The Perceptual Dimensions of Sounds

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The Perceptual Dimensions of Sounds
Tuesday November 23, 1999
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Dimensions of Sound

• PHYSICAL
• Frequency
• Amplitude
• Phase Harmonics
• Time

• PSYCHOLOGICAL
• Pitch
• Loudness
• Timbre
• Perceived time

What are the psychophysical relations between
these dimensions?
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Frequency and Pitch

• Pythagoras (6th Century B.C.)
• Spartan law forbade the people from making lyres
  with more than 7 strings.
• Dividing a string allowed a musician to play more
  than one note on a given string.
• Pythagoras developed a monochord a single string
  whose length could be changed by using a movable
  bridge.

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Monochord
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What Pythagoras Found

• Pythagoras found interesting relations between
  the length of a string, and the sound it
  produced.
• 11 Each half sounded at the same pitch.
• 12 The shorter half made a note an octave
  above the longer half.

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What Pythagoras Found

• When the length of the two halves were related by
  simple integer ratios, the two notes sounded good
  together.
• 12, 23, 34, etc...
• When they were related by complex ratios, they
  sounded bad together.
• 175214, 2198, 1643, etc...

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The Harmonic Series

• A complex waveform is said to be harmonic when
  the frequencies of the upper partials are integer
  multiples of the frequency of the fundamental.
• The harmonic series is a collection harmonically
  related partials
• 1 2 3 4 5 6 7 8 9 10

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Inharmonic (1 2.67 3.14)
Harmonic (1 2 3)
3
3
A
2
2
1
1
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
?
?
Upper partials are integer multiples of the
fundamental frequency ( 1).
Upper partials are not integer multiples of the
fundamental frequency ( 1).
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Math, Music and Morality

• Pythagoras revelled the relation between
  beautiful numbers and beautiful sound.
• This harmony was considered to extend to the
  social order.
• Boethius (6th Century A.D) story of Pythagoras
  and the angry teenager.
• Platos Republic Citizens only exposed to
  certain musical modes.

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Pythagoras
Pitch
String Length
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Pitch and Frequency

• Sound has long been considered to be comprised of
  vibrations in the air (Aristotle, Galen,
  Boethius).
• Fracastoro (16th Century) Resonating statue due
  to vibration of a specific frequency.

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Mersenne

• Examined rate of vibration in long (around 90
  feet) piece of rope.
• He found that the frequency of the ropes
  vibration was directly proportional to
• the length of the rope
• the square-root of the weigh which stretches it
• the square-root of the weight of the string
  itself
• the reciprocal of its diameter

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Galileo

• Observed the swinging of a pendulum.
• Rate of swing proportional to its length.
• Generalized this idea to strings.

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Galileo
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Hooke
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The Mel Scale(Stevens, Volkman Newman, 1937)

• A mel is a unit of perceived pitch.
• Once the perceptual unit is quantified, the
  question of the mathematical relation between
  physical and psychological units can be
  addressed.
• Differences in frequency units related to
  differences in mels.

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The Mel Scale
3000
2000
Mels
1000
1000
100
10
10000
Frequency (Hz)
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Amplitude and Loudness
Time
Low
Time
High
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The decibel scale

• Physical scale that describes the amplitude of a
  sound wave.
• Decibel is a measure of sound pressure, relative
  to background air pressure.
• dB 10 log I/I0

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dB 10 log I/I0
I0 Background Pressure .0002 dynes/cm3
Sound Pressure 5,000,000,000 dynes/cm3
I/I0 5,000,000,000 X .0002 1,000,000 log
I/I0 6 10 log I/I0 60 db
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Examples of dB Levels

• 0 dB Threshold of hearing
• 20 dB Soft whisper
• 40 dB Quiet office
• 60 dB Conversation at 1m
• 100 dB Heavy traffic
• 120 dB Large jet motor (at 22m)
• 180 dB Spacecraft launch

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• dB is a physical measure,what about a
  psychological measure?
• S.S. Stevens (1956) Classic scaling study
  established the sone scale, as a measure of
  perceived loudness.

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The Sone Scale
100
10
Sones
1
0.1
0
20
40
60
80
100
120
dB
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Perceptual Interactions between Intensity and
Frequency

• Fletcher-Munson Curves
• (Equal-Loudness Contours).
• Dependency of Pitch on Intensity
• S. S. Stevens (1935)
• -tones lt 1000 Hz decrease in pitch with
  increases in intensity
• -tones gt 2000 Hz increase in pitch with
• increases in intensity