You have not boken it correctly along phrase boundaries. The relevant phrase is not "on the note" but "(on the) note of optimism". com/us/definition/american_english/note ) This "note of ~" is a standard pattern, similar examples being "note of caution", "note of concern", "note of regret", and so on.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.