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Pitch and Frequency
·
Pitch and Frequency
A sound wave, like any other
wave, is introduced into a medium by a vibrating object. The vibrating object
is the source of the disturbance that moves through the medium. The vibrating
object that creates the disturbance could be the vocal cords of a person,
the vibrating string and sound board of a guitar or violin, the vibrating tines
of a tuning fork, or the vibrating diaphragm of a radio speaker. Regardless of
what vibrating object is creating the sound wave, the particles of the medium
through which the sound moves is vibrating in a back and forth motion at a
given frequency.
The frequency of a wave refers to how often the
particles of the medium vibrate when a wave passes through the medium. The
frequency of a wave is measured as the number of complete back-and-forth
vibrations of a particle of the medium per unit of time. If a particle of air
undergoes 1000 longitudinal
vibrations in
2 seconds, then the frequency of the wave would be 500 vibrations per second. A
commonly used unit for frequency is the Hertz (abbreviated Hz), where
1 Hertz = 1 vibration/second
As a sound wave moves through
a medium, each particle of the medium vibrates at the same frequency. This is
sensible since each particle vibrates due to the motion of its nearest
neighbor. The first particle of the medium begins vibrating, at say 500 Hz, and
begins to set the second particle into vibrational motion at the same frequency
of 500 Hz. The second particle begins vibrating at 500 Hz and thus sets the
third particle of the medium into vibrational motion at 500 Hz. The process
continues throughout the medium; each particle vibrates at the same frequency.
And of course the frequency at which each particle vibrates is the same as the
frequency of the original source of the sound wave. Subsequently, a guitar
string vibrating at 500 Hz will set the air particles in the room vibrating at
the same frequency of 500 Hz, which carries a sound signal to the ear of a listener,
which is detected as a 500 Hz sound wave.
The back-and-forth
vibrational motion of the particles of the medium would not be the only
observable phenomenon occurring at a given frequency. Since a sound wave is
a pressure wave, a detector could be used to
detect oscillations in pressure from a high pressure to a low pressure and back
to a high pressure. As the compressions (high pressure) and rarefactions (low
pressure) move through the medium, they would reach the detector at a given
frequency. For example, a compression would reach the detector 500 times per
second if the frequency of the wave were 500 Hz. Similarly, a rarefaction would
reach the detector 500 times per second if the frequency of the wave were 500
Hz. The frequency of a sound wave not only refers to the number of
back-and-forth vibrations of the particles per unit of time, but also refers to
the number of compressions or rarefactions that pass a given point per unit of
time. A detector could be used to detect the frequency of these pressure
oscillations over a given period of time. The typical output provided by such a
detector is a pressure-time plot as shown below.
Since a pressure-time plot
shows the fluctuations in pressure over time, the period of the sound wave can
be found by measuring the time between successive high pressure points
(corresponding to the compressions) or the time between successive low pressure
points (corresponding to the rarefactions). As discussed in an
earlier unit, the
frequency is simply the reciprocal of the period. For this reason, a sound wave
with a high frequency would correspond to a pressure time plot with a small
period - that is, a plot corresponding to a small amount of time between
successive high pressure points. Conversely, a sound wave with a low frequency
would correspond to a pressure time plot with a large period - that is, a plot
corresponding to a large amount of time between successive high pressure
points. The diagram below shows two pressure-time plots, one corresponding to a
high frequency and the other to a low frequency.
The ears of a human (and
other animals) are sensitive detectors capable of detecting the fluctuations in
air pressure that impinge upon the eardrum. The mechanics of the ear's
detection ability will be discussed later in this lesson. For now, it is sufficient
to say that the human ear is capable of detecting sound waves with a wide range
of frequencies, ranging between approximately 20 Hz to 20 000 Hz. Any sound
with a frequency below the audible range of hearing (i.e., less than 20 Hz) is
known as an infrasound and
any sound with a frequency above the audible range of hearing (i.e., more than
20 000 Hz) is known as an ultrasound. Humans are not alone in their ability to
detect a wide range of frequencies. Dogs can detect frequencies as low as
approximately 50 Hz and as high as 45 000 Hz. Cats can detect frequencies as
low as approximately 45 Hz and as high as 85 000 Hz. Bats, being nocturnal
creature, must rely on sound echolocation for navigation and hunting. Bats can
detect frequencies as high as 120 000 Hz. Dolphins can detect frequencies as
high as 200 000 Hz. While dogs, cats, bats, and dolphins have an unusual
ability to detect ultrasound, an elephant possesses the unusual ability to
detect infrasound, having an audible range from approximately 5 Hz to
approximately 10 000 Hz.
