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When playing any note on any instrument, it is said that we also hear this note's harmonic series.

What causes us to hear this? Does our ear "invent" those sounds, or are they actually produced by the instrument due to some physical phenomenon?

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We hear harmonics because they are physically produced by the instrument; they are not "invented" as some sort of illusion. In fact, we often aren't consciously aware of them, though we can hear their effect on an instrument's timbre, or tone quality.

They are caused because when an instrument such a string vibrates, it actually does so at more than one frequency at a time. Certain frequencies -- those whose wavelengths are integer multiples of the length of the instrument -- are able to create standing waves, and are then amplified. Thus the sound that is produced is a combination of these integer multiples of frequencies: f, 2f, 3f, 4f...

There is much more detail to be found here:

Note that some of the answers to those questions focus on guitar, but the same applies for any string instrument (harps, pianos, violins, cellos, etc...), and much of it also applies to vibrating columns of air (woodwinds, brass, organs...).

  • And if it wasn't for harmonics, brass instruments probably wouldn't be around, since they rely on these to produce most of their notes. The bugle and hunting horn being the most basic.And I guess that harmonics themselves have their own set of harmonics? – Tim Nov 24 '14 at 9:19
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    "They are not invented as some sort of illusion" - actually, they can be. This is the theory behind a 'resultant' organ stop, for example. http://en.wikipedia.org/wiki/Combination_tone. See also my answer to This question. – Widor Nov 24 '14 at 9:36
  • As far as I understand, all sound creates overtones except sine waves. I'm not sure if I have a misunderstanding but I have heard more along the lines of: Sound is a vibration of the air or another substance; this sound acts as a wave; the larger wave (the fundamental) causes subsequent waves at a higher frequencies. If this is a misunderstanding, it may be the result of an analogy from when I was learning about it. Harmonics were compared to dropping a pebble in the water. You will get one large wave and as the wave dissipates, several smaller waves follow. Is this not accurate? – Basstickler Nov 24 '14 at 13:42
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    @Tim: Interesting fact about the harmonic series: it also contains all the harmonics of all its members. For example, consider a fundamental with frequency f, and its harmonic 3f. Then the harmonics of 3f include: 6f, 9f, 12f, 15f, etc... All of which are also harmonics of f (since integers are closed under multiplication). – Caleb Hines Nov 24 '14 at 15:41
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    @Basstickler I wouldn't say that sound "creates" overtones, so much as it contains them -- the overtones are directly created by the same vibrating mechanism that creates the fundamental, not as some kind of secondary effect. The irregular shape of a (non-sinusoidal) sound wave is a result of the combination of these overtones. – Caleb Hines Nov 24 '14 at 16:07

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