Transcript Review - Stephen F. Austin State University

Review
Section IV
Chapter 13
The Loudness of Single and
Combined Sounds
Useful Relationships
 Energy and Amplitude, E  A2
 Intensity and Energy, I  E
 Energy and Amplitude, E  A2
Decibels Defined
 When the energy (intensity) of the
sound increases by a factor of 10, the
loudness increases by 10 dB
 b = 10 log(I/Io)
 b = 10 log(E/Eo)
 b = 20 log(A/Ao)
 Loudness always compared to the
threshold of hearing
Decibels and Amplitude
Amplitude vs. Loudness
Amplitude Ratio
10
8
6
4
2
0
0
5
10
Loudness (decibels)
15
20
Single and Multiple Sources
 Doubling the amplitude of a single speaker
gives an increased loudness of 6 dB (see
arrows on last graph)
 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  ...
Threshold of Hearing
 Depends on frequency
 Require louder source at low and high
frequencies
Perceived Loudness
 One sone when a source at 1000 Hz
produces an SPL of 40 dB
 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
Adding Loudness at
Different Frequency
As the pitch separation
grows less, the
combined loudness
grows less.
Note critical
bandwidth plateau for
small pitch
separation, growing
for lower frequencies.
Critcal Bandwidth
The sudden upswing
in loudness at very
small pitch
separation caused by
beats.
Upward Masking

Tendency for the loudness of the upper tone to be
decreased when played with a lower tone.
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
Multiple 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 )
Closely Spaced Frequencies
Produce Beats
Audible Beats
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
Adding Sinusoids
 Masking (one tone reducing the
amplitude of another) is greatly
reduced in a room
Stsp = S1 + S2 + S3 + ….
 Total sinusoidal partials (tsp versus tnp)
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.
Chapter 14
The Acoustical Phenomena Governing
the Musical Relationships of Pitch
Other Ways Of Producing And Using
Beats
 Introduce a strong, single frequency (say,
400 Hz) source and a much weaker,
adjustable frequency sound (the search
tone) into a single ear.
 Vary the search tone from 400 Hz up.
 We hear beats at multiples of 400 Hz.
A Variation in the Experiment
 Produce search tones of equal
amplitude but 180° out of phase.
 Search tone now completely cancels
single tone.
 Result is silence at that harmonic
 Each harmonic is silenced in the same
way.
 How loud does each harmonic need to be
to get silence of all harmonics?
Waves Out of Phase
Waves Out of Phase
1
0.8
Displacement
0.6
Superposition
of these waves
produces zero.
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Time
Loudness Required for Complete
Cancellation




400 Hz
800 Hz
1200 Hz
1600 Hz
95 SPL Source Frequency
75 SPL
75 SPL
75 SPL
 Harmonics are 20 dB or 100 times fainter
than source (10% as loud)
Start with a Fainter Source
 400 Hz
 800 Hz
 1200 Hz
 1600 Hz
89
63
57
51
SPL
SPL
SPL
SPL
Source – ½ loudness
¼ as loud as above
1/8 as loud as above
1/16 as loud as above
…And Still Fainter Source




400 Hz
800 Hz
1200 Hz
1600 Hz
75
55
35
15
SPL
SPL
SPL
SPL
Source
Too faint
Too faint
 This example is appropriate to music.
 Where do the extra tones come from?
 They are not real but are produced in the
ear/brain
Heterodyne Components
 Consider two tones (call them P and Q)


From above we see that the ear/brain will produce
harmonics at (2P), (3P), (4P), etc.
Other components will also appears as combinations of P
and Q
Original
Components
Simplest
Heterodyne
Components
Next-Appearing
Heterodyne
Components
P
(2P)
(3P)
(P + Q), (P – Q)
(2P + Q), (2P – Q)
(2Q + P), (2Q – P)
(2Q)
(3Q)
Q
Heterodyne Beats
 Beats can occur between closely
space heterodyne components, or
between a main frequency and a
heterodyne component.
 See the vibrating clamped bar
example in text.
Driven System Response
Natural
Frequency, fo
3rd Harmonic
is fo
2nd Harmonic
is fo
Other Systems
 More than one driving source
 We get higher amplitudes anytime
heterodyne components approach the
natural frequency.
 Non-linear systems
 Load vs. Deflection curve is curved
 Heterodyne components always exist
Harmonic and Almost Harmonic
Series
 Harmonic Series composed of integer
multiples of the fundamental
 Partial frequencies are close to being
integer multiples of the fundamental
 Always produce heterodyne components
 The components tend to clump around
the harmonic partials.
 May sound like an harmonic series but
“unclear”
Frequency - Pitch
 Frequency is a physical quantity
 Pitch is a perceived quantity
 Pitch may be affected by whether…
 the tone is a single sinusoid or a group
of partials
 heterodyne components are present, or
 noise is a contributor
The Equal-Tempered Scale
 Each octave is divided into 12 equal parts
(semitones)
 Since each octave is a doubling of the
frequency, each semitone increases
frequency by 12 2
 Each semitone is further divided into 100
equal parts called Cents
 The cent size varies across the
keyboard (1200 cents/octave)
Calculating Cents
 The fact that one octave is equal to 1200
cents leads one to the power of 2
relationship:
 Or,
 f2 
ln  
f1 

