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Physics 1251 The Science and Technology of Musical Sound

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Physics 1251The Science and Technology of Musical Sound

Unit 2

Session 13 MWF

Sound Intensity Level and dB

Foolscap Quiz:

What organ, located in the inner ear, is responsible for transforming mechanical vibrations into neural impulses? How does it discriminate different pitches?

The Cochlea. Different places on the Basilar Membrane are excited by different frequencies as revealed by the Nobel-prize-winning work of Bekesy.

Anatomy of Ear

Middle Ear

Outer Ear

Inner Ear

Cochlea

(micrograph)

“The Snail”

- o~ oval window
- r~ round window

2 mm

Structure of Cochlea

1. Spiral cone

2. Divided by

Basilar Membrane

3. In on top half

4. Out on bottom

5. “Sloshing “

Microstructure

of Cochlea

Basilar ——————→

Membrane

↑

Organ of Corti

↑

Auditory Nerve → →

Detail of Hair Cell

N.B. (Nota Bene):

Hair cells have nothingto do with hair that grows in the skin.

HC s are vibration senseorgans.

Outer Hair Cell in Cross Section

Cilia Displacement vs

Location on Basilar Membrane

2800 Hz

Relative Response

Position

Cilia Displacement vs

Location on Basilar Membrane

400 Hz

Relative Response

Position

Peak Frequency of Response vs

Location on Basilar Membrane

f [Hz]

Distance from Stapes (cm)

Georg von Békésy

(1899-1972)

1961 Nobel Prize in

Medicine

Demonstrated resonance

of Basilar Membrane

1′ Lecture:

The objective relative intensity level of sound is quantified as the Sound Intensity Level (SIL) and is measured in deciBel (dB), where

SIL = 10 Log( I / Ithreshold ) .

80/20The Intensity of a sound wave is the energy radiated per unit time per unit area. [W/m2]

FYI: Time Averaged Intensity < I >

< I > = 0.0012 pmax2

For air, with [ p ] = [Pa] and [ I ] = [W/m2]

< I >= < p‧ udisplacement >

= ½ pmax2/ ρv

“Resonant” Frequency vs

Location on Basilar Membrane

20 kHz

Range of Maximum Sensitivity

20 Hz

80/20The frequency range of detectability for humans is approximately 20 Hz to

20 kHz.

80/20Humans are most sensitive in the frequency range 2 kHz to 5 kHz.

80/20The lowest detectable intensity (the threshold of hearing Ithreshold ) is about 1 pW/m 2 or 1x10 -12 W/m 2 .

80/20The intensity at which one experiences pain (the threshold of pain) is about 1 W/m 2 .

Hair Cell (Stereocilium) is a Neuron

Hair Cell (Stereocilium) is a Neuron

- Neuron has threshold for simulation
- Neuron “fires” (is either on or off)
- Firing Neurons inhibit neighbors

Response (sensation)

Threshold of Hearing 10 -12 Watt/m2

Desensitization with greater stimulus

Stimulus (Intensity)

Consequence: Threshold

and Non-linear Response

Hair Cells in the Basilar Membrane

Neuronal Response of Hair Cells

Neuron fires and neighbors are inhibited

Larger

No increased response

Stimulus

No response

No larger responsebecause of inhibition

Attenuated response to much larger stimulus

Neuronal Response of Hair Cells

Response (sensation)

Stimulus (Intensity)

Consequence: Threshold

and Non-linear Response

Threshold of Pain1 Watt/m2

130

120

110

100

Ithreshold = 10 -12 W/m2

90

70

60

50

40

30

20

10

0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

Logarithm of Stimulus vs Response

SIL = 10 Log( I/Ithreshold)

Response

Log (Stimulus/ Ithreshold )

Mathematical Digression ☠ Peligro ☠

☠ Danger ☠ Logarithms Log(N)

☠ Lebensgefahr ☠ ☠ Attention

Logarithm = “ratio arithmetic”

Log (x) = L,such that 10 L = x.

The power of 10 that will produce a number x when 10 is raised to that power is called the Logarithm (on the base 10) of the number.

1.0

.9

.8

Log(x)

.7

.6

.5

.4

.3

.2

.1

0

0

1

2

3

4

5

6

7

8

9

10

x

The “LOG(x)” Function

Log(10) =1

Log(9) =0.95

Log(8) =0.90

Log(7) =0.85

Log(6) =0.78

Log(5) =0.70

Log(4) =0.60

Log(3) =0.48

Log(2) =0.30

Log(1) =0

Some Useful Facts about Logarithms:

Log( x ‧ y) = Log (x) + Log (y)

Log( x / y) = Log (x) – Log (y)

Log( x p) = p Log ( x )

Thus, in scientific notation:

Log ( z ‧ 10 n) = Log ( z ) + n

80/20Sound Intensity Level:

SIL = 10 Log ( I / I threshold ).

The Sound Intensity Level is 10 times the logarithm of the ratio of the intensity of a sound and the threshold of hearing.

The units of SIL are deciBel or dB.

I = I threshold 10 SIL/10

80/20Just Noticeable Difference (JND) is the limen of difference that elicits 75% correct answers in a Two Alternative Forced-Choice test (2AFC test).

Why 75%?

In 2 Alternative Forced Choice:

- 50% correct means random choice
- 100% means can always tell the difference.

80/20Just Noticeable Difference (JND) is the limen of difference that elicits 75% in a Two Alternative Forced-Choice test (2AFC test).

The limen of intensity is a ratio of about 1.26 which corresponds to a SIL difference of 1 dB.

10 Log( 1.26 ) = 1.0

Calculator Practice:

Log( 1) =

Log( 10 ) =

Log( 100 ) =

Log( 2 ) =

Log( 20 ) =

Log( 200 ) =

0

1

2

0.301

1.301

2.301

Calculator Practice:

SIL:

10 Log( 1) =

10 Log( 10 ) =

10 Log( 100 ) =

10 Log( 2 ) =

10 Log( 20 ) =

10 Log( 200 ) =

0 dB

10 dB

20 dB

3.0 dB

13.0 dB

23.0 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

The Mob Chorus:

- “Rhubard!”
- “Donnybrook!”
- “Dust up!”

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery, very loud:100 dB

Musical Dynamics

Pianissimo:ppvery soft:50 dB

Piano:psoft:60 dB

Mezzopiano:mpmedium soft: 66 dB

Mezzoforte:mfmedium loud:76 dB

Forte:floud:80 dB

Fortissimo:ffvery loud:90 dB

Fortississimo:fffVery,very loud:100 dB

Summary:

- The objective sound level is measuredby SIL in dB.
- SIL = 10 Log (I /Ithreshold) .
- 0 dB corresponds to an intensity of 1 pW/m2.
- The threshold of pain is 120 dB.