Homework #5
Instructions. This is due on November 7 at the beginning of
class. It is worth 20 points. Late papers lose 3 points per day late. You must
work by yourself, but feel free to consult with me. Explain what you did and
why you did it: good explanations are essential! Your paper should have an
introduction, a main body and a conclusion.
Introduction. Fourier series are heavily
used in engineering and physics. This assignment explores the use of Fourier
series in music synthesizers. A brief explanation of the general problem is
given on pages 611 and 612.
Problems. Recall from class that the Fourier series for the “square wave” function [f(x) = 1 for x³0 and f(x) = -1 for x<0] is given by the series
f(x) = 4/p (sinx + (1/3) sin3x + (1/5) sin5x + … )
A high-end music synthesizer manual summarizes this series by saying that the harmonic content of the square wave is 1/n for odd values of n. Note that this means that 1/5 is the fifth harmonic, even though it is the coefficient in the third term of the series. In this manual, the “sawtooth wave” is defined by the function f(x) = -x for x between -p and p. The manual states that the harmonic content of this wave is 1/n for all n. Verify this by finding the Fourier series of f(x) = -x. (Note that this is the only part of this assignment where you are asked to compute a Fourier series. The rest of the assignment just asks you to write down series.)
Write out the Fourier series for a wave that has harmonic content 1/n for every third n. Take the first four terms of this series and graph this. Write out the Fourier series for a wave that has harmonic content 1/n for n = 1, 2, 3, 4 and harmonic content 0 for n > 4. Graph this.
The action of many of the control knobs on a music synthesizer can be explained by their effect on the harmonic content of the resulting wave. For example, the “cutoff frequency” setting has a dramatic effect on the timbre (tone quality) of the tone produced. When the cutoff frequency is set at some value c, the harmonic content for all n larger than c is set equal to 0. (The second example in the paragraph above corresponds to a cutoff frequency setting of 4.) Graph the square wave with cutoff frequency settings of (a) 7 and (b) 3. Briefly discuss whether the tone in (a) or (b) would sound more like a pure flute tone.
The “resonance” setting also changes timbre significantly. When this is set at 1, you get the basic wave (e.g., square wave). When resonance is set at 2, the first four harmonics are divided by 2, the fifth and sixth harmonics are the same, the seventh harmonic is divided by 2, and all other harmonics are 0. Write out and graph the Fourier series for the sawtooth wave with resonance setting of 2. Note that this is starting to look like the saxophone waveform shown on page 611.