Transcript Chapter 13

Chapter 13
The Loudness of Single and Combined
Sounds
Four Important Musical
Properties
 Pitch (Chapter 5)
 Tone Color (Chapter 7 and others)
 Duration (Chapter 10 and 11)
 Loudness
Piston Experiment
D
Atmospheric
Pressure
L
More than Atm.
Pressure
 Clearly P 1/V

V = (¼pD2)L
Same Experiment with Sound
 At the threshold of hearing for 1000 Hz
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D = 0.006 cm (human hair)
L = 0.01 cm
Change in volume of our “piston” of one part in
3.5 billion.
Developing a Sense of Scale
100X Threshold of
Hearing
Tuning Fork at 9 in
10,000 X Threshold
Messo-Forte
1,000,000X
Threshold
Threshold of Pain
Energy and Intensity
 Energy is the unifying principle

heat, chemical, kinetic, potential, mechanical (muscles),
and acoustical, etc.
 For vibrational processes, energy is proportional to
amplitude squared, or E  A2
 On the receiving end intensity is proportional to
energy, or I  E

I  A2
Loudness
 When the energy (intensity) of the sound increases
by a factor of 10, the loudness increases by 1 bel
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
Named for A. G. Bell
One bel is a large unit and we use 1/10th bel, or decibels
 When the energy (intensity) of the sound increases
by a factor of 10, the loudness increases by 10 dB
Decibel Scale
 For intensities

b = 10 log(I/Io)
 For energies

b = 10 log(E/Eo)
 For amplitudes
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b = 20 log(A/Ao)
Threshold of Hearing
 The Io or Eo or Ao refers to the intensity, energy, or
amplitude of the sound wave for the threshold of
hearing
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Io = 10-12 W/m2
Loudness levels always compared to threshold

Relative measure
 SPL (Sound Pressure Level)
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2.833 10-10 atm. = 0.000283 dynes/cm2
One part in 3.53 billion
Common Loud Sounds
160
Jet engine - close up
150
Snare drums played hard at 6 inches away
Trumpet peaks at 5 inches away
140
Rock singer screaming in microphone (lips on mic)
130
Pneumatic (jack) hammer
Planes on airport runway
Cymbal crash
120
Threshold of pain - Piccolo strongly played
Fender guitar amplifier, full volume at 10 inches away
Power tools
110
Subway (not the sandwich shop)
100
Flute in players right ear - Violin in players left
ear
Common Quieter Sounds
90
Heavy truck traffic
Chamber music
80
Typical home stereo listening level
Acoustic guitar, played with finger at 1 foot away
Average factory
70
Busy street
Small orchestra
60
Average office noise
50
Quiet conversation
40
Quiet office
30
Quiet living room
20
Conversational speech at 1 foot away
10
Quiet recording studio
0
Threshold of hearing for healthy youths
Loudness/Amplitude Ratios
Loudness
(Decibels)
0
1
2
3
4
5
6
7
8
9
10
11
12
Amplitude
Factor
1.000
1.122
1.259
1.413
1.585
1.778
1.995
2.239
2.512
2.818
3.162
3.548
3.981
Loudness
(Decibels)
Amplitude
Factor
13
14
15
16
17
18
19
20
40
80
120
4.467
5.012
5.623
6.310
7.079
7.943
8.913
10.000
100.000
10,000
1,000,000
Amplitude vs. Loudness
Amplitude vs. Loudness
Amplitude Ratio
10
8
6
4
2
0
0
5
10
Loudness (decibels)
15
20
Quantifying the Sense of Scale
Sound Level
(at 1000 Hz)
Amplitude
Ratio
Loudness
Threshold of
Hearing
Tuning Fork
1
0 dB
100
40 dB
Mezzo-Forte
10,000
80 dB
1,000,000
120 dB
Threshold of Pain
Loudness Arithmetic
 To get the loudness at, say 97 dB
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Split into 80 + 17
From table 80 dB is an amplitude ratio of 10,000
17 dB is an amplitude ratio of 7.079
97 dB corresponds to 7.079*10,000 = 70,790
amplitude ratio
Adding loudspeakers
 Doubling the amplitude of a single speaker
gives an increased loudness of 6 dB (table)
 Two speakers of the same loudness give an
increase of 3 dB over a single speaker
 For sources with pressure amplitudes of pa,
pb, pc, etc. the net pressure amplitude is
p
net
 p a2  p 2b  p c2  ...
Example
 Let pa = 5, pb = 2, and pc = 1
p
net
 5 2  2 2  12  25  4  1  30  5.48
Only slightly greater than the one source at 5.
Threshold of Hearing
Hearing Response
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Horizontal axis in octaves
Low frequency response is poor
The range of reasonable sensitivity is 250 - 6000 Hz
Young people tend to have the same shaped curve,
but the overall levels may be raised (less sensitive)
 The high frequency response is worse as we age
 Curve for threshold of pain looks the same, 120 dB
the threshold of hearing
Perceived Loudness
 One sone when a source at 1000 Hz produces
an SPL of 40 dB

