chapter 13 the loudness of single and combined sounds
TRANSCRIPT
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
Clearly P 1/V V = (¼D2)L
Atmospheric Pressure
D
More than Atm. Pressure
L
Same Experiment with Sound
At the threshold of hearing for 1000 Hz 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 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 = 10 log(I/Io)
For energies = 10 log(E/Eo)
For amplitudes = 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 Io = 10-12 W/m2
Loudness levels always compared to threshold Relative measure
SPL (Sound Pressure Level) 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 awayTrumpet peaks at 5 inches away
140 Rock singer screaming in microphone (lips on mic)
130
Pneumatic (jack) hammer
Cymbal crash
Planes on airport runway 120 Threshold of pain - Piccolo strongly played
Fender guitar amplifier, full volume at 10 inches away
Power tools 110
Subway (not the sandwich shop) 100Flute in players right ear - Violin in players left ear
Common Quieter Sounds
90
Heavy truck traffic
Chamber music 80Typical home stereo listening levelAcoustic guitar, played with finger at 1 foot away
Average factory
70
Busy street
Small orchestra
60 Conversational speech at 1 foot away
Average office noise 50
Quiet conversation 40
Quiet office 30
Quiet living room 20
10 Quiet recording studio
0 Threshold of hearing for healthy youths
Loudness/Amplitude Ratios
Loudness Amplitude(Decibels) Factor
0 1.0001 1.1222 1.2593 1.4134 1.5855 1.7786 1.9957 2.2398 2.5129 2.81810 3.16211 3.54812 3.981
Loudness Amplitude(Decibels) Factor
13 4.46714 5.01215 5.62316 6.31017 7.07918 7.94319 8.91320 10.00040 100.00080 10,000120 1,000,000
Amplitude vs. Loudness
Amplitude vs. Loudness
0
2
4
6
8
10
0 5 10 15 20
Loudness (decibels)
Am
plit
ud
e R
ati
o
Quantifying the Sense of Scale
Sound Level(at 1000 Hz)
Amplitude Ratio
Loudness
Threshold of Hearing
1 0 dB
Tuning Fork 100 40 dB
Mezzo-Forte 10,000 80 dB
Threshold of Pain 1,000,000 120 dB
Loudness Arithmetic
To get the loudness at, say 97 dB 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 p p net
p 2c
2b
2a
Example
Let pa = 5, pb = 2, and pc = 1
48.5301425 125 net
p 222
Only slightly greater than the one source at 5.
Threshold of Hearing
Hearing Response
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 ANumber 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 Add by square root rule Need 10 to double the loudness
One player who can vary loudness is more effective than fixed loudness players
Building a Narrow BandNoise Source
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.
280 284 290 294 300 304 310 314 320
Frequency
Am
plit
ud
e
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…
f = 1 octave L = 24 sones
f = ½ octave L = 20 sones
f = 0 L = 16 sones
Adding Loudness atDifferent Frequency
Lower tone 300 Hz Lower tone 200 Hz Lower tone 100 Hz
Notes
The plateau at small pitch separation is interesting 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 Apparent Loudness
1200 13 sones
1500 4 sones
17 sones
900 13 sones
1200 6 sones
19 sones
600 13 sones
900 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…
)S S 0.2(S 0.3S 0.5S 0.5S 0.75S S S 87654321tnp
Example
For the five harmonically related noise partials – each with loudness 13 sones 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 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 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 Loudness200 8.5 sones400 10 sones630 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 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)
Stnp = 12 + (0.75*13.5) + (0.5*13) = 29 sones
Different formulas are needed for noise and sinusoidal waves
Central Frequency Perceived Loudness
200 12 sones
400 13.5 sones
630 13 sones
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 Sound Q produced with regular mouthpiece Sound R produced with a modified mouthpiece
Different Mouthpieces
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5 6 7 8 9
Harmonic Number
So
un
d P
res
su
re A
mp
litu
de
Tone Q(original)
Tone R(modified)
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 Loudness Q R1 17 122 19 223 9 114 3 65 2 76 2 57 2.0 3.58 0.3 3.09 0.0 2.5
Total 54.3 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 Weights are chosen to model the ear response to an SPL
of 40 dB Type B
Weights are chosen to model the ear response to an SPL of 70 dB
Type C 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