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.

      On this page, you can first hear a tune played with a simple sine wave, then the same tune played with the first nine harmonics. In the third sound file, the first three harmonics have been removed:


     The timbre is somewhat different, but you hear basically the same tune at the same pitch.

     So 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".

     You can hear and see this phenomenon explained and demonstrated here:


     Another demonstration:


     Here is Wikipedia's entry on the Missing Fundamental, with a sound file:


     To help you visualize what's going on, the figure below shows a spectrogram of the same pitch with the lowest frequency successfully removed. If you like, you can try making a spectrogram of the sound yourself with Praat to confirm the harmonic structure of the tones.      

     Additionally, when you hear two pure tones, even when they do not have the same harmonic structure, the ear and brain again subtract one frequency from the other, and you "hear" a lower-pitched tone with a frequency equivalent to this difference.

     For example, if you've ever played with a two-tone whistle (e.g. a
(London bobby whistle another example), you may remember that when blowing it, you heard a third, lower tone in addition to the whistle's two original tones. This is called a Tartini tone or combination tone.

     If the difference between the two pitches is about 70 Hz or less, you usually hear a pitch that is an average of the frequencies of the two tones, with a "beat". The rate of the beat will be determined by how big the difference in frequencies is.

     When the difference is greater than 70 Hz, however, you will hear a lower third pitch equivalent to the difference between the two frequencies.

     More information on missing fundamentals and difference frequencies with a two-tone whistle:

     Actually, the same thing happens with "sum tones", i.e. if you play two simple tones, you will also hear a tone with a frequency of the sum of the two tones, e.g. you will hear a faint tone at 900 Hz if you simultaneously play two pitches at 400 Hz and 500 Hz. 

     See this video for more details: How To Play Notes That Aren't There      

     The difference tone 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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