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I kind of see where you're going with this but it doesn't quite make sense to me. The E12 has plenty of output power. If power output is the obvious difference, are you saying that if I use a sensitive 32 Ω headphone, a weaker amp should make the headphones sound "less full" than with the E12 given every other aspect of the amp is exactly the same and playing music at the same volume level without clipping, and without adding extra noise? How does output power affect how "full" a headphone sounds?
Basically if the amplifier can output enough voltage and current to deliver enough power to your 'phones without clipping at an actual listening level then you are half way there! And I'm not talking about an amp that can deliver 20 times more power than your Grados actually require!
Obviously you also need a decent signal to noise ratio and low enough distortion. Not just low THD, also low IMD.
Throw in a quiet power supply.
I have a few headphone amps, only one is quiet enough to drive my Shure IEMs.
I was surprised to find out that even some of my battery operated headphone amps are too noisy for my IEMs. I had assumed that if you get off the power grid them you should get reduced background hash. Guess what?
One of my battery powered amps creates too much low level buzz. And this is when it runs off batteries! i.e. charger not connected, source not connected.
Oh, and Flat frequency response throughout the audio bandwidth.
Circuit board needs to be laid out properly.
The right components have to be chosen and applied properly.
Want more?
Talk to someone who actually designs audio gear for a living.
Someone who has designed a few products which have sold a few thousand pieces with negligible warranty claims.
Can't find an audio designer in your neighbourhood?
Try reading one of the books written by noted audio designer Douglas Self.
The rest of us really only understand a small part of what separates the good from the great.
I've worked as a professional electrical engineer for many years as a power engineer. I even designed analog signal processing gear for a year. But I know enough to know that I am not an expert in audio signal design.
I'll stop ranting now.....