Sones are usually additive
Response at constant SPL
Observations
 Broad peak (almost a level plateau) from 250 - 500
Hz
 Dips a bit at 1000 Hz before rising dramatically at
3000 Hz
 Drops quickly at high frequency
 The perceived loudness of a tone at any frequency
about doubles when the SPL is raised 10 dB
Equalizer Settings
Single and Multiple Sources
Relative Amplitude for Curve A
Number of Sources for Curve B
Notes
 Need to almost triple the amplitude of a single
source before the perceived loudness reaches two
sones
 The four-sone level occurs for an amplitude
increase of 10X
 Curve B adds multiple one sone sources
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Add by square root rule
Need 10 to double the loudness
 One player who can vary loudness is more effective
than fixed loudness players
Amplitude
Building a Narrow Band
Noise Source
280
284
290
294
300
304
Frequency
310
314
320
 Make a number a
sinusoidal tones closely
spaced in frequency.
 The loudness is equal to
that of a single
sinusoidal source of the
same SPL at the central
frequency.
Adding Two Narrow Band
Noise Sources
 We have two noise sources – one at 300 Hz the
other at 1200 Hz or more, each at 13 sones

Since the frequencies are far apart, they add to give 26
sones
 As frequencies move closer together…
Df = 1 octave L = 24 sones
Df = ½ octave L = 20 sones
Df = 0
L = 16 sones
Adding Loudness at
Different Frequency
Lower tone 300 Hz
Lower tone 200 Hz
Lower tone 100 Hz
Notes
 The plateau at small pitch separation is interesting
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
We process closely spaced pitches as though they are
indistinguishable in perceived loudness
Called Critical Bandwidth – notice that it grows at low
frequency
Frequency
Critical Bandwidth
> 280 Hz
1/3 octave (major third)
180 - 280
2/3 octave (minor sixth)
< 180Hz
1 octave
Adding a Harmonic Series
 Consider the set of frequencies – each at 13 sones
300
600
Fifth (half octave) - these combine to 19.5 sones
900
Perfect fourth (five semitones) - these combine to 19 sones
1200
Major third (four semitones) - these combine to 17 sones
1500
Upward Masking
 The upper tone's loudness tends to be masked
by the presence of the lower tone.
Examples
Frequency
1200
1500
Apparent Loudness
13 sones
4 sones
17 sones
900
1200
13 sones
6 sones
19 sones
600
900
13 sones
6.5 sones
19.5 sones
Notice that upward
masking is greater at
higher frequencies.
Upward Masking Arithmetic
 Rough formula for calculating the loudness
of up to 8 harmonically related tones
 Let S1, S2, S3, … stand for the loudness of
the individual tones. The loudness of the
total noise partials is…
Stnp  S1  0.75S2  0.5S3  0.5S4  0.3S5  0.2(S6  S7  S8 )
Example
 For the five harmonically related noise partials –
each with loudness 13 sones
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300 Hz (13 sones)
600 Hz (0.75*13 sones = 9.75 sones)
900 Hz (0.5* 13 sones = 6.5 sones)
1200 Hz (0.5* 13 sones = 6.5 sones)
1500 Hz (0.3*13 sones = 3.9 sones)
 Stnp = 13 + 9.75 + 6.5 + 6.5 + 3.9 = 39.65 sones
Closely Spaced Frequencies
Produce Beats
Open two instances of the Tone Generator on the
Study Tools page. Set one at 440 Hz and the other
at 442 Hz and start each.
Notes on Beats
 Beat Frequency = Difference between the
individual frequencies = f2 - f1
 When the two are in phase the amplitude is
momentarily doubled that of either
component
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gives an increase in loudness of 50%
Notice increase in loudness on Fig. 13.6 as pitch
separation becomes small
Beat Loudness
Increase Pitch Separation
 When the frequency difference reached 5 15 Hz, the beat frequency is too great to hear
the individual beats, but we hear a rolling
sound with loudness between 16 and 19.7
sones.
Beats – Two Sources
 One or the other component may dominate in
certain parts of the room
 Beats are more prominent than in the single
earphone experiment
 Some will be able to hear both tones and the
beat frequency in the middle

