13. The Case of the Missing Fundamental
We have learned that when we hear a periodic sound, the pitch we perceive is based on the fundamental frequency of the sound, rather than on any of the harmonics (also called "overtones" or "partials") which may also be present in the signal. We also know that the fundamental frequency is the lowest in frequency of the harmonics (there are exceptions to this which we will disregard for now) and it also has the greatest amplitude of all the harmonics. But the reason we perceive only the pitch of the fundamental frequency is not due simply to its greater amplitude.
We know that the harmonics of a sound occur at progressive multiples of the fundamental frequency, e.g. 100, 100 x 2 = 200, 100 x 3 = 300, 100 x 4 = 400, and so on. But what would happen if we were to remove only the fundamental frequency from a sound and keep all of the other harmonics? What would you "hear" as the "pitch" of such a sound? You may find the answer a bit surprising.
You can read about it and hear it demonstrated at the URL below. A regular pattern of nine harmonics is played, then another tone is played which is the same as the first, except that its fundamental frequency has been removed. Then the next harmonic up is removed, then the next, and so on. The figure below the link is a spectrogram made with WASP of a similar audio file. If you like, you can try making a spectrogram of the sound yourself with Praat to confirm the harmonic structure of the tones.
Here is Wikipedia's entry on the Missing Fundamental, with a sound file:
More information on missing fundamentals and difference frequences with a two-tone whistle:
Page with sound samples:
The fundamental and first two overtones above it are missing!
The pitch your ear and brain "hear" is in each case not based on the harmonic with the lowest frequency; you "hear" rather the tone as having the pitch of the original fundamental frequency, even when it is not physically present in the signal! Why does this happen? Well, very simply, it would seem that it is the harmonic structure that determines our perception of pitch, rather than simply the frequency of the lowest harmonic that is physically present in the signal. It is as though our brains calculate the difference in Hertz from one harmonic to the next to decide what the real "pitch" of the tone is. This is called a "difference tone". When you hear two pure tones, the ear and brain subtract one frequency from the other, and you "hear" a tone with a frequency of this difference. As a further example: if you've ever played with a two-tone whistle you may remember that when blowing, you heard a third, lower tone in addition to the whistle's two original tones. Amazing?
This phenomenon is exploited in designing telephone systems and small stereo speakers. How? Go on to the next page and find out!
Next: Forry, wrong number! I: The frequency ranges of speech and hearing
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