What does "You would play upon me; you would seem to know my stops" mean in Hamlet?

This quote means that Hamlet is aware that Guildenstern and Rosencrantz are trying to manipulate him for Claudius’s benefit.

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In act 3, scene 2 of William Shakespeare’s Hamlet, the eponymous antihero tells Guildenstern, “You would play upon me; you would seem to know my stops.” Hamlet says this in a somewhat snarky manner. The sentence lets Guildenstern and his partner, Rosencrantz, know that he is aware of what they’re doing. He’s cognizant that they’re manipulating him and serving as spies for Claudius. Although they are supposed to be chums from school, Guildenstern and Rosencrantz aren’t around to help their friend deal with his grief. They’re working for Hamlet’s nemesis. Their job is to manipulate Hamlet and gain valuable information that Claudius can use to his advantage.

Thus, when Hamlet says “play,” he likely means manipulation. Guildenstern and Rosencrantz “play upon” Hamlet the way that one plays upon a pipe. Hamlet seems to be slyly comparing himself to the musical instrument that Guildenstern can’t play. Cheekily, Hamlet notes that Guildenstern won't try to play (manipulate) a wind instrument and make a sound come out of it, yet he can try to play (manipulate) Hamlet and make sound (potentially useful intel) rise out of him.

The second clause—“you know my stops”—speaks to the misplaced confidence that Guildenstern and Rosencrantz have when it comes to their job. They’re supposedly familiar enough with Hamlet to be able to push the right buttons in order to make him talk. Unfortunately for them, Hamlet can't be played so easily. He shows no mercy for their attempts at manipulation. Later on, Hamlet engages in his own manipulation, which gets Guildenstern and Rosencrantz killed.

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Hamlet has been deeply suspicious of his former college friends Rosencrantz and Guildenstern since they arrived in Denmark. Early on, he compels them to admit that his step-father/uncle, King Claudius, sent for them, and so Hamlet believes that they are reporting on him to the king. They are, but they believe that they are doing so to help Hamlet; Claudius has asked them to find out what's bothering the prince so that he can help Hamlet, but Hamlet believes their motives are less altruistic. They are sent to tell Hamlet that he has shocked his mother, Gertrude, and that she wants to see him, but Hamlet is pretty unfriendly to them. Rosencrantz is saddened by Hamlet's behavior, saying, "My lord, you once did love me" (3.2.310). Likewise is Guildenstern, who says, "O my lord, if my duty be too bold, my love is too unmannerly" (3.2.322-323). In other words, he means that if he is forgetting his manners around Hamlet, it is only because he is worried about the prince. However, Hamlet does not believe them.

When the actors enter with the recorders (a simple type of wind instrument), Hamlet asks Guildenstern to play one. Hamlet insists and Guildenstern declines, again and again, saying that he does not know how. Hamlet then uses a metaphor, comparing himself to one of the recorders, saying,

Why, look you now, how unworthy a thing you make of me!
You would play upon me. You would seem to know my
stops. You would pluck out the heart of my mystery. You
would sound me from my lowest note to the top of my
compass. And there is much music, excellent voice, in this
little organ, yet cannot you make it speak? 'Sblood, do you
think I am easier to be played on than a pipe? Call me what
instrument you will, though you can fret me, yet you
cannot play upon me. (3.2.336-344)

Hamlet believes that Rosencrantz and Guildenstern are trying to manipulate him, to convince him to tell them all his secrets. However, he says, they cannot even make a recorder, a much simpler instrument than he, and yet they believe that they can make him talk? Hamlet insists, using some wordplay, that it is easier to "play" a recorder than it is to "play" him (to trick him into revealing his "mystery"). He is angry, of course, because he believes these one-time friends are now attempting to manipulate him, as though he were simple or stupid.

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This play on words that Hamlet utters to Rosenkrantz and Guildenstern (IIIii, ll. 364-5) shows that he is not fooled by their false friendship.  The whole speech is a musical metaphor;"pluck" refers to lutes, a recorder (flute) has "stops" or "frets" --holes that vary the pitch of the notes.  When Hamlet says "Call me what instrument you will, though you fret me, yet you cannot play upon me" he is continuing the metaphor, using the players' presence and recorder to make his point.  ("Fret" is a play on words also.)  He is saying that, if they cannot even play"this little organ," a recorder, how can they expect to manipulate as complex a person as Hamlet? The metaphor was introduced earlier (IIIii, ll. 70-71) to Horatio, a true friend: "That they are not a pipe for Fortune's finger/ To sound what stop she please."

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