For all you younger folk, Pitch is irrelevant (except when it is grossly different). Those of us who are very senior citizens have found that middle C has dropped to about A when we go for the songs. ( And I will leave this list for a few days to spend a long weekend with my fellow dotards singing our group songs of the late fifties. No, not Elvis, the Princeton Tigertones of the late fifties.).
As you all know I'm new to lute notation, but as I look at the French notation I see no absolute at to a pitch, nor any key signature (I'm sure I'll be corrected on this). The old German notation had a key called H (and I have a modern fugue written in the sequence BACH, in honor of the composer, I'd have to pull out the music to see what note the H was, but it was in the Western chromatic scale. As I'm playing with string lengths and guages for a new thing I'm doing I'd guess that the change of materials may have changed the base pitches (not on organs, of course). Perhaps what happened was that instruments were made, and stringed with what was available, and the instrument/string combination defined the pitch. But then when ensemble, or orchestral, music came in vogue there had to be a standard made. Pure speculation, interested in comments. Best, Jon