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