Recommended: live jazz NYC venue, 'Bar Next Door'

It's not in N.Y.C. But if you are ever in Los Angeles there is a place called Vibrato that is owned by the legendary Herb Alpert. (He even plays there once in awhile) It has wonderful acoustics and has live Jazz 6 nights a week. Great venue.

Tiny place on Christopher St. The 55 Bar.This place is really excellent for local badass jazzbos...David Bowie scouted a few guys from this club and used them on his new Black Star record!! Check it out if in NYC.

Scott,
Live acoustic jazz in an intimate environment is a terrific template  for natural sound.  I've been frequenting these types of venues for 25 years in the Detroit metropolitan area. Jazz is alive and well and  I'm very grateful for this musical rich environment. Your ears will certainly adopt and recognize true tone and sound of instruments and human vocalists. 
You'll discover  as a result of this exposure that to your ears some components /products get closer to that presentation than others.  Best of all you get to enjoy and appreciate the beauty and emotional power of talented musicians expressing themselves. I can't get enough of these  kinds  of experiences. 😃😃
Charles, 

In the west village is a place called Smalls. In a basement, about 25 seats. Excellent acoustics. Very little amplification used.

Good for you; and exactly right.  If you've never been, be sure to check out The Village Vanguard.  Historic venue that reeks of jazz with an amazing vibe.  Nothing like live music.