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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