Download Chapter 15 - SFA Physics and Astronomy

Chapter 15
Successive Tones: Reverberations,
Melodic Relationships,
and Musical Scales
Audibility of Decaying
Sounds in a Room
 The first of the tone we hear is the
directly propagated wave.
 Because of the precedence effect, the
direct wave will combine with the most
direct reflections (within 30 to 50
milliseconds) and be perceived as one.
Picture of a Clearly Heard Tone
Decay – similar
to the attack
Attack – heard as
one because of
Precedence Effect
Reverberation Time
 The time required for the sound to
decay to 1/1000th of the initial SPL
 Audibility Time
 Use a stopwatch to measure how long
the sound is audible after the source is
cut off
 Agrees well with reverberation time
 It is constant, independent of frequency,
and unaffected by background noise
Why does Audibility Time Work?
 Threshold of hearing temporarily
shifted to 60 dB below a loud tone?
 60 dB is 1000 times in SPL which then
matches the definition of Reverberation
Time
 Measurements show that this happens,
but only for a few tenths of a second
 Not long enough to make audibility time
work
Why does Audibility Time Work?
 The ear is responding to the rate of
change of loudness?
 Look at example on next slide
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
 Instruments assume uniform decay of
the sound, which may not be the case
Device to Study Successive Tones
Tone Generator 1
1
2
Tone Generator 2
Switch
Amplifier
Speaker
Notes on Tone Switcher
 Tone generators produce fundamental
plus a few harmonics to simulate real
instruments
 Switching cannot be heard
 Reverberation time at least ⅓ sec.
Experiment
 Start with TG1 on C4
 Switch to TG2 and adjust
 At certain frequencies the decaying
TG1 will form beats with the partials
or heterodyne components of TG2
 The beats will be most audible when the
amplitudes are equal.
Using Reverberation
 These experiments show that we can
use reverberation as an aid in
performing
 It is easier to perform in a live room
(shower)
 Noise can mask the decaying partials
and make pitch recognition more difficult
Conclusions
 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
 We will use the Tone Switcher to help
find intervals that produce beat-free
relationships to the fundamental.
 The fact that the frequency generators
contain harmonics makes this possible
 Notice that the octave is a doubling of
the frequency and the next octave would
be four times the frequency of the
fundamental
First Important Relationship
 Three times the fundamental less an
octave
 3f/2 or an interval of 3/2 or a fifth
 Fundamental will have harmonics that
contain the fifth
 Five such relationships can be found in
the first octave
Just Intervals (with respect to C4)
Chromatic Scales
Listed
Interval
Interval
Computed
Cent
Frequency
Name
Ratio
Frequency
Difference
(equal-tempered)
(beat-free)
C
261.63
E
329.63
3rd
5/4
327.04
14
F
349.23
4th
4/3
348.84
2
G
392.00
5th
3/2
392.45
-2
A
440.00
Major 6th
5/3
436.05
16
C
523.25
octave
2/1
523.26
0
Relationships Among Five Principles
Note
Frequency
(equal-tempered)
Interval
Ratio
Interval
Name
Resulting
Frequency
Note
F
349.23
3/2
5th
523.85
C
E
329.63
4/3
4th
439.51
A
G
392.00
4/3
4th
522.67
C
F
349.23
5/4
3rd
436.54
A
E
329.63
6/5
Minor 3rd
395.52
G
A
440.00
6/5
Minor 3rd
528.00
C
Finding the Missing Steps
 Notice the B and D are not
harmonically related to C
 Finding B
 A fifth (3/2) above E gives 490.56 Hz
 A third (5/4) above G gives 490.00 Hz
 Difference is 2 cents – sensibly equal
The Trouble with D
 A Fourth (4/3) below G gives 294.34
Hz
 A Fifth (3/2) below A gives 290.70 Hz
 Difference is 22 cents or 1¼%
 Sounded together these “D’s” give
clear beats
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.
Putting numbers to the Ab/G#
Problem
From
at
Interval
Ratio
Giving
E
327.04
Third
5/4
408.80
C
523.26
Third
5/4
418.61
The Problem with F#
C
D
E
F
3rd
G
A
B
min3
3rd
Other discrepancies exist but these highlight
the problem.
C
Saving the Day
 As the speed increases discrepancies
in pitch are more difficult to detect.
 The sound level is greater at the
player’s ear than the audience. He
can make small adjustments. He is
always better tuned than the
audience demands.
Working Toward Equal
Temperament
 The chromatic (Just) scale uses intervals
which are whole number ratios of the
frequency.
 Scales have unequal intervals

E
327.04 F 348.84
1.0666
16/15
B
490.5
1.0667
16/15
C 523.26
but
C#
279.07
D1
290.7
1.0417
F
348.84
F#1
363.38
1.0417
Making the Interval Equal
 An octave represents a doubling of the
frequency and we recognize 12 intervals in
the octave.
 Make the interval 12 2  1.059463
 Using equal intervals makes the
cents division more meaningful
 The following table uses
Breaking Up One Interval
Interval in Cents
Frequency Ratio
Frequency
Note
0
1.00000
261.63
C4
10
1.00579
263.15
20
1.01162
264.67
30
1.01748
266.20
40
1.02337
267.75
50
1.02930
269.30
60
1.03526
270.86
70
1.04126
272.43
80
1.04729
274.00
90
1.05336
275.59
100
1.05946
277.19
D4
Comparison
Frequency
Ratio
Musical Interval
Cents
(Just)
Cents
(EqualTempered)
1/1
Unison
000
000
2/1
Octave
1200
1200
3/2
Fifth
702
700
4/3
Fourth
498
500
5/3
Major sixth
884
900
5/4
Major third
386
400
6/5
Minor third
316
300
8/5
Minor sixth
814
800
Pitch Discrepancy Groups
 When pitch discrepancies exist in a
scale, the cent difference from the
equal-tempered interval cluster into
three groups
Low Group
Middle Group
High Group
12 cents low
Equal-tempered
frequency
12 cents high
 Each group has a range of about 7 cents
 If a player is asked to sharp/flat a tone, (s)he
invariably goes up/down about 10 cents,
moving from one group to another.
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
Indian Music Comparisons
Indian music uses a generalize seven note scale like the do
re mi of Western music.
Indian
sa
re
ga
ma
pa
dha
ni
sa
Western
do
re
mi
fa
sol
la
ti
do
Letter
C
D
E
F
G
A
B
C
The Reference Raga
 The rag is the most important
concept of Indian music.
 The Hindi/Urdu word "rag" is derived
from the Sanskrit "raga" which means
"color, or passion". It is linked to the
Sanskrit word "ranj" which means "to
color".
The Alap
 An Indian piece will usually open with
an alap, notes going up and down the
scale to establish position and
relationship.
 They will play around a tone, the tone
evasion becoming very elaborate.
 It becomes a game between the player
and the listeners.
 Jazz has similar variations.
Indian Modes
Play Bilawal
Play Kafi
Pitch Variations
 In Western music we have similar
pitch wanderings (vibrato, for
example) that the Indian musician
would find strange.
 We almost always make abrupt
transitions from one note to the next
without the slides of Indian music.