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I know when a string is plucked on a violin you can hear many overtones at harmonic frequencies, but where do these overtones come from?

Does an ideal string vibrating in a vacuum vibrate at the fundamental and harmonic frequencies? Are the harmonic overtones caused by the shape of the resonating body? Are they from the materials used?

"in a vacuum" will not make sound, but yes your thoughts are on the right track.
All vibrating bodies, strings, plates, beams, etc, have a natural set of harmonics (or overtones as described by some). These are usually determined by the boundary conditions on the vibrating object and a related to the fundamental tone by a simple relationship. For the ideal model of a string fixed at two ends the relationship is

f_n = n*f_1 (f_1 is sometimes called f0, the fundamental).

The fundamental is the lowest frequency of vibration supported by the object and is heard as the natural tone (for example on a properly tuned guitar the open string are named for the fundamental tone).

What excites the harmonics is the attack. Plucking a string at different points will produce completely different sets of allowed harmonics (only those supported by the boundary conditions will show up). Tapping a string or bowing a string will produce different harmonic content and this is what is heard as "tone" by listeners and musicians. Twangy versus warm, smooth, etc. are all adjectives that describe harmonic content.

As for materials, the specific materials do come into play for determining the fundamental but once that is known the harmonic sequence is fixed. For plates and beams the overtones are NOT related by a simple relationship for all boundary conditions. Some vibrating systems can have dissonant overtones. The same applies to pipes, a.k.a horns, and percussion instruments.

What you observe is a physical property of all resonators. In the case of vibration every resonator has different modes of vibration. In the case of a drum head or a cymbal these modes are not harmonic, in the case of strings or air columns the modes of vibration wich are noticeable are harmonic.

Think of a string: It is fixed at both ends if you start and draw on paper the possibilities the string has to vibrate you´ll get one top or valley as the first possibility, one valley and one top as the second, "top valley top" or "valley top valley" as the next one and so on. (In reality a string of a bowed instrument is not moving this way, but thats not important for now)

Here is an illustration of four of the basic modes of vibration of that string:

These are the basic possibilities the string can move and if you don´t take special measures always when the first one is possible, the second and third and ... are also possible.

If you take this fact into account then the question reverses. How would we expect when we excite the first mode of vibration that there is not even a little energy transfered to the second, third and so on? After all all parts of the string are connected which each other, there are strands in the core and wraps and a bow is not a laser in a super cooled atomic trap so these modes are mechanically coupled.

If you manage to only excite the fundamental mode, the fundamental mode will excite the other ones and so on, simply because its possible to happen.

And thats a basic physical principle: When you transfer energy (e.g. with the bow) all accessible states (modes) will receive there share of it. The one with the lowest energy receives the most, the next one less, the next next one less less and so on. Which one receives how much depends on technique, location of the excitation, material, construction, and so on.

This principle holds true for every kind of excitation. Radio waves, Colors and so on.

In the case of stringed instruments you can make the higher modes of vibration more unfavourable by slightly touching the string effectively reducing the amount of energy that those modes receive that have strong movement in that position and killing those overtones or harmonics. Look for harmonic glissando.

Last but not least heres an animation how the motion of that theoretical string consisting of the 4 modes of vibration would look like:

That said a real string of a bowed instrument does not move in this way because of the stick/slip effect of the bow which creates the characteristic timbre of bowed instruments compared to e.g. a harp:

The answer to this actually depends on the instrument. There are three essentially different reasons why an instrument's overtones might be (more or less) harmonic, and at least two of them might be in play for stringed instruments.

1. The vibrating element resonates at a harmonic set of frequencies (as a string does), as described in the other answers. An ideal (perfectly elastic) string resonates harmonically, but in practice real strings have some stiffness which causes them to deviate somewhat from harmonicity, which is why piano tuning is hard. In general this is the complete explanation for the harmonicity of instruments with plucked or hammered strings.




2. The vibrating element is being continually excited (as in a violin being bowed, a wind instrument being blown into, or a larynx being sung through). In this case, over time it will tend to settle into a periodic state (this is a form of "mode locking", though if you search for that term online you will mostly find people talking about lasers). Because periodic waveforms correspond to harmonic partials, this means that instruments can be made to vibrate harmonically even when the underlying resonant frequencies aren't harmonic. The end result is much closer to being truly harmonic than it is under reason 1 (e.g., a violin played arco is much more harmonic than one played pizzicato), but it also is a lot more fragile and technique-dependent (it's harder to get good tone bowing a violin than it is plucking a violin).




3. Fakery. For the most part, pitched percussion instruments (xylophones, bells, timpani, and so on) don't really have harmonic overtones. However, they've been carefully tuned so that the bottom few overtones are as nearly harmonic as possible. This is good enough to trick us puny humans into hearing them as (more or less) pitched.

Some further notes:

• Reasons 1. and 2. often operate together. Most orchestral instruments, for example, have vibrating elements which are either strings or columns of air (and thus are roughly harmonic for reason 1.) but which are then coerced to be exactly harmonic because of reason 2. On the other hand, the harmonicity of the human voice is entirely a consequence of reason 2.; the human body doesn't contain any intrinsically harmonic vibrating elements, but we are capable of vibrating periodically anyway!


• Often 3. is helped by choosing shapes which don't have very many overtones to start with. In a perfectly rigid freely vibrating one-dimensional rod, the second partial is roughly 2.8 times the frequency of the fundamental, and the third partial is roughly 5.4 times the fundamental. The shapes of xylophone bars are tweaked so that these partials will be at 3 and 6 times the fundamental instead, but it's certainly helpful that there were only two partials in that frequency range to tweak (where there would have been five for an actually harmonic instrument).

If you're specifically concerned with string instruments, then — as my examples above have hopefully suggested — you still need to worry about both 1. and 2. in general, but probably not 3.

xerotolerant is correct that "overtones" is the correct word in this case. Harmonic and overtone can sometimes be used interchangeably, but not in this case.

This is a big topic, but the short answer is the shape of the wave, which is called the "wave form."

We often depict waves as being simple, smooth curves, but they are, in fact, much more complicated. If you digitally record a sound and then zoom in really, really close, you will see that the wave form is very jagged and irregular. This shape is what gives each instrument its unique characteristics, including the levels of the various harmonics.

So, to answer your specific questions:

Does an ideal string vibrating in a vacuum vibrate with harmonics?

Yes and No, the vibrating string will still create a wave, but the wave will over no medium (i.e. air) to travel through.

Are the harmonics caused by the shape of the resonating body? Are they from the materials used?

Yes to both. All of the properties, e.g. the shape, size, material, performance technique, etc., combine to create the unique sound that you hear.