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Combination of tones
(Road to discuss harmony)
1. Linear superposition
• If two driving forces are applied simultaneously, the response will
be the sum of the responses to the driving forces individually.
• For instance: doubling the driving force doubles the response.
• In linear systems independent signals do not influence each other.
Linear systems (examples):
Loudspeakers, microphones and amplifiers should be linear to some extent.
Is ear a linear system?



Linear addition of two sound waves:  (r , t )  1 (r , t )  2 (r , t )
Please review lecture 6 (interference)
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2. Beats
Slightly mismatched frequencies cause audible “beats”
fbeats  f 2  f1
f1  f 2
Question: The beat frequency between tones with frequencies f1 and f2 is
2.0 Hz. In order to increase the beat frequency, one must __.
A.
B.
C.
D.
E.
increase f1
increase f2
decrease f1
decrease f2
There is not enough
information to choose
Second-order beats are the beats between two tones whose frequencies are
nearly but not quite in a simple ratio. They are also called beats between mistuned
consonances.
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Example
f1 = 16 Hz
f2 = 18 Hz
f beats  f 2  f1
Tbeats  1 f beats  1 f 2  f1  0.50s
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2a. Beats (calculations - optional)
y1  A cosk1 x  1t 
y 2  A cosk 2 x   2 t 
1  2
       
cos   cos   2 cos
 cos

2
2

 

k1  k 2
 1   2
y  y1  y2  2 A cos
t
2
 2
k 
 
y  2 A cos
t
x  sin t  kx
2 
 2
beats    2  1  2mod
fbeats  f 2  f1
1   2 / 1  1
k1  k 2
  1   2
x  sin 
t
2
  2

x

  1   2  / 2
k  k1  k 2  / 2
  1   2   
k  k1  k 2   k
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3. Consonance and Dissonance
Consonance - sounds that are pleasant
Consonant intervals in descending order of consonance:
λ2:λ1
1:1
1:2
2:3
3:4
3:5
4:5
5:6
5:8
f2:f1
1:1
2:1
3:2
4:3
5:3
5:4
6:5
8:5
examples
unison
octave
perfect fifth
perfect fourth
major six
major third
minor third
minor six
(C,C)
(C,C)
(C,G)
(C,F)
(C,A)
(C,E)
(C,Eb)
(C,Ab)
or
or
or
or
or
or
(F,C)
(G,C)
(Eb,C)
(Ab,C)
(A,C)
(E,C)
# of half steps
0
12
7
5
9
4
3
8
5
f  v/
Octave (C/C)
1
C : C '  1 : 2;
1:2
fC : fC '  2 :1
Perfect fifth (C:G)
1
G : C  2 : 3;
2:3
fG : fC  3 : 2
Perfect fourth (C:F)
1
3:4
F : C  3 : 4;
f F : fC  4 : 3
Octave (C’/C)
2:3
32 1

43 2
 G  C '   3  2  1
 
      ;
 C  G   4  3  2
 f G  f F   3  4 
        2
 f C  f C   2  3 
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4. Helmholtz Theory (1877)
• Dissonance occurs when partials of the two tones produce 30-40 beats
per second
• The more partials of a tone coincide with the partials of another, the less
chance that beats in the range will produce roughness
• This explains why simply frequency windows define most of the
consonant intervals
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4a. Consonance and Dissonance between two pure tones
•When two pure tones are sounded together, consonance or dissonance
depends upon their frequency difference rather than on their frequency ratio
•If the frequency difference is greater than a critical band, they sound
consonant
•If the frequency difference is less than a critical band, they sound dissonant
•According to Plomp and Levelt (1965) the maximum dissonance occurs at
¼ the critical bandwidth
•According to Kameoka and Kuriyagowa (1969) it also depends on the
sound pressure level: Δf = 2.27(1 + (Lp–57)/40)f 0.447
(f is the frequency of the primary tone, and Lp is sound pressure level)
•The critical bandwidth changes depending on the octave of the two tones
•The higher the octave, the closer two notes could be and still be consonant
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4b. Consonance & Dissonance between two complex tones
•In this case we have to consider the roughness between the fundamental
notes as well as between the harmonics
•This is what explains why some intervals are more consonant than others
•In the case of the perfect fifth the two lower harmonics coincide and the two
produce frequency differences within the critical bandwidth
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5. Musical Scales and Temperament
Musical scale – a succession of notes in ascending or distending order
•In Western music octave is divided in 12 semitones
•Chromatic scale - all 12 semitones
•Most music makes use of 7 selected notes (major or minor scales)
•There are many ways to construct musical scales
•Different scales are different ways of dividing octave (almost always)
“Standard” scales:
•Pythagorean scale
•Mean-tone temperament
•Scale of just intonation
•Equal temperament
Tuning – an adjustment of pitch in any instrument so that it corresponds to an
accepted norm (scale)
Temperament – a system of tuning in which the intervals deviate from acoustically
pure (Pythagorean) intervals
Intonation – the degree of accuracy with which pitches are produced
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Scales and logarithms
• When we go from octave to octave up, each time we multiply frequency by 2
Examples:
1) If we go 3 octaves up, frequency is 2x2x2 = 23 = 8 times higher
2) If we go 7 octaves up, frequency is 27 = 128 times higher
• On keyboards and on musical staff distance between notes is changed
linearly
Examples:
1) If we go 3 octaves, it is 3 time as much as one octave
2) If we go 7 octaves, it is 7 time as much as one octave
• This means that keyboard and musical staff have logarithmic scale: distance
between keys and notes is proportional to the logarithm of the frequency
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Equal temperament
(All semitones are the same)
Octave is divided into 12 equal semitone intervals
Semitone ratio:
x12  2  x  21 12  1.05946
(~6% up)
Whole tone:
x 2  21 6  1.12246  1.05946
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Advantage:
•5th
and
4th
are reasonably good
2
 1.4983
3 : 2  1.5000
7 12
•3d
and
6th
are OK
2  1.6818
5 : 3  1.6667
9 12
•Modulation from key to key is easy
A
440
B flat
466
B
494
C
523
C sharp
554
D
587
D sharp
622
E
659
F
698
F sharp
740
G
784
A flat
831
A
880
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