PH 105

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PH 105
Fourier Analysis, Timbre, Envelope
Dr. James van Howe Lecture 10

Time
2
Name the artist

• Black-eyed Peas
• Outkast
• A Tribe Called Quest
• De La Soul

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True or False
Most of the sound generated by a violin or guitar
comes from vibration of the strings
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Resonance in Tubes
Which picture accurately shows the displacement
antinodes for the fundamental mode in a closed
tube?
A
C
B
D
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What is the spectrum if I play both sine waves at
the same time?
Spectrum
A.
1
0.8
0.6
Amplitude
0.4

0.2
0
440
880
Frequency (Hz)
B.
1
0.8
0.6
Amplitude

0.4
0.2
0
440
880
Frequency (Hz)
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Open Tube
Fundamental
L
l/2
Loudest at displacement node Pressure antinode
l
3l/2
In general
similar to the string next mode next harmonic
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Closed Tube
L
l/4
3l/4
5l/4
In general
only odd harmonics exist
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End Correction
Brief note every open end of a tube is
acoustically longer
End correction add 0.61r where r is the radius,
of the tube to any open end to get the acoustic
length
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Helmholtz Resonator Cavity
Cross sectional area of neck, A
m mass of air in neck

k springiness of air in cavity
Volume of Air, V
Resonant frequency
What needs to happen to the volume of air to get
a higher pitch? Think about blowing across a pop
bottle that is empty and one that is half full
(less air).
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Examples of Helmholtz Resonators

• Late 19th century spectrum analyzer way to
  select and listen to (measure) one frequency
• Air resonance of a guitar or violin sound box
• Reflex port of loudspeaker cabinet neck into
  cabinet allows air resonance
• Automobile mufflers side branch absorbs sound

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Fourier Analysis/Synthesis
Fouriers Theorem Any periodic vibration can be
built up from a series of simple vibrations (sine
waves) whose frequencies are harmonics of the
fundamental
Pure tone the simplest vibration a sine wave
Fourier Analysis Studying a complex waveform by
looking at the simple pure tones
Fourier Synthesis Constructing a waveform out of
the simple pure tone building blocks
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Frequency Spectrum
Plotting out the strength of the individual sine
waves (pure tones) that make up a complex
waveform (sound)


Time
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Bb3 on Clarinet
Spectrum
1
0.8
0.6
Amplitude
0.4

0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
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Harmonic Number
Special recipe of pure tone harmonics that make
the clarinet unique!

Time
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Synthesis
Pure tones alone sound like when you drive a
speaker with a sine wave. Added together in the
right recipe they make a complex waveform and
unique sound
Pipe organ constructed from pure tones
Piano constructed from pure tones
Chimes constructed from pure tones
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Timbre quality of a complex tone tone color
Musical definition
Timbre the auditory sensation by which a
listener can judge two sounds with the same
loudness and pitch as dissimilar.
ANSI Definition
Example concert A on a violin vs. on a basson
Timbre The unique recipe of pure tones that
makes a sound unique
My definition
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Timbre Scales
Subjective rating for sound quality
Pg 136, Rossing
Dull Brilliant
Cold Warm
Pure Rich
Dull Sharp
Compact Scattered
Full Empty
Colorful Colorless
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Timbre and Fourier Analysis

• Simple tones (tuning fork, widely stopped organ
  pipe) soft pleasant sound, free from roughness,
  dull at low frequencies
• Musical tones up to the sixth harmonic (piano,
  French horn, human voice) sound richer than
  simple tones, remain sweet and soft if higher
  harmonics absent
• Tones consisting mainly of odd harmonics (narrow
  stopped organ pipes, clarinet) sound hollow if
  many harmonics present, nasal. If fundamental
  present, tone is rich tone is poor when
  fundamental weak
• Tones with strong harmonics above sixth or
  seventh are rough and cutting, very distinct

Helmholtzs rules for Timbre
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Envelope affects Timbre
Remember beats?
DP
Pressure, DP
time, t
If and ,
Envelope
we hear a sine wave at
whose volume goes up and down (modulated) at

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Wiggly stuff inside Just beats
Dt
Dt
t
t
t
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Neil Armstrongs missing a
Armstrong claims to have said Thats one small
step for a man, one giant leap for Mankind not
as we hear, Thats one small step for man
Analyzing the Envelope of the time trace, Peter
Ford, a computer programmer working on software
for communication disorders, found the missing
a.
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Attack Envelope of Instrument
Attack
Sustain (periodic)
When playing a note, it takes some time for the
signal to become periodic
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Attack sometimes defines the timbre of an
Instrument
Sometimes the recipe of pure tones (spectrum)
isnt enough to define the instrument
Attack more important
Spectrum more important
Equally important

• Oboe
• Clarinet
• Bassoon
• Tuba
• Trumpet

• Trombone
• French Horn

• Flute

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What is the spectrum if I play both sine waves at
the same time?
Spectrum
A.
1
0.8
0.6
Amplitude
0.4

0.2
0
440
880
Frequency (Hz)
B.
1
0.8
0.6
Amplitude

0.4
0.2
0
440
880
Frequency (Hz)
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If I model the Clarinet as a tube closed at one
end, the spectrum should contain

• Even Harmonics
• Odd Harmonics
• Both Even and Odd Harmonics

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Homework
For Monday Chapter 10 R3, R6
R review questions E Excercise