The answer to your question is yes and no.
Residential single phase power works like this... you have two hot wires and either a neutral/ground or a separate neutral and ground. For now, we'll forget about the ground, so you have two hots and a neutral. The neutral is electrically centered between the two hots.
In the breaker panel, 120v branch circuits connect to one hot and a neutral, giving you 120v. When something like an air conditioner, dryer, stove, or hot tub requires 220v, it connects to the two hot wires, giving it 220v.
Now here is where things get tricky...
Certain appliances require BOTH 120 and 240v internally to operate. For example, in a stove, the oven's elements are 240v, but the burners are 120v. In a dryer, the heating elements are 240v but the drive motor, drum light, etc are 120v. In these setups, the circuit brings all 3 wires (4 if there is a separate ground). Then the appliance breaks it out internally and connects internal components as appropriate.
Other appliances, such as air conditioners, only require 240v. In those cases, only the two hots and a ground are brought out to the unit, as the neutral doesn't get connected to anything so it would be a waste of wire.
Your hot tub... I have no idea whether it's 240 only or 120/240. If it's 120/240, there is a neutral conductor present in the box and what you want to do is possible. If it's only 240, then you are SOL, EXCEPT if your house's service entrance (the main wires from the power company) is a 3 wire (which gives you 2 hots and a ground/neutral shared).
See, the hot tub will definitely have a ground there. If your house has a separate neutral and ground and a neutral wasn't run to the tub, you CANNOT use the ground as a neutral. It might technically work, but it's dangerous and a very serious code violation. If your house uses a shared neutral/ground, then you CAN do it because the ground will definitely be there, and it's legal to use a ground as a neutral in this scenario.
I hope I'm still making sense.
Long story short, if you aren't up to snuff on electricity and the electric codes, then you probably want to have an electrician come in and evaluate your setup. He (or she) will need to look at your service entrance panel and your hot tub wiring. If it's possible to do, a sub-panel will have to be installed outside. Your hot tub probably has a 30-50a breaker in the main breaker panel. You can't just tap off of a 50a circuit and install a 20a outlet, as a 20a circuit needs a 20a breaker. So, new breaker will need to be installed outside for the hot tub, along with branch breakers for your tank. All of this will be expensive, perhaps more expensive than just running a new circuit for your tank.
That said, 1100w is not bad. A 15 amp circuit can supply around 1875w, and I doubt your entertainment center draws more than 700w, at least not continuously (bass punches during loud movies and such pull little spikes, but circuits are designed to handle that).
