I had a recent issue with my high e string where the saddle was adjusted as short as possible but the twelfth fret was still very flat. I read online that a new set of strings may fix this, and that turned out to be a perfect solution. My question now is, from a physical standpoint, why does string age affect intonation? I can understand why old strings won't hold their tuning as long, but I don't understand how age can throw a strings intonation off.
I've been curious about this too. I imagine this is the same phenomenon that violinists refer to as a string "going false." The effect is the location of pitches becomes inconsistent along the length of the string, especially noticeable on a fretless instrument; i.e. if you tried to play, on two strings, a pitch a perfect fourth higher than the open string, the placement would have to be slightly different on the two strings (which is of course impossible with frets). I've also noticed, especially with an unwound wire string like the violin's high E, that plucking a false string can produce "two pitches," as if an E and another slightly detuned E were being played at the same time.
This page simply says "A 'false' string is one with uneven thickness and mass along its length." It suggests that it involves the "flattening" of the round cross-section of the string. This discussion thread is rambly and largely speculative, but I imagine much of that speculation is true, that it also involves changes to the elasticity of the metal. The fact is, for a wound string, there are so many parts that there could be a lot of factors. The string's core is perhaps a number of braided steel wires, which could become inconsistent in thickness, shape, or elasticity, and the winding could become deformed or perhaps have its "coils" spread apart or together.
Of course, if the new strings hadn't solved the problems, there could be other suspects as well. The wood of fretboards wears grooves between the frets over time, and frets themselves could become worn and be inconsistent from one string to another.
I can understand why old strings won't hold their tuning as long...
Actually, I would expect (and find, in practice) the opposite. New strings always have a "break-in period" during which they stretch and have to be retuned regularly (the period varies in length and intensity depending on the material), but once they've "settled in," they generally hold their pitch quite well (unless maybe eventually some mechanical flaw like we've been discussing impacts their stability).
tl,dr: Finger gunk.
It's the same reason why strings become dull over time. When playing. the oil, sweat and salt from your fingers interacts with the string and results in corrosion and accumulation.
This is more pronounced where you fret the most and this results in an uneven mass distribution and the center of gravity moves slightly away from the 12th fret. Hence it's out of intonation.
Probably the already given answers are also right, in the sense that the mentioned factor physically modify the harmonic response of the string. But I think soaking in ethanol for 24 hours will not make the string new, because of a physical phenomenon called "fatigue", that I guess plays the main role here.
As you know every material have his own elastic constant E (young modulus), in the case of steel it is around E=210 GPa. The fatigue is a phenomenon that ""make the material weaker"" when the same point of material is subjected repeatedly to many cycles of the same load. https://en.wikipedia.org/wiki/Fatigue_(material) You can see that e.g. the "main" LA string goes 440 Hz (around? sorry i'm just half musician), so it does 880 cycles from minimum tension to max every second that it vibrate. Every material have its own characteristic number of allowed cycles before suffering of fatigue (see Whaler function). From this number you can compute by simple division the number of seconds that the string can vibrate before suffering of fatigue. This phenomenon modify the value of this Young coefficient E (and also others like Poisson coeff., more complicated, just doesn't matter, but the idea is the same). But as you know the vibrating string doesn't have the same tension everywhere: every harmonic have his own wavelength. Assuming the string is vibrating sinusoidally, in a full period we have one max and one min points (the one with horizontal tangent line, i.e. derivative 0, for those who knows a bit of math) and those points will be the most strained ones. Instead, the nodes of the mode will be less strained ( this because the differential equation of the vibrating string is Yxx=k*Ytt where k basically is modified by tuning. xx and tt are partial derivations). https://en.wikipedia.org/wiki/Normal_mode To conclude, the string will fatigue not in a constant way, but differently in every point. If you add this to all imperfections (like that touching the string non uniformly will consume it non uniformly for example), after a while the string doesn't perform as new.