The
sensation of a frequency is commonly referred to as the pitch of a sound. A high
pitch sound corresponds to a high frequency sound wave and a low pitch sound
corresponds to a low frequency sound wave. Amazingly, many people, especially
those who have been musically trained, are capable of detecting a difference in
frequency between two separate sounds that is as little as 2 Hz. When two
sounds with a frequency difference of greater than 7 Hz are played
simultaneously, most people are capable of detecting the presence of a complex
wave pattern resulting from the interference and superposition of the two sound waves.
Certain sound waves when played (and heard) simultaneously will produce a
particularly pleasant sensation when heard, are said to be consonant. Such sound waves form the
basis of intervals in
music. For example, any two sounds whose frequencies make a 2:1 ratio are said
to be separated by an octave and
result in a particularly pleasing sensation when heard. That is, two sound
waves sound good when played together if one sound has twice the frequency of
the other. Similarly two sounds with a frequency ratio of 5:4 are said to be
separated by an interval of a third; such sound waves also sound good when played
together. Examples of other sound wave intervals and
their respective frequency ratios are listed in the table below.
|
Interval
|
Frequency Ratio
|
Examples
|
|
Octave
|
2:1
|
512
Hz and 256 Hz
|
|
Third
|
5:4
|
320
Hz and 256 Hz
|
|
Fourth
|
4:3
|
342
Hz and 256 Hz
|
|
Fifth
|
3:2
|
384
Hz and 256 Hz
|
The ability of humans to
perceive pitch is associated with the frequency of the sound wave that impinges
upon the ear. Because sound waves traveling through air are longitudinal waves
that produce high- and low-pressure disturbances of the particles of the air at
a given frequency, the ear has an ability to detect such frequencies and
associate them with the pitch of the sound. But pitch is not the only property
of a sound wave detectable by the human ear. In the next part of Lesson 2, we will investigate the
ability of the ear to perceive the intensity of a sound wave.
Every musical note is associated with a unique frequency. The
two widgets below allow you to investigate the relationship between a musical
note and the associated frequency.
Check Your Understanding
1. Two musical notes that
have a frequency ratio of 2:1 are said to be separated by an octave. A musical
note that is separated by an octave from middle C (256 Hz) has a frequency of
_____.
|
a. 128 Hz
|
b. 254 Hz
|
c. 258 Hz
|
|
d. 345 Hz
|
e. none of these
|
|
Sound Waves and Music - Lesson 2 - Sound Properties and Their
Perception
Intensity and the Decibel
Scale
·
Intensity and the Decibel Scale
Sound waves are introduced
into a medium by the vibration of an object. For example, a vibrating guitar
string forces surrounding air molecules to be compressed and expanded, creating
a pressure disturbance consisting of an alternating pattern of compressions and
rarefactions. The
disturbance then travels from particle to particle through the medium,
transporting energy as it moves. The energy that is carried by the disturbance
was originally imparted to the medium by the vibrating string. The amount of
energy that is transferred to the medium is dependent upon the amplitude of
vibrations of the guitar string. If more energy is put into the plucking of the
string (that is, more work is done to displace the
string a greater amount from its rest position), then the string vibrates with
a greater amplitude. The greater amplitude of vibration of the guitar string
thus imparts more energy to the medium, causing air particles to be displaced a
greater distance from their rest position. Subsequently, the amplitude of
vibration of the particles of the medium is increased, corresponding to an
increased amount of energy being carried by the particles. This relationship between
energy and amplitudewas
discussed in more detail in a previous unit.
Sound Intensity and
Distance
The amount of energy that is
transported past a given area of the medium per unit of time is known as
the intensity of
the sound wave. The greater the amplitude of vibrations of the particles of the
medium, the greater the rate at which energy is transported through it, and the
more intense that the sound wave is. Intensity is the energy/time/area; and
since the energy/time ratio is equivalent to the quantity power, intensity is simply the
power/area.
Typical units for expressing
the intensity of a sound wave are Watts/meter2.
As a sound wave carries its
energy through a two-dimensional or three-dimensional medium, the intensity of
the sound wave decreases with increasing distance from the source. The decrease in intensity with increasing distance is
explained by the fact that the wave is spreading out over a circular (2
dimensions) or spherical (3 dimensions) surface and thus the energy of the
sound wave is being distributed over a greater surface area. The diagram at the
right shows that the sound wave in a 2-dimensional medium is spreading out in
space over a circular pattern. Since energy is conserved and the area through which
this energy is transported is increasing, the intensity (being a quantity that
is measured on a per areabasis) must decrease.
