Inharmonicity

A guitar or other fretted string goes out of tune as it is played up the neck, and that is investigated for AW2 guitars and their strings. A way of fixing this is to move the bridge away from the nut a little, so that the second harmonic (12th fret) is exactly twice the open string first harmonic (or fundamental).    It turns out tension in the string is really important, and sometimes string stiffness is important.  “Sometimes” because it depends on what happens at the string ends.   If the ends are clamped then it really matters.  If they pinned (the same as hinged) it doesn’t really matter.  If one end is clamped and the other is hinged or pinned, it is somewhere in between. 

Guitar strings are complicated.  They oscillate up and down, and side to side, and in between.  That changes during a note.  And when you fret, does your finger clamp the string, or pin it?  Ask that question for both up and down, and side to side, motion. No one knows for sure.  So we set out to find out. 

It looks like they are between pinned (“doesn’t matter”) and up to half way between pinned and clamped (it matters). There is a suggestion that strings without an overwinding (like top e) are very close to being pinned, whereas the overwound strings (like low E) are very close to being halfway between pinned and clamped.  

Here are some technical notes on how that was deduced.