Joe, Jeri & others:

In olden times, when instruments were actually tuned mainly by using harmonics and beats of one note against another, tuning was nearly always to the "harmonic" scale(s). In this system, there is a very real difference when you change the key of a tune.

With enharmonic tuning, a Bb in the key of F is not necessarily the same frequency as the A# in the key of B.

When Beethoven "pushed" the equal tempered scale into a degree of acceptance, the distinctions - for ET tuned instruments - became much less important.

Harmonic scale tuning still is used - on antique organs and in a cappella singing (e.g. barbershop), but for the most part the world works to the ET scale, and key changes don't have the impact they once did.

As a personal opinion -- t.i.c., persons who make a fetish of cross-tuning their git's to all of the "right" harmonics automatically make themselves "out of tune" if they change keys: but if they cain't hear that, then we'll let them do their thing.

Listen to a good barbershop or other unaccompanied group, to get a feel for how things work in the enharmonic world. In the real world, pitch varies so much with fret locations, bridge compensations, aggressiveness of attack, and dozens of other things, that "close enough" is really all that is obtainable - or required.

It has to sound good - and be fun.

John