The mathematical relationship between intensity and distance is sometimes referred to as an inverse square relationship. The intensity varies inversely with the square of the distance from the source. So if the distance from the source is doubled (increased by a factor of 2), then the intensity is quartered (decreased by a factor of 4). Similarly, if the distance from the source is quadrupled, then the intensity is decreased by a factor of 16. Applied to the diagram at the right, the intensity at point B is one-fourth the intensity as point A and the intensity at point C is one-sixteenth the intensity at point A. Since the intensity-distance relationship is an inverse relationship, an increase in one quantity corresponds to a decrease in the other quantity. And since the intensity-distance relationship is an inverse square relationship, whatever factor by which the distance is increased, the intensity is decreased by a factor equal to the square of the distance change factor. The sample data in the table below illustrate the inverse square relationship between power and distance.
The mathematical relationship between intensity and distance is sometimes referred to as an inverse square relationship. The intensity varies inversely with the square of the distance from the source. So if the distance from the source is doubled (increased by a factor of 2), then the intensity is quartered (decreased by a factor of 4). Similarly, if the distance from the source is quadrupled, then the intensity is decreased by a factor of 16. Applied to the diagram at the right, the intensity at point B is one-fourth the intensity as point A and the intensity at point C is one-sixteenth the intensity at point A. Since the intensity-distance relationship is an inverse relationship, an increase in one quantity corresponds to a decrease in the other quantity. And since the intensity-distance relationship is an inverse square relationship, whatever factor by which the distance is increased, the intensity is decreased by a factor equal to the square of the distance change factor. The sample data in the table below illustrate the inverse square relationship between power and distance.
|
Distance
|
Intensity
|
|
1
m
|
160
units
|
|
2
m
|
40
units
|
|
3
m
|
17.8
units
|
|
4
m
|
10
units
|
The Threshold of Hearing
and the Decibel Scale
Humans are equipped with very
sensitive ears capable of detecting sound waves of extremely low intensity. The
faintest sound that the typical human ear can detect has an intensity of 1*10-12 W/m2. This intensity corresponds
to a pressure wave in which a compression of the particles of the medium
increases the air pressure in that compressional region by a mere 0.3 billionth
of an atmosphere. A sound with an intensity of 1*10-12 W/m2 corresponds to a sound
that will displace particles of air by a mere one-billionth of a centimeter.
The human ear can detect such a sound. WOW! This faintest sound that a human
ear can detect is known as the threshold of hearing (TOH). The most intense
sound that the ear can safely detect without suffering any physical damage is
more than one billion times more intense than the threshold of hearing.
Since the range of
intensities that the human ear can detect is so large, the scale that is frequently
used by physicists to measure intensity is a scale based on powers of 10. This
type of scale is sometimes referred to as a logarithmic scale. The scale for
measuring intensity is the decibel scale. The threshold of hearing is assigned a sound
level of 0 decibels (abbreviated 0 dB); this sound corresponds to an intensity
of 1*10-12 W/m2. A sound that is 10 times
more intense ( 1*10-11 W/m2) is assigned a sound level
of 10 dB. A sound that is 10*10 or 100 times more intense (1*10-10 W/m2) is assigned a sound level
of 20 db. A sound that is 10*10*10 or 1000 times more intense (1*10-9W/m2) is assigned a sound level
of 30 db. A sound that is 10*10*10*10 or 10000 times more intense (1*10-8 W/m2) is assigned a sound level
of 40 db. Observe that this scale is based on powers of 10. If one sound is 10x times more intense than
another sound, then it has a sound level that is 10*x more decibels than the
less intense sound. The table below lists some common sounds with an estimate
of their intensity and decibel level.
|
Source
|
Intensity
|
Intensity Level
|
# of Times
Greater Than TOH |
|
Threshold
of Hearing (TOH)
|
1*10-12 W/m2
|
0
dB
|
100
|
|
Rustling
Leaves
|
1*10-11 W/m2
|
10
dB
|
101
|
|
Whisper
|
1*10-10 W/m2
|
20
dB
|
102
|
|
Normal
Conversation
|
1*10-6 W/m2
|
60
dB
|
106
|
|
Busy
Street Traffic
|
1*10-5 W/m2
|
70
dB
|
107
|
|
Vacuum
Cleaner
|
1*10-4 W/m2
|
80
dB
|
108
|
|
Large
Orchestra
|
6.3*10-3 W/m2
|
98
dB
|
109.8
|
|
Walkman
at Maximum Level
|
1*10-2 W/m2
|
100
dB
|
1010
|
|
Front
Rows of Rock Concert
|
1*10-1 W/m2
|
110
dB
|
1011
|
|
Threshold
of Pain
|
1*101 W/m2
|
130
dB
|
1013
|
|
Military
Jet Takeoff
|
1*102 W/m2
|
140
dB
|
1014
|
|
Instant
Perforation of Eardrum
|
1*104 W/m2
|
160
dB
|
1016
|
Investigate!
