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Hey there fellow physicists & musicians!

I have a question both physics and music related.

How does the string tension affect the sound spectrum? More precisely, how do the respective amplitudes of the nodes (fundamental vs harmonics) change with string tension, like a population distribution?

Imagine a vibrating string with different lengths and tensions, yet with the same frequency (i.e. a long vs short neck guitar tuned the same).

From my feeling the sound shifts towards higher frequencies for more tense string, appears somewhat more compact mid-focused, while more loose strings sound a lot bassier (at the same frequency). Of course intrument, wood, etc. do have effects but I was looking for a more fundamental approach.

Any ideas?

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The physics in the 1st approximation are worked out here:

https://en.wikipedia.org/wiki/String_vibration

with the result that frequency depends on the length, tension, and mass density as:

$ f = \frac{1}{2L}\sqrt{\frac{T}{\mu}} $

The 1st factor is why bass strings are long and treble are not (on a piano). The lower 3 strings on your guitar have a tension producing string wrapped in wire (dead weight) so that you can increase $\mu$ while still maintaining a reasonable tension (to low: it sounds bad, to high: snap goes the guitar).

Of course, strings on instruments are not infinitely thin--and their finite width shifts the harmonics up from their canonical frequencies (inharmonicity). The n-th frequency is:

$ f_n = (nf_1)\times [1+(n^2-1)\frac{r^4\kappa}{TL^2}] $

were r is the string radius and $\kappa$ a material modulus (with units of pressure).

Longer strings are less affected by this, so concert grands are really long.

See the Railsback curve to see how this affects piano tuning in surprising ways. (For a physicist, it as fascinating "fail" of the 1st approximation).

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