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How we hear pitch and tone

The relation of frequency to pitch


Blowing a flute produces an almost pure tone at a definite pitch. The waveform is almost a pure sine wave. The sound wave from a violin or viola is more complex, arising from the saw-tooth force with which the bow pulls at the violin string. It is well known that any periodic waveform can be built up from a numbers of sine and cosine waves each of whose frequencies is a multiple of a fundamental -- this is Fourier's theorem. The multiples are called harmonics or overtones. A note on the violin and viola may have 12 or more harmonics. The figure below shows the waveform in time of the open C string of a viola and under it its frequency spectrum from 0 to 2000 Hz.

The story is more complicated because the wooden box parts of violin and viola are not able to radiate sound at low frequencies. As a consequence the fundamental frequency is missing from all notes played on the bottom (C) string of the viola, and is similarly weak on the G string of a violin. You can see in the spectrum above that there is no peak at 131 Hz -- the first noticable peak is at 262 Hz, the first harmonic at Middle C. How can it be, therefore, that we still hear the sound as being low in pitch? After all, a descending scale played from Middle C (C4) on the viola down an octave to C3 does not start sounding an octave higher as the player moves down the string.

The answer to this paradox lies in the way our brains make sense of a spectrum of frequencies in terms of 'pitch'. Listen to this first sound clip. You will hear 13 notes each lasting 2 seconds. The first and the last are the bowed viola open C string. The second is the pure fundamental f0=131Hz. (You need a good bass on your speakers for this.) The next has the first harmonic added, so is f0+f1. The third note has the second harmonic added -- f0+f1+f2, and so on with one higher harmonic being added until f10 is there too. In any note all harmonics have the same amplitude. The overall amplitude has been set to the same level for all the notes.

The diagrams below show the waveforms and frequency spectra of four of these notes.

Although the tone of these notes is different -- they become brighter and have more 'edge' as harmonics are added -- you may accept that they all have the same pitch. Look at the last waveform above, corresponding to 5 peaks in the spectrum. It is clear thart this has the same repeat period as the pure sine wave.

Now listen to this second sound clip. You will hear five 2-second notes: 1) f0+f1+...+f9+f10 as before, 2) f1+...+f9+f10 -- the fundamental f0 has been silenced, 3) f1+3*f2+f3+...+f9+f10 -- fundamental absent and harmonic f2 three times greater than the others, 4) f1+3*f2+f3+...+f9+f10 with amplitudes made similar to those in the viola spectrum above, 5) the open C note on an actual viola.

When the fundamental at 131 Hz is silenced, we feel that the note has lost its bass fullness, but you may still agree that the note has the same pitch. We get a clue to why from the waveform below which has only f1+f2+f3 present in equal strength. The waveforn is not dissimilar from that above for f0+4 harmonics. Essentially it has the same period as the fundamental even though the fundamental itself is missing. Remarkable! So we hear the note at the pitch of the fundamental even when the fundamental is not actually there. The brain must be making sense of the complex sound, with several frequencies playing at once, by picking out the period of repetition and interpreting that as the perceived pitch.

Actually, I think it is more complicated. If we just heard the viola note in isolation, without being told what it was, we might be uncertain whether its pitch was C3 or C4. I suspect that the musical context guides our hearing.

We can also ask what happens when the partial frequencies are not simple integer multiples of a fundamental, whether present or not. Such sounds are called 'anharmonic'. This third sound clip has two 2-second tones, the first harmonic,the second anharmonic. The frequencies in the first are 262, 392.5, 523, 664, 784 Hz, being exactly the 1st to 5th harmonics of C3 (131 Hz -- not itself present). For the second tone these freqencies have been alternately raised and lowered by 8 Hz to 270, 384.5, 531, 646 and 793 Hz. I hear a slight rise (sharpening) in pitch by about half a semitone (3%). Or do you hear two notes -- a two-note chord? Below are the waveform and spectrum of this anharmonic tone. The waveform is not periodic, but the peaks and troughs are correlated roughly every 1/135 seconds. This suggests that the brain identifies the pitch of complex sounds with correlation times in the wave amplitude.

Read here my longer article on the missing fundamental and how our ears hear the sound of a violin. It describes sound amplification within the outer ear, transmission from air to fluid by the middle ear, and how frequency in selected at the basilar membrane of the cochlea in the inner ear, and amplfied and filtered further by the hair cells.

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