We have seen that by adding two pure tones, with frequencies corresponding to the fundamental and second harmonic from the harmonic series, we make a more complex waveform with a period equal to the period of the fundamental pure tone. Now, we can ask a more sophisticated question: if we start with pure tones from a harmonic series, how can we synthesize a specific complex wave? Alternatively, what we're really asking what is the harmonic content of an arbitrary complex wave? What we're really aiming to get is a quantitative description of the complexity of a periodic waveform. With that description, we can then give a reason why two sounds with the same pitch and loudness sound different!
Again, let's deal with specific examples rather than trying to answer the more general question. There is indeed an answer to the general question, but it involves a great deal of mathematics called Fourier analysis. We'll not talk about the sophisticated math, instead we'll deal with the consequences. For our example, let's take the sawtooth wave shown below
From the waveform, we can see that the period is 4 ms, meaning that the fundamental harmonic has a frequency of 250 Hz, and the harmonic series we should work with has frequencies 250 Hz, 500 Hz, 750 Hz, 1000 Hz, ... We can also see that the amplitude of the sawtooth waveform is p = 2 x 10-2 N/m2. The amplitude is, of course, related to the perceived loudness of the sound.
Fourier analysis tells us what amplitudes to assign to the different harmonics to synthesize the sawtooth wave. The answer is that the amplitude of the nth harmonic should be
This means that we can synthesize the sawtooth wave by superimposing (adding) pure tones from the harmonic series, with the fundamental frequency of 250 Hz, choosing the amplitude of the fundmental to be 0.64 p (the factor of 0.64 is just the ratio 2 / 3.14159...); the amplitude of the second harmonic to be -0.32 p; the amplitude of the third harmonic to be 0.16 p; the amplitude of the fourth harmonic to be -0.08 p; etcetera. If we start with a superposition of just the first two harmonics, we already get a reasonable representation of the sawtooth waveform, as shown below.
Shown in the top part of the picture are the first- and second-harmonic pure tones with amplitudes equal to 0.64 p = 0.0128 N/m2 and -0.32 p = -0.0064 N/m2, respectively. Note that the second harmonic starts off going to negative pressure differences whereas the first harmonic starts off going to positive pressure differences. This just reflects the different sign for the two amplitudes. Consequently, there is partial destructive interference between the two pure tones for times between 0 and 1 ms. This switches to constructive interference for times between 1 and 3 ms.
As we add together more and more pure tones from the harmonic series, with amplitudes determined from the Fourier analysis, we get a better and better representation of the sawtooth waveform. Below, is shown the superposition of the first four harmonics compared to the original sawtooth waveform.
One objective of establishing the harmonic content of a complex waveform is to quantify the shape of the wave in some fashion, independent of its amplitude and frequency. Recall, we have already established a connection between the measured amplitude of a wave and the qualitative attribute of sound called loudness. As well, the measured frequency of a wave is related to the qualitative attribute called pitch. A quantification of the shape will enable us to say why sounds of different timbre are different. How can we quantify the complexity of a waveform? The answer to this question is to establish the amplitudes of the higher harmonics contained within the Fourier analysis of the waveform. It's better if we factor out from the superposition of harmonics the amplitude of the fundamental (this will just affect the loudness). If we do that, we are left with coefficients, An, multiplying the mathematical expressions for each harmonic pure tone. For the sawtooth wave, the "weights" of each harmonic pure tone are
1 (fundamental) ; -1/2 (2nd harmonic); +1/3 (3rd harmonic); -1/4 (4th harmonic) ; ...
This sequence of weights of the different elements of a harmonic series will always produce a sawtooth waveform, independent of the frequency (given by the frequency of the fundamental harmonic and related to the pitch of the sound) or the amplitude (related to the loudness of the sound)! Different shaped complex waveforms will have different weighting coefficients (An) determining their harmonic content. This means that we can quantify the complexity of different wave shapes using a Fourier analysis. The complexity of the wave form is then related to the sound quality called timbre.
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Last updated: 30 Oct 1999
Comments: bland@indiana.edu