1. (Road to discuss harmony)
Combination of tones
(Road to discuss harmony)
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:
Please review lecture 6 (interference)

2. 2. Beats Slightly mismatched frequencies cause audible “beats”
Question: The beat frequency between tones with frequencies f1 and f2 is 2.0 Hz. In order to increase the beat frequency, one must __.
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.

3. Example
f1 = 16 Hz
f2 = 18 Hz

4. 2a. Beats (calculations - optional)

5. 3. Consonance and Dissonance
Consonance - sounds that are pleasant
Consonant intervals:
λ2:λ1
f2:f1
Interval’s name
Examples
# of half steps
1:1
unison
(C, C)
1:2
2:1
octave 12
2:3
3:2
perfect fifth
(C, G) or (F, C) 7
3:4
4:3
perfect fourth
(C, F) or (G, C) 5
3:5
5:3
major six
(C, A) or (Eb, C) 9
5:6
6:5
minor third
(C, Eb) or (A, C) 3
4:5
5:4
major third
(C, E) or (Ab, C) 4
5 :8
8:5
minor six
(C, Ab) or (E, C) 8

6. Octave (C/C) 1
1:2
Perfect fifth (C:G) 1
2:3
Perfect fourth (C:F) 1
3:4
Octave (C’/C)
2:3

7. 4. Helmholtz Theory (1877)
Dissonance occurs when partials of the two tones produce 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

8. 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

9. 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

10. 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

11. Scales and logarithms
When we go from octave to octave up, each time we multiply frequency by 2
Examples:
If we go 3 octaves up, frequency is 2x2x2 = 23 = 8 times higher
If we go 7 octaves up, frequency is 27 = 128 times higher
On keyboards and on musical staff distance between notes is changed linearly
If we go 3 octaves, it is 3 time as much as one octave
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

12. (All semitones are the same)
Equal temperament
(All semitones are the same)
Octave is divided into 12 equal semitone intervals
Semitone ratio: 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
880
(~6% up)
Whole tone:
Advantage:
5th and 4th are reasonably good
3d and 6th are OK
Modulation from key to key is easy