The deliberately dense servant

Grumio takes on many characteristics of the stock characters from the Commedia dell'Arte known as the ‘zanni’ (the servants). MORE on the zanni?
More on the zanni: These characters were clowns who enjoyed practical jokes and intrigue but could also be quarrelsome, cowardly, and treacherous. The horse-play and clowning that takes place when Petruchio and Katherina arrive home after a long journey in terrible weather is seen as the traditional antics of the ‘zanni’.     
Grumio is an ‘ancient’ sturdy, grouchy commentator on life, akin to the Porter in Macbeth, who resents his life of service. He has endured years of ‘trusty’ service with Petruchio and enjoys gaining the upper hand by frustrating his master (and later mistress) when possible. From the moment of his arrival in Padua, with a comic routine in which he misunderstands Petruchio’s instruction to ‘knock me here’ (on the door not on the head), Grumio is a source of humour for the audience. He is described by Hortensio as ‘ancient’, whilst Curtis jokes at his shortness of stature.

Cynical commentator

Because of Grumio’s frequent asides, the audience is drawn into colluding with his cynically astute observations on the other characters. He acts as a foil for Petruchio, 
bearing the brunt of his master’s short temper and mercurial spirit, whilst trying to copy Petruchio’s verbal dexterity. However, being of a lower status than Tranio, Grumio’s wit usually results in his come-uppance. 
When Grumio does have the chance to exert power, as when he has news of Petruchio’s arrival for the waiting servants in Act 4 Scene 1, he relishes it. His wry observations make him a good story-teller, entertaining his fellow servants with fine, comic detail on the mishaps of the journey made by Petruchio and Katherina: 
But hadst thou not crossed me, thou shouldst have heard how her horse fell and she under her horse; thou shouldst have heard in how miry a place, how she was bemoiled, how he left her with the horse upon her, how he beat me because her horse stumbled, how she waded through the dirt to pluck him off me, how he swore, how she prayed, that never prayed before, how I cried, how the horses ran away, how her bridle was burst…
Act 4 Scene 1     
He clearly enjoys the esteem of the lower status servants, even if not of his master.

Power play

Throughout the play the curmudgeonly Grumio manipulates situations wherever he can find advantage. In Act 4 Scene 3 he refuses to give food to Katherina and, while he may be simply a servant copying his master, there are hints of a more vindictive personality as he taunts Katherina with the food he refuses to give her:
I fear it is too choleric a meat.
How say you to a fat tripe finely broil'd?
Act 4 Scene 3    
Later he tries to assert superiority over the Tailor and to avoid being blamed for his part in any confusion. However, he ends the play, ‘in his place’ serving at the wedding breakfast but with no lines to say.
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