The HD 650s ( Massdrop HD 6xx is identical except paint and cable) are an audiophile staple. If you're willing to spend $300+ on an amp for them they live up to their reputation. I have HD 600s and HD 650s. With lower end equipment up to an O2/ODAC I like the 600 better. Feed them with my class A Aune X7S with a balanced cable and the HD 650 takes the lead. Neither the 600 nor the 650 does well with bass heavy music though.
If you want neutral - warm with about the most amazing bass you can get in a headphone then check out the Massdrop x Fostex TH-X00 Ebony. The only slight weakness is soundstage but the imaging is amazing and makes up for it.
[video="https://www.youtube.com/watch?v=3pFM_hW33bw"][/video]
https://www.massdrop.com/buy/massdrop-fostex-tr-x00-ebony-headphones
I have both the Mahogany and the Ebony and in my opinion the Ebony plays with headphones that cost twice as much. It's as good with London Grammar as it is with Liquid Soul. The Mahogany is more V or W shaped and the vocals can get lost in the sub bass and treble. Both are very easy to drive but will really come alive when amped correctly.
Add an Aune X1S for a VERY nice DAC/ solid state amp combo and you're golden. Massdrop is a one stop shop here.
https://www.massdrop.com/buy/aune-x1s-32bit-384-dsd128-mini-dac
Now if you're willing to maybe break that budget limit a LITTLE bit the Massdrop x Focal Elex is an all around headphone that's more refined then the Fostex. It's basically a Focal Elear with Focal Clear pads on it for less money. I don't have this one but I do have an Elear and it's beautiful.
https://www.massdrop.com/buy/massdrop-x-focal-elex-headphones
You could pair it with a cheap amp/dac now as it's not hard to drive and then get something worthy of it as your budget allows. And it wouldn't be wrong to spend an equal amount as the headphone cost or even more on an amp and DAC. It's not as 'fun' as the Ebony Fostex but it's more refined and detailed. Bass extends down to an insane 5hz. This is HiFi tier. I'd put the Ebony right on the cusp of HiFi as well. The HD600 and HD650 are probably HiFi as well but they require a good chain to give you that sound quality and even there there are holes in what they do well.