# What are the most common overtones of a vibrating string? [duplicate]

EDIT: You can tell the degree of uniqueness of this question by it's very noticeable contrast in answers compared to the "duplicate". The answers to the question this "duplicates" are vastly different. The simple test for duplicates is: are the answers the same, which they are not. Perhaps the answer to this question could be gleaned from the answers to the other question, but simply asking a less broad question does not make it a duplicate of more broad ones or vice versa.

I'd like to know what the most common overtones of a vibrating string are. My guess is that in terms of frequency ratio and order of amplitude, they are:

``````2.0, 4.0, 1.5, 8.0, 2.25, 16.0, 1.333

Translated to 12 tone equal temperment:
oct, 2oct, P5th, 3oct, 10th, 4oct, P4th
``````

This is just my guess, based off of nothing but my perception of consonance. What I wish I had was a study of different strings in different instruments being hit in different ways followed by a simplified frequency spectrum listing the 6-10 loudest frequencies in order and with amplitude and frequency relative to the loudest.

I understand that this is may be a very open ended question, but I also believe that there are some things which are always true ie. octaves are more resonant than diminished 5ths.

Also, I found a link sort of explaining piano overtones, but only briefly: https://en.wikipedia.org/wiki/Piano_acoustics#Inharmonicity_and_piano_size

• Why are you guessing "based on nothing" when there are hundreds of music sites (not to mention Physics sites) which explain overtone theory in complete detail? Oct 7, 2016 at 11:38
• Nice try. First of all, there's no such thing as non-integer overtones. Wherever you got those numbers is misleading at best. Second, there's no such thing as "degree of uniqueness". Third, there's absolutely no unambiguous "more resonant" concept. Resonances' strengths depend on the mechanical body involved -- and how the vibrating source was energized in the first place. Oct 7, 2016 at 19:16
• 2.0 is a more common strong overtone than 59.0 . It was a question I had, as noobish as it may be, so I asked. Thanks for the link to related link though. It's a good read. Oct 7, 2016 at 19:24
• Seph - we have a Be Nice rule here. Remember this please. I have deleted that comment. Oct 7, 2016 at 22:12
• I'm totally down for being nice. I would like to point out that trying to make a person look dumb for asking a question might be out of those bounds, though it is not unheard of here. Oct 9, 2016 at 1:58

To a reasonable approximation the overtones of the vibrating strings used in musical instruments are integer multiples of the fundamental. In a tablular format similar to the OP:

``````1.0, 2.0, 3.0, 4.0 ...
``````

These correspond to the string modes:

The relative amplitudes of the degree to which these modes are excited depends on the details of how the string is excited. The first is position along the string: the modes with the larger amplitude at the site of excitation will be more excited. E.g. striking the the center of the string will only excite the odd numbered frequencies. The nature of the excitation: whether it is bowed, or pulled and then released or struck with a hammer affect the relative amplitude of the modes too.

The body of the instrument has its own resonances too; these end up affecting the relative amplitudes as energy is transferred between the string and the body.

Fundamentals of Musical Acoustics by A. Benade provides several chapters that explores the physics of strings, and addresses your question in more detail, in a way that does not go too far afield into complex mathematics.

Even when you consider piano strings, which are thick and stiff enough to exhibit inharmonicities, the usual way to think of the situation is to start with the "ideal string" approximation, and then worry about the deviations away from this as a 2nd order effect.

• ` striking the the center of the string will only excite the odd numbered frequencies` Woah! So the hammers in pianos could all be placed at exactly the center and essentially create a square wave? Oct 6, 2016 at 19:53
• @SephReed square wave would require the right amplitude (A ~ 1/N) and phase relationships; I wouldn't expect that to actually occur.
– Dave
Oct 6, 2016 at 19:58
• As you can see with @Dave's nice illustration, the integer divisions are the basis for the harmonic series (i.e., fractions 1/2, 1/3, 1/4, ..., 1/n) and THAT series is the basis for just intonation. Math and music have some interesting relationships. See en.wikipedia.org/wiki/Just_intonation for more details. Oct 6, 2016 at 21:50
• I read the Just Intonation Primer a few years ago. I suppose I kind of forgot about the string part. Oct 7, 2016 at 17:49