Download Fourier Series and Harmonics in Wind Instruments

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Professor Naala Brewer
Spring 2011
Fourier Series and Harmonics in Wind Instruments
By Amy Blatt
The French mathematician Jean-Baptiste-Joseph Fourier (1768-1830) is
famous for his representation of a mathematical function as a sum of sine or cosine
waves, known as the Fourier Series. This trigonometric series analyzes periodic
functions and can be applied to music and harmonics. All musical instruments
have periodic audio signals which represent tones. These audio signals are the
cause of pressure vibrations that are we perceive as sound.
In a study through Johns Hopkins University by Michael Ross (Spring 2004)
based on an earlier version by Kevin Rosenbaum (Fall 1995), the Fourier Series
was used to analyze complicated tones. Before understanding a complicated tone,
a pure tone must first be defined. It can be described by the function
x(t)  a cos( ot   ) , where a is the amplitude ( a  0 ),  o is the frequency in
radians/second (  o  0 ), t is the time in seconds, and  is the phase angle in
radians. Frequency can also be expressed in Hertz, given by the function
(“Listen to Fourier Series”)
fo 
o
2 .
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To the ear, tones separated by an octave (a factor of two in frequency) have
very similar pitches. Since an octave is divided into 12 notes, equally spaced on a
logarithmic scale, there is a relationship between a pure tone and frequency. A, the
first note of the octave below middle C, has a fundamental frequency of 220 Hz.
1
Each successive note, A#, B, C, C#, D, D#, etc. has a frequency that is 2 12 times
greater than the previous note. This relationship can be derived from the following
steps:
220Hz  2b  2b  2b  2b  2b  2b  2b  2b  2b  2b  2b  2b  440Hz
220Hz  212b  440Hz
212b  2
log(212b )  log(2)
12b log(2)  log(2)
12b  1
b
1
12
Therefore,
1
1
1
1
1
1
1
1
1
1
1
1
220Hz  212  212  212  212  212  212  212  212  212  212  212  212  440Hz
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More complicated tones are represented by a Fourier Series. This series is a
summation of pure tones whose frequencies are integer multiples, or harmonics, of
a fundamental frequency represented by the following equation:
x(t)  a1 cos( ot  1 )  a2 cos(2 ot  2 )  a3 cos(3 ot  3 )     .
The Fourier Series can represent natural and synthetic tones. There is a special
case of the Fourier Series for a square wave (a computer-generated tone) and the
clarinet. A square wave is produced by the computer flipping a simple device from
on to off, instead of a steady oscillation in sound pressure. The approximation of a
square wave using the Fourier Series is described in John Smith’s honors contract
from Fall 2010, “Fourier Series Approximation of Periodic Square Waves.” A
visual representation of the square wave is shown in the “Fourier Series Applet”
below.
This shows a Java applet that allows the user
to hear each note and view its corresponding
signal. (“Listen to Fourier Series”)
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The clarinet has a uniquxe Fourier Series representation because it only contains
odd harmonics. According to Brian Harvey from the University of California,
Berkeley, “If a clarinet is playing [the note] A above middle C, the waveform
includes frequencies of 440 Hertz, 1320 Hertz (3 times 440), 2200 Hertz (5 times
440), and so on. But the waveform does not
This is a graphical representation of the
odd harmonics.
includeclarinet’s
frequencies
of 880 Hertz (2
(“Clarinet Sounds”)
times 440), 1760 Hertz (4 times 440), and so on.” This occurs because the
clarinet’s resonator is cylindrical, therefore suppressing the even-numbered
harmonics, producing a purer tone. (“Harmonic Series (music)”)
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The following is a Fourier Series representation for the clarinet:
x(t)  0.382 cos(2 195.0t  1.35)  0.237 cos(2 584.9t  0.48)  0.169 cos(2 974.8t  0.30) 
0.151cos(2 1754.6t  1.35)  0.066 cos(2 2534.5t  1.41)  0.061cos(2 2144.6t  2.40) 
0.057 cos(2 2339.5t  0.40)  0.041cos(2 1364.7t  1.32)
(Fessler)
In this case,
x(t)  a cos(2 ot   )  a cos(2 3 ot   )  a cos(2 5 ot   )  a cos(2 9 ot   )
a cos(213 ot   )  a cos(211 ot   )  a cos(212 ot   )  a cos(2 7 ot   )
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Although it may seem like the “squiggles” would produce an inconsistent
sound, they actually form a perfectly steady tone. That is why wind instruments,
when played correctly, are pleasing to the ear and rich in sound. The oscillations
in the above picture only represents about 1/100 of a second. Therefore, there are a
few hundred repetitions of the cycle in one second of a musical note. (Harvey)
This shows a Java applet that allows the user to choose a tone and view the corresponding x(t)
and amplitude spectrum as well as play the tone. The signal shown in this picture represents the
tonal wave of a clarinet and its amplitude spectrum (in red). (“Listen to Fourier Series”)
To try making your own tones with square, sine, and cosine waves, refer to
the applet “Fourier Synthesis” shown below.
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The Fourier Series is useful for analyzing music and harmonics, but it can
also be applied to many other aspects of mathematics and engineering.
Works Cited
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"Clarinet Sounds." Michigan Technological University - Department of Physics.
Web. 11 May 2011. <http://www.phy.mtu.edu/~suits/clarinet.html>.
Fessler, Jeffrey A. "Musical Signal Processing (Signal Spectra)." Electrical
Engineering and Computer Science at the University of Michigan. Web. 11
May 2011. <http://eecs.umich.edu/~fessler/course/100/l/l04-lab3-fft1.pdf>.
"Fourier Series Applet." Paul Falstad. Web. 11 May 2011.
<http://www.falstad.com/fourier/index.html>.
"Fourier Synthesis." NTNUJAVA Virtual Physics Laboratory. Web. 11 May 2011.
<http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=17>.
"Harmonic Series (music)." Wikipedia, the Free Encyclopedia. Web. 11 May 2011.
<http://en.wikipedia.org/wiki/Harmonic_series_(music)>.
Harvey, Brian. "Computer Science Logo Style Vol 2 Ch 13: Example: Fourier
Series Plotter." Computer Science Division | EECS at UC Berkeley. Web. 11
May 2011. <http://www.cs.berkeley.edu/~bh/v2ch13/fourie.html>.
"Listen to Fourier Series." Johns Hopkins University. Web. 09 May 2011.
<http://www.jhu.edu/signals/listen-new/listen-newindex.htm>.
Smith, John. "Fourier Series Approximation of Periodic Square Waves." 2010.
Web. <http://math.asu.edu/~nbrewer/studentprojects/>.