cents  1200
ln(2)
Frequency Value of Cent
Through the Keyboard
3.0
2.5
2.0
Hz/cent
1.5
1.0
0.5
0.0
0
1000
2000
3000
Frequency
4000
5000
The Unison and Pitch Matching
 Consider two tones made up of the
following partials
Harmonic
1
2
3
4
Tone J
250
500
750
1000
Tone K
252
504
756
1008
2
4
6
8
Beat Frequency
 Adjust the tone K until we are close
to a match
Notes on Pitch Matching
 As tone K is adjusted to tone J, the
beat frequency between the
fundamentals becomes so slow that it
can not easily be heard.
 We now pay attention to the beats of
the higher harmonics.
 Notice that a beat frequency of ¼ Hz in
the fundamental is a beat frequency of 1
Hz in the fourth harmonic.
Add the Heterodyne Components
 In the vicinity of the original partials,
clumps of beats are heard, which
tends to muddy the sound.




Eight frequencies near 250 Hz
Seven near 500 Hz
Six near 750 Hz
Five near 1000 Hz.
Results
 A collection of beats may be heard.
 Here are the eight components near 250
Hz sounded together.
The Octave Relationship
Tone P
Tone Q
200
400
401
600
800
802
1203
 As the second tone is tuned to
match the first, we get harmonics
of tone P, separated by 200 Hz.
 Only tone P is heard
1604
The Musical Fifth
 A musical fifth has two tones whose
fundamentals have the ratio 3:2.
Tone M
200
400
600
800
Tone N
301
602
903
1204
 Now every third harmonic of M is
close to a harmonic of N
Results
 We get clusters of frequencies separated
by 100 Hz.
 When the two are in tune, we will have the
partials…
200 300 400



600
800 900
1200
This is very close to a harmonic series of 100 Hz
The heterodyne components will fill in the
missing frequencies.
The ear will invariably hear a single 100 Hz tone
(called the implied tone).
Chapter 15
Successive Tones: Reverberations,
Melodic Relationships,
and Musical Scales
Audibility Time
 Use a stopwatch to measure how long
the sound is audible after the source
is cut off
 Agrees well with reverberation time
 Time for a sound to decay to 1/1000th
original level or 60 dB
 It is constant, independent of
frequency, and unaffected by
background noise
Advantages of Audibility Time
 Only simple equipment required
 Many sound level meters can only
measure a decay of 40-50 dB, not the
60 dB required by the definition
 Sound level meters assume uniform
decay of the sound, which may not be
the case
Successive Tones
 We can set intervals easily for successive
tones (even in dead rooms) so long as the
tones are sounded close in time.
 Setting intervals for pure sinusoids (no
partials) is difficult if the loudness is small
enough to avoid exciting room modes.
 At high loudness levels there are enough
harmonics generated in the room and ear
to permit good interval setting.
 Intervals set at low loudness with large
gaps between the tones tend to be too wide
in frequency.
The Beat-Free Chromatic
(or Just) Scale
Chromatic Scales
Interval Name
Interval Ratio
C
Frequency (beat-free)
261
E
3rd
5/4
327
F
4th
4/3
349
G
5th
3/2
392
A
Major 6th
5/3
436
C
octave
2/1
523
Harmonically Related Steps
C
D
E
F
G
A
B
Notice the B and D are not
harmonically related to C
C
Intervals with B and D
5th
3rd
C
D
E
F
G
4th
5th
A
B
C
Filling in the Scale
3rd
C
E
D
F
3rd
G
A
B
C
4th
3rd
3rd
Minor 6
Notice that C#, Eb, and Bb come into the scheme, but
Ab/G# is another problem.
Finding F#
C
D
E
F
3rd
3rd
G
min3
A
B
C
Equal Temperament
 An octave represents a doubling of the
frequency and we recognize 12 intervals in the
octave. The octave is the only harmonic
interval.
 Make the interval
12
2  1.059463
 Using equal intervals makes the
cents division more meaningful
 The following table uses
Complete Scale Comparison
Interval
Ratio to Tonic
Just Scale
Ratio to Tonic
Equal Temperament
Unison
1.0000
1.0000
Minor Second
25/24 = 1.0417
1.05946
Major Second
9/8 = 1.1250
1.12246
Minor Third
6/5 = 1.2000
1.18921
Major Third
5/4 = 1.2500
1.25992
Fourth
4/3 = 1.3333
1.33483
Diminished Fifth
45/32 = 1.4063
1.41421
Fifth
3/2 = 1.5000
1.49831
Minor Sixth
8/5 = 1.6000
1.58740
Major Sixth
5/3 = 1.6667
1.68179
Minor Seventh
9/5 = 1.8000
1.78180
Major Seventh
15/8 = 1.8750
1.88775
Octave
2.0000
2.0000
Chapter 16
Keyboard Temperaments and
Tuning: Organ, Harpsichord,
Piano
Notes on the Just Scale
Major Scale
The D corresponds to the upper D in the pair found in
Chapter 15. Also, the tones here (except D and B) were the
same found in the beat-free Chromatic scale in Chapter 15.
Minor Scale
Here we use the lower D from chapter 15 and the upper Ab.
Notes on the Equal-Tempered Scale
 The fifth interval is close to the just fifth

