Good budget sub for music?

Rel Q Series or a REL Storm III if your budget will allow. Very easy to integrate, very musical. The Q's won't take you to the bottom octaves, but are a very nice, fast, musical sub which it sounds like what you are looking for. I don't know your speakers, so I can say whether the Rel subs will be the right speed for them, but if you are crossing over at 80, you should be fine.

Paradigm PDR 12

DIY is your best option. I'd suggest one of these 308SWR51XXLS or 268SWR5XXLS woofers from Peerless and a heavily braced sealed birch wood box - appropriately sized of course - for example the 308 works with 2.6 cu ft in a sealed configuration ( I'd also stay away from reflex or passive radiators if you like high quality musical transient response).

You can get the whole shebang here for less than 500 bucks

This indeed will outperform most all subwoofers on the market - but you'll have some work to do. I'd go for the damped pulp paper cones that Solen suggest over the metal/aluminium unless you are willing to risk audible out of band ringing noise from the bell-like tendencies of a metal driver...remember metal only looks cool...and you are not going to enjoy your precious midrange if the sub is audibly ringing...

This will work well with your Peerless woofer/midrange. Bracing the box internally will be very important - you don't want cabinet waffle. Good Luck. And don't ask me details aboiut woodwork - I am not retired yet and simply don't have time for this - one day perhaps...

Mirage S-10. I have been using its predecessor, the LF-10, for 3-1/2 years in a 2.1-channel music-only system. It's a front-firing 10" with dual ports, 150w internal amp with 400w peak power, continuously adjustable output, continuously adjustable crossover from 40Hz to 120Hz, and a 0-180 deg. phase switch.

I mention all this because the Mirages are exceptionally easy to blend with satellites. I've placed Mirage subs in two of my systems and also in a sat/sub 2.1-channel setup in my neighbor's living room. The Stereophile test CD demonstrates that these subs extend down to the mid-20s.

In the 3-1/2 years I used this arrangement, the Mirage sub was unfailingly musical and lively, blending seamlessly with the satellites. It never sounded like a home theater LFE "thump machine" sub, but always supplied an agile and timbre-correct foundation to the music.

And I played every type of music with this setup--folk, acoustic and electric blues, acoustic folk, acoustic jazz, big band, orchestra, chamber music, pop, rock, opera, bluegrass, electrified country, and choral music on SACD, CD, and LP.

I recently replaced it and the sats with a pair of full range floorstanders because I wanted to increase the overall radiating area to fill the open-architecture living space better. HOWEVER, I can't say that doing so particularly improved the bass-to-midrange continuity over the Mirage sat/sub arrangement. Either I'm uncommonly lucky or they're that good at blending musically.