Finest subwoofer for music under $1,000?

A used REL strata 3 is quite good for that price. You might even get one of their larger models. RELs are very subtle ... to the point where I often can't tell that mine is doing anything until I turn it off and the music loses all depth. Integration is that good. Subtlety and integration are the keys with classical music.
You really need to tell us your main speakers and your room size, and whether you have a preamp output to which to connect the subwoofer in order for others to make other suggestions.

As suggested REL is unbeatable when it comes to music and seamless integration. REL matches with any speaker you have provided you hook them up to the power amp output with your speakers ( kind of biwire which on the sub is called hi level). You run your main speakers full range and get the sub to mate with it, which means no nasty crossover glitches or distortion is introduced in the signal chain.

Strata III is a sealead unit which is even easier to blend with the room and speakers. Strata II was ported. All the models above Strata are ported. If you have a small to medium room sealed is better.

vmps -take a look

Its a stretch but for around 1K(used-usually abot 1200.00) you can get a Martin Logan Depth. It integrates with 2 channel as good as I've ever heard,and for surround its incredible.