I don't have HD650s, just the ancient HD600s and HD580s that have been hugging my head over the last 10 years. The only real A/B comparisons I've done with my Doge have been against my Pico DAC+Amp, and against my Isabellina HPA (and the the Doge was using the Isabellina as the DAC). I took the Doge to a Head-Fi meet and listened to it and tons of other gear that blur together in my mind to the point of not being useful for real comparison unfortunately. (First meet paralysis - hopefully the one coming up in a couple weeks here will be less of a blur to me.
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I think I'm split about 60/40 between the Isabellina and Doge when I'm at home (also, I've only had the Isabellina for a few weeks, so my ears are still getting used to the new sound presentation). The Isabellina has a lot more clarity and a wider soundstage, and for some music it's absolute bliss. But it's sometimes a bit sterile sounding to me (I'm hoping the Isabellina's LFP-V upgrade helps with that), so I switch to the Doge until I hit other music that I think would sound better on the Isabellina... If I had to choose just one, I'd keep the Isabellina no question, but the Doge unit still gets a good bit of playing time here.
I've had the Pico and Doge for about the same amount of time, but they were typically separated in space (the Pico mostly lives at the office with the HD580s, the Doge at home with the HD600s). I don't think I have a strong preference one way or the other between the Pico and Doge; the Pico doesn't have the same impact as the Doge, but it has the edge on absolute clarity. I actually think that the Doge comes out ahead in my mind and with my slight tube bias. Too bad I can't strap the Doge to my iPod.
Interestingly, the Doge is actually quiet enough after it's warmed up to be usable with the JH16s, which surprised me - a lot. There's a definite background hum, but it's low enough to only be a slight distraction during quieter passages of music. With an appropriate music selection, it's definitely a fun combination.
I only have two sets of tubes that I got when I bought the Doge on the forums here, and pretty quickly settled on one set. I expect the Doge could be cleaned up a bit with better tubes, but that's an experiment for another time for me...
Also, I'm probably not the typical Head-Fi'er: I don't have the proverbial golden ears - I'm much more interested in how the music affects me than I am about being able to hear the guy on the third balcony crinkling his cough drop wrapper, and left alone, I'm not likely to notice (for example) a 2 dB gain at 6kHz unless I perceive it throwing off the overall balance of what I'm listening to. That definitely plays into how I rate my gear, so take my thoughts with an appropriate self-prescribed dosage of salt grains.