When I hear a song or a melodious tune, I can often figure out the relative pitches and play the tune on the piano. But I usually only play it on C major (the simplest scale for me--occasionally G major , because I often can't decide between "mi fa" and "si do" (E-F versus B-C). So if I were to write its score, I would write it in C major.
My question is: what about professional musicians? If they are to jolt down the score or play by ear when hearing a new music piece, how do they know or decide what key signature to write it in? Why would one write it in any one other than the simple C major?
Example 1: from hearing the first few notes (first 3 bars?) of the last movement of Cesar Franck's Violin Sonata (the violin part), my immediate reaction is either:
(a) "so, la, fa, mi-re, do", and play it on "G, A, F, E-D, C";
(b) "re, mi, do, si-la, so", and play it on "D, E, C, B-A, G"
But I guess neither is the original score.
Example 2: Auld Lang Syne:
I always thought the score were: (c) "so, do...do, do, mi, re...do-re, mi, do...do, mi, so, la---" = G C...C C E D...C-D E C...C E G A---
But I just read the score is actually: (d) "do, fa...fa, fa, la, so...fa--so, la, fa...fa, la, do, re---" = C F...F F A G...F-G A F...F A C D---
(Upon reflection, the keys in (d) are exactly the solfege in (c) but uttered in F major.)