By "the close", Shakespeare means what we usually call the cadence - the conclusion of a musical "sentence" or of a whole piece. He makes John of Gaunt say that a piece of music, like a man's life, is more appreciated as it draws to an end than at any time before. How sadly true! Well, by Shakespeare's rule this piece should be well appreciated, for it includes an improbable number of pretty "closes", increasingly lingering and nostalgic.