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Blatt 1 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 . Blatt 2 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 Blatt 3 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”) Blatt 4 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)”) Blatt 5 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(213 ot ) a cos(211 ot ) a cos(212 ot ) a cos(2 7 ot ) Blatt 6 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. Blatt 7 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 Blatt 8 "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/>.