It is possible to represent bird sound as approximate notes on a musical score. However, the notes would generally be so high, the chords so complex, the pace so rapid, and the subtleties so intricate that there is little reason to do so!
Instead, bird song is shown visually as sonograms, produced by a sonograph (or sound spectrograph). The frequency (effectively the pitch of the sound) is shown on the vertical axis, the time on the horizontal, so it is graphically similar to a musical score.
While not being so instantly interpretable as a piece of music, the lines and squiggles that represent the sounds show the structure of bird song with, for a bird such as a Wren, 100 notes packed into four or five seconds. The sonograph proved revolutionary in picking out things such as dialects in bird song, and in working out how adults and young birds can recognise each other's calls when, to the human ear, every one sounds identical.
Visit www.wildsong.co.uksonagrams.html to learn - and hear - more about them.