Budget for headphone is $400USD. I don't mind open headphones. I will buy a headphone amplifier that is suited for the headphone afterwards.
The Etymotic ER-4P has a very flat frequency response and is very detailed.
My musical tastes range from classical to death metal. But most of it is jazz fusion, rock and classical.
For well under you budget--AKG Q701 and Beyer DT 880 (+/- Pro)--250 or 600 Ohm version.
For more HiFi HE-400, Sony MDR-SA5000 (hard to find and needs no amp).