Knowing
the intensity of a sound wave allows one to calculate the deciBel (dB) level of
that sound. Use the DeciBel Calculator widget to
determine the deciBel rating from any intensity in Watt/meter2. Enter intensities
using scientific notation - for example, enter 5e-5 for 5.0x10-5.
While the intensity of a
sound is a very objective quantity that can be measured with sensitive
instrumentation, the loudness of
a sound is more of a subjective response that will vary with a number of
factors. The same sound will not be perceived to have the same loudness to all
individuals. Age is one factor that affects the human ear's response to a
sound. Quite obviously, your grandparents do not hear like they used to. The
same intensity sound would not be perceived to have the same loudness to them
as it would to you. Furthermore, two sounds with the same intensity but
different frequencies will not be perceived to have the same loudness. Because
of the human ear's tendency to amplify sounds having frequencies in the range
from 1000 Hz to 5000 Hz, sounds with these intensities seem louder to the human
ear. Despite the distinction between intensity and loudness, it is safe to
state that the more intense sounds will be perceived to be the loudest sounds.
Investigate!
As
mentioned in the previous paragraph, even the frequency will affect our
perception of the loudness of a sound. For instance, a 100 Hz sound at 60 dB will
not sound as loud as a 1000 Hz sound at 60 deciBel. Fletcher–Munson curves or equal loudness curves are often used
to demonstrate the perceived loudness of a sound for a given frequency. Use
the widget to investigate the effect of the frequency upon the
perceived loudness and to view the equal loudness curves.
Check Your Understanding
1. A mosquito's buzz is often rated with a
decibel rating of 40 dB. Normal conversation is often rated at 60 dB. How many
times more intense is normal conversation compared to a mosquito's buzz?
|
a. 2
|
b. 20
|
c. 100
|
d. 200
|
e. 400
|
See Answer
2. The table at the right represents the decibel level for
several sound sources. Use the table to make comparisons of the intensities of
the following sounds.
How many times more intense
is the front row of a Smashin' Pumpkins concert than ...
a. ... the 15th row of the
same concert?
b. ... the average factory?
c. ... normal speech?
d. ... the library after
school?
e. ... the sound that most humans can just
barely hear?
See Answer
See Answer
Frequencies for equal-tempered scale, A4 =
440 Hz
Other
tuning choices, A4 =
|
440
|
|
Note
|
Frequency
(Hz)
|
Wavelength
(cm)
|
|
C0
|
16.35
|
2109.89
|
|
C#0/Db0
|
17.32
|
1991.47
|
|
D0
|
18.35
|
1879.69
|
|
D#0/Eb0
|
19.45
|
1774.20
|
|
E0
|
20.60
|
1674.62
|
|
F0
|
21.83
|
1580.63
|
|
F#0/Gb0
|
23.12
|
1491.91
|
|
