J.D. Flood and D.W. Follet 1 An Introduction to Fourier Analysis and Fourier Synthesis Goals: 1. to understand the relationship between musical ratios and the harmonic series. 2. to be able to relate these ideas to the physics of waves, particularly Fourier analysis and synthesis. 3. to relate the harmonic series to the problem of minor articulated by Rameau. We will be working at the website developed by Paul Falstad. http://www.falstad.com/mathphysics.html Press “CTRL” and click on the link above and look for “Signal Processing”. Click on “Fourier Series Applet”. 1. Getting acquainted with the software a. The sine wave and its components. Start by clicking on the button “Sine”, found in the list of buttons on the right. The waveform you see in the top panel is a sine wave. This is the shape that would be traced out by the motion of a simple pendulum or small segment of a vibrating string (in a single mode) if you plotted its position as a function of time. Click on the box next to “Sound” to hear this wave. b. Adding components You should see “Mag/Phase View” directly below “Sound”. Click on the box to activate this view. The middle panel, labeled “Magnitudes” shows the spectrum for the “sine” waveform. The spectrum of a sound shows each frequency, and the strength of that frequency, making up the sound. The process of obtaining the spectrum is called Fourier analysis. Each dot in the series represents a member of the harmonic series built on the fundamental frequency (or first harmonic) of 220.5 Hz. Placing the pointer on one of the dots will highlight it in yellow, while displaying the frequency it represents above the word “Magnitudes”. Add the frequency you have chosen to your sound (Fourier synthesis) by dragging the dot up to about ½ (Magnitudes of about 0.5) the height of the dot representing 220 Hz . The length of the resulting line represents the amplitude of that frequency. Sketch the waveform and spectrum on the axes provided below. Be sure to label the frequencies on the spectrum. Waveform Spectrum
1 J.D. Flood and D.W. Follet 2 2. Synthesizing waves Before you begin the following exercise, clear the waveform by pulling all the frequencies in the spectrum down to zero amplitude (magnitude). a. Add together two pure sine waves of frequencies 440 Hz and 660 Hz. You can control which sine waves you add in by dragging the dots representing each frequency, so select equal amounts of 440 Hz and 660 Hz. Sketch the resulting waveform and spectrum below. Label the frequencies in the spectrum. b. What is the pattern repetition frequency of the resulting complex wave? Use the software to help you figure out the repetition frequency of the resulting wave by selecting a frequency, in order starting with 220 Hz, and giving it a small amplitude (so it will not significantly affect the waveform), then comparing the repetition cycle of the synthesized wave to the repetition cycle of the selected sine wave. c. If you were to try to Fourier analyze this complex wave, what would be the frequency of the fundamental? Explain the reasoning you used to arrive at your answer. d. Which harmonics do 440 Hz and 660 Hz represent? How did your determine the harmonic numbers? e. Listen to the sound. What interval do you hear? f. Divide the higher frequency by the lower frequency. What ratio of small numbers is this close to? This ratio is associated with the interval you just heard.
2 J.D. Flood and D.W. Follet 3 Major and Minor Triads 3. a. Make a wave with the two frequencies 880 Hz and 1320 Hz. Listen to the interval. How does it compare to the interval you heard in 2e? What is the pattern repetition frequency? b. If you now add a sine wave with frequency 1100 Hz, what is the pattern repetition frequency? c. Identify the frequency of the fundamental for the wave you constructed in 3b and draw the spectrum to illustrate which Fourier components are present in this wave. Be sure to label the frequencies present in the sound. If you could hear this wave as a sound, it would be the major triad A6­C#7­E7, as played by three tuning forks, or other instruments that do not have overtones. Listen to this sound. d. Identify the fundamental frequency present in this chord, which is the value you calculated in part c. e. Form a wave by combining equal amplitudes of 2200 Hz, 2640 Hz and 3300 Hz. What is the pattern repetition frequency? f. Identify the frequency of the fundamental and draw the spectrum to illustrate which Fourier components are present in this wave. Which harmonics do these correspond to? g. Identify the chord in part 3f. Is it musically sensible to try to define a fundamental for this chord in the same way as we did in 3d? Describe and explain any differences.
3 J.D. Flood and D.W. Follet 4 h. Comment on these calculations in relation to the basic properties of major and minor chords. How are your observations related to Rameau’s problem of minor, described on page 105 in your text? Fourier Synthesis: Building a square wave 4. a. Look for the “Square” button and click on it to see a square wave. Sketch the waveform and spectrum below. The heights of the bars in the spectrum graph tell you how much of each harmonic is in the square wave so draw them carefully. You will need these values for part b. b. Clear the screen. Try to synthesize a square wave by adding harmonics with the correct amplitude. As you add each odd harmonic adjust the “Phase” markers in the bottom panel to halfway down for each harmonic you use. Which harmonics are present? Compare the harmonics present in your wave to those represented in the spectrum of the square wave you sketched above.
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