 2
12
7
= 1.49831 whereas the just fifth is 1.5
 Only fifths and octaves are used for
tuning
 Perfect fifth is…

Three times the frequency of the tonic reduced
by an octave – f5th = 1.5 fo

3*fo = 2*f5th

Equal-tempered fifth is reduce 2 cents from the
perfect fifth
Tuning by Fifths
 Recall that the tonic contains the perfect
fifth as one of the partials
 We tune by listening for beats
 Ex. The equal-tempered G4 is 392 Hz
7
12

2 *C4
 
 Use perfect fifth rule
 3(261.63) – 2(392.00) = 0.89 Hz
 This difference would be zero for a perfect fifth
 We tune listening for a beat frequency of slightly less
than 1 Hz
Just and Equal-Tempered
Interval
Just
EqualTempered
Tonic
1.00
261.63
261.63
Major 2nd
1.13
294.33
293.67
-4
Major 3rd
1.25
327.04
329.63
14
Major 4th
1.33
348.84
349.23
2
Major 5th
1.50
392.45
392.00
-2
Major 6th
1.67
436.05
440.01
16
Major 7th
1.88
490.56
493.89
12
Octave
2.00
523.26
523.26
0
Minor 3rd
1.20
313.96
311.13
-16
Minor 6th
1.60
418.61
415.31
-14
Cent Diff.
Use of the Previous Table
 It is there to compare the two scales, not to
memorize.
 Know how to generate the frequencies in the
table
 Just frequencies come by multiplying by whole
number ratios
 Equal-tempered frequencies come by
multiplying by a power of 12 2
Notes
 Certain intervals sound smoother (or
rougher) than others.
 Notice particularly the Major 3rd, 6th, and
7th and the minor intervals
The Circle of Fifths
Pythagorean Comma
 Start from C and tune perfect 5ths all the
way around to B#.
 C and B# are not in tune.
 A perfect 5th is 702 cents.
 702+702+702+702+702+702+702+702+702+
702+702+702= 8424 cents
 An octave is 1200 cents.
 1200+1200+1200+1200+1200+1200+1200=
8400 cents
 8424 - 8400 = 24 cents = Pythagorean
Comma
Well-Tempered Tuning
 Many attempts distribute the
Pythagorean Comma problem around
the circle of Fifths in different ways to
make the problem less obvious
 Werckmeister III – shrunken fifths
 The interval is found by dividing the
Pythagorean Comma into four equal parts
(23.46/4 = 5.865). So instead of the
perfect fifths being 702 cents, they are
696.1 cents.
Werckmeister Circle of Fifths
 Numbers in the
intervals refer to
differences from
the perfect
interval.
 The ¼ refers to ¼
of the Pythagorean
Comma.
 Result is that
transposing yields
different moods
Physics of Vibrating Strings
 Flexible Strings
 Frequencies of the harmonics depend directly on
the tension and inversely on the length, density,
and thickness of the string.
 fn = nf1
 Hinged Bars
 Frequencies of the harmonics depend directly on
thickness of the bar and inversely on the length
and density.
 fn = n2f1
Real Strings
 We need to combine the string and bar
dependencies
f n (stiff string under tens ion)  f n2 (flexible)  f n2 (bar)
Physics of Vibrating Strings
The Termination


Strings act more like
clamped connections
to the end points
rather than hinged
connections.
The clamp has the effect
of shortening the string
length to Lc. The effect
of the termination is
small.
Physics of Vibrating Strings
The Bridge and Sounding Board
We use a model where the string is firmly
anchored at one end and can move freely on a
vertical rod at the other end between springs
FS is the string natural
frequency
FM is the natural
frequency of the block
and spring to which the
string is connected.
The string + mass acts as a simple string would that is elongated
by a length C.
The slightly longer length of the string gives a slightly lower
frequency compared to what we would have gotten if the string
were firmly anchored.
 Piano and harpsichord tuning is not marked
by beat-free relationships, but rather
minimum roughness relationships.
 The intervals not longer are simple
numerical values.
 Larger sounding boards have overlapping
resonances, which tend to dilute the
irregularities.
 Thus grand pianos have a better harmonic
sequence than studio pianos.