G0
|
24.50
|
1408.18
|
|
G#0/Ab0
|
25.96
|
1329.14
|
|
A0
|
27.50
|
1254.55
|
|
A#0/Bb0
|
29.14
|
1184.13
|
|
B0
|
30.87
|
1117.67
|
|
C1
|
32.70
|
1054.94
|
|
C#1/Db1
|
34.65
|
995.73
|
|
D1
|
36.71
|
939.85
|
|
D#1/Eb1
|
38.89
|
887.10
|
|
E1
|
41.20
|
837.31
|
|
F1
|
43.65
|
790.31
|
|
F#1/Gb1
|
46.25
|
745.96
|
|
G1
|
49.00
|
704.09
|
|
G#1/Ab1
|
51.91
|
664.57
|
|
A1
|
55.00
|
627.27
|
|
A#1/Bb1
|
58.27
|
592.07
|
|
B1
|
61.74
|
558.84
|
|
C2
|
65.41
|
527.47
|
|
C#2/Db2
|
69.30
|
497.87
|
|
D2
|
73.42
|
469.92
|
|
D#2/Eb2
|
77.78
|
443.55
|
|
E2
|
82.41
|
418.65
|
|
F2
|
87.31
|
395.16
|
|
F#2/Gb2
|
92.50
|
372.98
|
|
G2
|
98.00
|
352.04
|
|
G#2/Ab2
|
103.83
|
332.29
|
|
A2
|
110.00
|
313.64
|
|
A#2/Bb2
|
116.54
|
296.03
|
|
B2
|
123.47
|
279.42
|
|
C3
|
130.81
|
263.74
|
|
C#3/Db3
|
138.59
|
248.93
|
|
D3
|
146.83
|
234.96
|
|
D#3/Eb3
|
155.56
|
221.77
|
|
E3
|
164.81
|
209.33
|
|
F3
|
174.61
|
197.58
|
|
F#3/Gb3
|
185.00
|
186.49
|
|
G3
|
196.00
|
176.02
|
|
G#3/Ab3
|
207.65
|
166.14
|
|
A3
|
220.00
|
156.82
|
|
A#3/Bb3
|
233.08
|
148.02
|
|
B3
|
246.94
|
139.71
|
|
C4
|
261.63
|
131.87
|
|
C#4/Db4
|
277.18
|
124.47
|
|
D4
|
293.66
|
117.48
|
|
D#4/Eb4
|
311.13
|
110.89
|
|
E4
|
329.63
|
104.66
|
|
F4
|
349.23
|
98.79
|
|
F#4/Gb4
|
369.99
|
93.24
|
|
G4
|
392.00
|
88.01
|
|
G#4/Ab4
|
415.30
|
83.07
|
|
A4
|
440.00
|
78.41
|
|
A#4/Bb4
|
466.16
|
74.01
|
|
B4
|
493.88
|
69.85
|
|
C5
|
523.25
|
65.93
|
|
C#5/Db5
|
554.37
|
62.23
|
|
D5
|
587.33
|
58.74
|
|
D#5/Eb5
|
622.25
|
55.44
|
|
E5
|
659.25
|
52.33
|
|
F5
|
698.46
|
49.39
|
|
F#5/Gb5
|
739.99
|
46.62
|
|
G5
|
783.99
|
44.01
|
|
G#5/Ab5
|
830.61
|
41.54
|
|
A5
|
880.00
|
39.20
|
|
A#5/Bb5
|
932.33
|
37.00
|
|
B5
|
987.77
|
34.93
|
|
C6
|
1046.50
|
32.97
|
|
C#6/Db6
|
1108.73
|
31.12
|
|
D6
|
1174.66
|
29.37
|
|
D#6/Eb6
|
1244.51
|
27.72
|
|
E6
|
1318.51
|
26.17
|
|
F6
|
1396.91
|
24.70
|
|
F#6/Gb6
|
1479.98
|
23.31
|
|
G6
|
1567.98
|
22.00
|
|
G#6/Ab6
|
1661.22
|
20.77
|
|
A6
|
1760.00
|
19.60
|
|
A#6/Bb6
|
1864.66
|
18.50
|
|
B6
|
1975.53
|
17.46
|
|
C7
|
2093.00
|
16.48
|
|
C#7/Db7
|
2217.46
|
15.56
|
|
D7
|
2349.32
|
14.69
|
|
D#7/Eb7
|
2489.02
|
13.86
|
|
E7
|
2637.02
|
13.08
|
|
F7
|
2793.83
|
12.35
|
|
F#7/Gb7
|
2959.96
|
11.66
|
|
G7
|
3135.96
|
11.00
|
|
G#7/Ab7
|
3322.44
|
10.38
|
|
A7
|
3520.00
|
9.80
|
|
A#7/Bb7
|
3729.31
|
9.25
|
|
B7
|
3951.07
|
8.73
|
|
C8
|
4186.01
|
8.24
|
|
C#8/Db8
|
4434.92
|
7.78
|
|
D8
|
4698.63
|
7.34
|
|
D#8/Eb8
|
4978.03
|
6.93
|
|
E8
|
5274.04
|
6.54
|
|
F8
|
5587.65
|
6.17
|
|
F#8/Gb8
|
5919.91
|
5.83
|
|
G8
|
6271.93
|
5.50
|
|
G#8/Ab8
|
6644.88
|
5.19
|
|
A8
|
7040.00
|
4.90
|
|
A#8/Bb8
|
7458.62
|
4.63
|
|
B8
|
7902.13
|
4.37
|
(To convert lengths in cm to inches, divide by 2.54)
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