Old strings are a different matter. Strings ideally obey Hook's law for small perturbations and undergo simple harmonic oscillation with standing waves (overto

Old strings are a different matter. Strings ideally obey Hook's law for small perturbations and undergo simple harmonic oscillation with standing waves (overtones) that have nodes at the neck and bridge (or fret and bridge). With old strings in particular, the fundamental of the string changes over time (as the string sounds). Audibly so! My mental model is that the "length" of the string is a really fuzzy concept for [worn] strings in motion. Plucking the string deforms it, and the string slowly retracts back to the stable length (at rest) as it dissipates energy. I haven't done any math or seen any, but I don't think its linear, either. You can easily verify this using an electric guitar if you plug it into a tuner thing with a visible meter - the arm will swing around. Also, I haven't done any physics in over a decade because I write software now...