Mendelssohn famously expressed the view that his musical thoughts were too definite to be put into words. It is true that music can leave us lost for words, but that is probably because we are lost for thoughts. If we were thinking clearly, we would find the words. We should not blame words for our indefinite thoughts.
Music has no equal in expressing the play of emotions. The alternation of joy and sorrow, of stress and release, can be managed within the twinkling of an eye. Or ear! In a musical score, sign and sound are fused (and not confused!). A powerful musical score is immediate in its impact. One can tell at a glance that a work has character.
What is the link between skill in notation (or a flair for notation) and good thinking? One would seem to support the other. A notational problem can highlight a musical weakness. A notational tweak can lift a piece from the doldrums.
Notation packs a punch in calling for both rationality and formality. We can have the former without the latter, as when we improvise. We are thinking musically when we improvise, and Mendelssohn would have smiled on our efforts, as he admired the improvising genius of J.S.Bach.
In conversation with friends, we are free to be irrational or to lob gobbets of good sense into platitudinous seas, more pallid than incarnadine! We may even fly off at a tangent, leaving our listeners hopelessly, delightedly or frustratedly, in the lurch. We can be casual, or we can strive to raise our game; not out of vanity, but for the general good.
Writing, though, has the edge, by virtue of formality. The pen or pencil, virtual or real, can bring heat and light to the most half-baked of idle thoughts. Passionately, and fastidiously, we are moved to edit, trip up, go headlong, and find a fresh slant.
Similarly, we achieve a measure of precision through formal musical notation, and this encourages us to aim for further definition through refinement, those tweaks and adjustments prompted by the highlighting of weaknesses. Intuition plays a part, given the complexity of musical problems, but analysis is the best test of any solution. We can trust in our intuitions but we are the poorer if we make no attempt to account for them. Such an attempt, or analysis, must begin with observation and description. We observe and describe, using words to do so.