To give a specific example of how key and temperament affect notes on the dulcimer, if you had 3 dulcimers built to be tuned (EBE) and
one was "equal temperament"
one was "mean tone temperament"
one was "just intonated"
G# would be flatter on the "just intonated" dulcimer than it is on the equal temperament dulcimer
G# would be sharper on the "meantone" dulcimer than it would on the equal temperament dulcimer.
That is not a common tuning, but it is an example of one where G# is quite different for each temperament, since you mentioned G#.

It depends on what temperament is being used, as Skip also mentioned. In equal temperament, G# is the same as Ab. There are other temperaments (systems of tuning used at different points of history and/or in different cultures) where G# may not be the same as Ab.

In addition to that, players of fretless or other instruments capable of microtones may choose to inflect notes depending on what follows them. For example, leading tones (like G# in the key of A) might be pushed sharper in order to accentuate that leading to the tonic.

To complicate things further, due to acoustical physics of various instruments, there are often tuning peculiarities regardless of what temperament the instrument maker was going for. Clarinets, for example, overblow an interval of a 12th- but some fingerings end up producing wider or narrower 12ths, even on high end instruments. It's up to the player to be able to compensate for that. And of course some ears are more sensitive than others to even notice.

More than you probably wanted to know for your question!

It boils down to the fact that the frequencies of the notes are not exact/even and do not double [times 2] as is commonly accepted. This is the basic reason the frets are not evenly spaced and results in the 'Rule of 18' and '12th root of 2' for fret placement. I think 'wolf notes' are also part of this.

When the "G#" frequency approximately doubles, the frequency is not the same as the doubled frequency of the Ab. This difference is/can be audible.

When 'equal temperment' was developed, this difference was, essentially, eleminated.

Most of the other 'temperments' are attempts to modify this difference.

updated by @skip: 09/01/23 04:02:23PM

I though I understood the whole sharp/flat thing but now I'm confused. I have always heard for example a G sharp was the same as a A flat. But now I hear that pitch wise they are different at least on a string instrument but not on a keyboard. Can someone help me understand? Thank you! 

updated by @homer-ross: 09/02/23 12:49:28AM