PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 44
Sound
Sound Intensity
Intensity Levels
Levels
Based
Based in
in part
part on
on Sam
Sam Matteson’s
Matteson’s Unit
Unit 22 Session
Session 13
13
Sam
Trickey
Sam Trickey
Jan.
Jan. 24,
24, 2005
2005
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Challenge to our thinking: How should we
quantify and measure how loud or soft a
sound is?
Possible choice: Amplitude of oscillations. But ,
displacement, or velocity or pressure Amplitude?
Pressure is easiest but there is a problem of smallness.
Human ears can detect ∆p ≈ 2 x 10-10 atm, i.e. a sound
pressure of about 2 x 10-5 Pa. The threshold of pain is
roughly 106 higher, still 10-4 atm.
Huge range of small
small numbers!
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Since we know that sound moves membranes
(drum heads, ear drum, microphone diaphragms, speaker cones, etc.) we might focus on
energy instead of amplitude.
Energy and amplitude aren’t the same.
SHO again: at max extension or compression,
U= Pot Energy = ½KA2 [A = amplitude]
3 Waves: AY=2 AX; UZ=2 UX
then UY=4 UX ≠ UZ
PHYPHY-2464
Pres. 4 Sound Intensity Levels
However, the energies in question involve time.
Example: measure total energy delivered at a point
due to a steady source. The value simply grows with
time. Not very useful.
Power = ∆E/ ∆t
= amount of energy / time interval of delivery
As said on TV –
But wait, there’s more !
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
The “more”: Area!
We really need to know how much energy is delivered
to an area per time interval. This is defined to be
INTENSITY
I = Power/area = (Energy/time)(1/area)
Example: 12 watts delivered to 0.6 m2 is
I = 12/0.6 = 20 W/ m2
Intensity = constant x Amplitude2
So we still have the huge range of values
problem
PHYPHY-2464
Pres. 4 Sound Intensity Levels
The Intensity of a sound wave is the energy radiated (or
delivered) per unit time per unit area. [W/m2]
FYI: Time Averaged Intensity < I >
< I > = 0.0012 pmax2 ≈ pRMS2/400
The numerical values come from
< I >= < p x udisplacement > = ½ pmax2/ ρv
with [ p ] = [Pa] and [ I ] = [W/m2] and evaluating ρv for
air at normal atmospheric pressure and room temperature
density.
By convention pRMS = pmax/√2 is what we mean by “the
pressure”
pressure” of a sine wave.
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Dealing with big ranges of variation –
Usually we don’t care about absolute sound
pressures or intensities, only about their
ratios, p1/p2 or I1/I2
(If we agree on some reference level, then
everything else is ratios.)
Logarithms are the arithmetic of ratios.
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Mathematical Interlude
Logarithms Log(N)
Pick a positive number x. The power of 10 that will produce x
when 10 is raised to that power is called the Logarithm (on the
base 10) of the number x.
Log (x) = L means that 10 L = x.
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
The Function“LOG (x)”
Log(9)
=0.95
Log(8)
=0.90
Log(7) =0.85
Log(6) =0.78
Log(5) =0.70
Log(10) =1
Log(4) =0.60
Log(3) =0.48
Log(x)
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
PHYPHY-2464
Log(2) =0.30
Log(1) =0
0
1
2
3
4
5
x
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Pres. 4 Sound Intensity Levels
Some Useful Facts about Logarithms:
Log( a x b) = Log (a) + Log (b)
Log( a / b) = Log (a) – Log (b)
Log( x p) = p Log ( x )
Thus, in scientific notation:
Log ( z x 10 n) = Log ( z ) + n
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Sound Intensity Level:
LI = 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)
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Definition (“convention”):
I threshold = 10-12 W/m2
LI = SIL = 10 Log ( I / 10-12)
= 10 Log I + 120
Actually measured: Sound PRESSURE Level
Lp = SPL = 20 Log ( p/ pthreshold)
=20 Log (p/2 x 10-5) ≈ 20 Log p + 94
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Just Noticeable Difference (JND) is the limen of
difference that elicits 75% correct answers in a Two
Alternative Forced-Choice test (2AFC test).
“limen” – term used in psychology to denote
threshold
“2AFC test” – “which lens is better, A or B?”
Why 75%?
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Why 75%?
In 2 AFC
• 50% correct means random choice
• 100% means can always tell the difference.
•Thus, 75% is halfway between random and certainty.
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
„Empirically the limen of intensity is a ratio of about
1.26
„This corresponds to a SIL shift of 1 dB.
10 Log( 1.26 ) = 1.0
.
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Musical Dynamics
Pianississimo:
ppp really soft
Pianissimo:
pp very soft
Piano:
p
soft
Mezzopiano:
mp medium soft
Mezzoforte:
mf medium loud
Forte:
f
loud
Fortissimo:
ff
very loud
Fortississimo:
fff
really loud
50 dB
60 dB
65 dB
70 dB
76 dB
80 dB
85 dB
90 dB
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Musical Dynamics:
We have a problem –
the composers mark the range
can the performers deliver it?
Is it possible for an arbitrary, un-amplified
orchestral or band instrument to be played over a
40 dB range?
Does a trained human voice have that dynamical
range?
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Combined Sound Levels:
Two independent sources
Add INTENSITIES (not SIL or SPL)
Example
Iclarinet= 2 x 106 Ithresh
Iflute= 106 Ithresh
Itot= 3 x 106 Ithresh
SILflute = 10 log Iflute / Ithresh = 60 dB
SILclarinet = 10 log Iclarinet / Ithresh = 63 dB
SILtogether = 10 log Itot / Ithresh = 64.8 dB
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Combined Sound Levels:
Two sources at (or almost) same frequency:
Interference must be taken into account
Example : A1= 3A2 → I1= 9I2 → Itot = 10 I2
Independent: SILtot = SIL2 + 10 log 10
= SIL2 + 10 dB
Constructive: AC= 3A2 + A2 = 4A2 → IC= 16 I2
SILC = SIL2 + 10 log 16 = SIL2 + 12 dB
Destructive: AD= 3A2 - A2 = 2A2 → ID= 4 I2
SILD = SIL2 + 10 log 4 = SIL2 + 6 dB
PHYPHY-2464
Pres. 4 Sound Intensity Levels
Summary:
•The objective sound level is measured as SIL or SPL
in dB.
•SIL = 10 Log (I /Ithreshold)
•SPL = 20 Log (p/pthreshold)
•0 dB corresponds to an intensity of 1 pW/m2.
• Add intensities to get combined SIL; beware
interference.
• Remark - Un-amplified orchestral music rarely
exceeds 90 dB; the threshold of pain is 120 dB.
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