Only the beat frequency is heard with earphone
experiments
Sinusoidal Addition
 Masking (one tone reducing the amplitude of
another) is greatly reduced in a room
Stsp = S1 + S2 + S3 + ….

Total sinusoidal partials (tsp versus tnp)
Experimental Verification
 Two signals (call them J and K) are adjusted
to equal perceived loudness
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Sound J is composed of three sinusoids at 200,
400, and 630 Hz, each having an SPL of 70 dB
(see Fig 13.4)
Frequency
Perceived Loudness
200
8.5 sones
400
10 sones
630
8.5 sones
Stsp = 8.5 + 10 + 8.5 = 27 sones
Sound K
 Sound K is composed of three equal-strength
noise partials, each having sinusoidal
components spread over 1/3-octave
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Central frequencies of 200, 400, and 630 Hz
Adjust K to be as loud as J
Measured loudness 75 dB
Again using Fig 13.4
Sound K (cont’d)
Central Frequency
200
400
630
Perceived Loudness
12 sones
13.5 sones
13 sones
 Stnp = 12 + (0.75*13.5) + (0.5*13) = 29 sones
 Different formulas are needed for noise and
sinusoidal waves
Notes
 Noise is more effective at upward masking in room
listening conditions
 Upward masking plays little role when sinusoidal
components are played in a room
 The presence of beats adds to the perceived
loudness
 Beats are also possible for components that vary in
frequency by over 100 Hz.
Saxophone Experiment
 Note written G3 has fundamental at 174.6 Hz
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Sound Q produced with regular mouthpiece
Sound R produced with a modified mouthpiece
Different Mouthpieces
Sound Pressure Amplitude
1.2
1.0
0.8
Tone Q
(original)
0.6
Tone R
(modified)
0.4
0.2
0.0
0
1
2
3
4
5
6
7
Harmonic Number
8
9
Results
 Original instrument showed strong
harmonics out to about 4 and then falling
rapidly
 Modified mouthpiece shows a weakened first
harmonic, very strong second, and then
strong harmonics 5, and 6
Perceived Loudness
Harmonic
1
2
3
4
5
6
7
8
9
Total
Loudness
Q
17
19
9
3
2
2
2.0
0.3
0.0
54.3
R
12
22
11
6
7
5
3.5
3.0
2.5
72.0
The new
mouthpiece
makes the sax
1.33 times as
loud (72/54)
Sound Level Meter
Design Specs
 Mimics what our ears receive
Frequency
(Hz)
Reduction
from Original
- Type A
Reduction
from Original
- Type B
Reduction
from Original
- Type C
100
0.1
0.56
1.0
200
0.28
0.28
1.0
500 - 2000
1.0
1.0
1.0
5000
0.7
0.7
0.6
Three Types
 Type A
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Weights are chosen to model the ear response to an SPL
of 40 dB
 Type B
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Weights are chosen to model the ear response to an SPL
of 70 dB
 Type C
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Weights are chosen to model the ear response to an SPL
of 100 dB
Meter Shortcomings
 Cannot account for upward masking
 Cannot account for beats
 It measures dB, not sones (not necessarily
one-to-one)
dB Compared to Sones