beats between mistuned consonances.
Physics 198
Homework 15
Due October 22
6. The lowest fifth on a piano is from A0 (27.5 Hz) to E1 (41.2 Hz). By applying the criteria of
roughness, show why is sounds less pleasant than the fifth from, A4 (440 Hz) to E1 (569 Hz),
for example. (See Fig. 5.10)
Solution:
If the frequency difference between two pure tones is greater than the critical band, theysound
constant; if it is less than the critical band they sound dissonant.
From A0 (27.5 Hz) to E1 (41.2 Hz): Δf = 41.2 Hz – 27.5 Hz = 13.7 Hz .
The critical band is about 90Hz (See Fig. 5.10).
Δf =13.7Hz < 90Hz , so it sounds less pleasant.
From A4 (440 Hz) to E1 (569 Hz): Δf = 660 Hz – 440 H z = 220 Hz .
The critical band is about 110 Hz (See Fig. 5.10).
Δf =110Hz < 220Hz , so it sounds more pleasant.
7. Pure tones with frequencies 440 Hz and 448 Hz are sounded together. Describe what is heard
(pitch of the fused tone, frequency of beats). Do the same for tones with frequencies 440 and
432 Hz.
Solution:
Pitch of fused tone is  f1  f 2  2 . Frequency of beats is fbeats  f1  f 2
(1) For two tones of frequencies 440 Hz and 448 Hz:
 f1  f2  2  440Hz  448Hz  2  444Hz
fbeats  f1  f 2  440Hz  448Hz  8Hz
(2) For two tones with frequencies 440 Hz and 432 Hz
 f1  f2  2  440Hz  432Hz  2  436Hz
fbeats  f1  f 2  440Hz  32Hz  8Hz