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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?

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    @Tim Harmonics refer specifically to integer multiples of the fundamental frequency. Overtones refer to any resonant frequency above the fundamental frequency. An overtone may or may not be a harmonic - taken from here – xerotolerant Dec 13 '18 at 18:04
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    @xerotolerant Okay, if you're talking about the string instrument technique harmonics, then fine, overtone is not interchangeable. However, OP is clearly not referring to the technique ,OP refers here to the natural overtones/harmonics present in every non-sine-wave note, and as such, the two are equivalent. – user45266 Dec 13 '18 at 23:17
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    Thank you for all of the feedback on my question. I'm sorry my original question was unclear. I've edited it to hopefully get past some of the pedantic differences between harmonics and overtones, and the misunderstanding of harmonics as related to other musical concepts (e.g. guitars) – nanotek Dec 13 '18 at 23:49
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    @jdjazz It seems to me that Op is asking about just plucking a note normally. – user45266 Dec 14 '18 at 5:26
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"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.

  • Comments are not for extended discussion; this conversation has been moved to chat. – Dom Dec 14 '18 at 17:43
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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.

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    "which is why piano tuning is hard": it's also one reason why longer pianos have a better sound in the bass. The strings can have a lower mass per length, so they are wound less heavily and are not as stiff, so they behave more like ideal strings than do the bass strings of shorter pianos. – phoog Dec 14 '18 at 16:19
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What you observe is a physical property of many resonators / oscillators. 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 something linke that: (In reality a string of a bowed instrument is not moving this way, but thats not important for now)

enter image description here

These are the basic possibilities the string can move.

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.

The excitation by the bow simply isn´t precisely fitting to the fundamental so it excitates so harmonics too. This can also be seen by the fact that changes in the way the string is "plucked" (or bowed) also change the timbre, that is the relative composition of the vibration spektrum and thus the overtones.

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:

enter image description here

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:

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    “a physical property of all resonators” – um, not really. Of all oscillators with a continuous space extend, perhaps. Also I don't agree with what you're saying about modes that are excited “simply because they can”. I hesitate to downvote this answer because in particular the end is spot-on (stick/slip effect), but most before that is pretty much bogus physically. – leftaroundabout Dec 14 '18 at 12:22
  • Why are the first four "basic"? I don't understand that. Other than that I like your animation. – ggcg Dec 14 '18 at 12:40
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    How does the fundamental excite the other modes? In a linear system this is impossible. Can you edit your post to describe this better. Are you assuming non-linearity of the instrument? This certainly does not happen in the ideal system model. – ggcg Dec 14 '18 at 12:42
  • The characteristic tone of bowed instruments as opposed to the harp and other plucked strings also depends on the point at which the string is bowed or plucked. A harp string is typically plucked near its center, which results in a tone with minimal odd harmonics, giving a sound that has always seemed to me somewhat hollow. The same effect can be achieved on a guitar by plucking at the twelfth fret. Similarly, bowing closer to or farther from the bridge changes the tone of a bowed instrument (sul ponticello or sul tasto, respectively). – phoog Dec 14 '18 at 16:15
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    @ggcg Yep! Thats what I wanted to say. In no way special, just the first 4. – DrSvanHay Dec 14 '18 at 19:47
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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.

  • The OP cites the correct definitions of harmonics and overtones in the comments above. This shows that they understand the difference between harmonics and overtones, no? The phrase "ideal string" makes me think the OP does want to ask about harmonics specifically, and not overtones generally. – jdjazz Dec 13 '18 at 23:42

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