The issue is that headphones doesn't always present an identical load to the amp, so when the load varies it'll interact differently with the amp if it approaches the point where damping factor is sacrificed or altered... Basically the manufacturer provided impedance measurement is an average, not an absolute constant figure. So essentially, you might be at the 1/8 ratio from 300Hz thru 10KHz but then at a different frequency you might be well below 1/8. Doesn't always have to be reflected in the bass.
The least affected headphones are planar magnetics since they have a completely flat impedance curve, so it's never gonna react differently to the amp depending on output impedance regardless of the frequency played (tho I'm unsure if electrical damping is still a factor overall, with planars specifically). AKG's Q/K 70x line has a relatively flat impedance curve, which is why I think it's often mentioned in the same context (plus it's high amping requirements).
It's not totally flat tho and still shifts by about 10-20 ohm at 10KHz, at least according to the InnerFidelity chart I'm reading... I guess that's still a relatively small shift tho, compared to headphones whose impedance jumps by 50-200 ohm depending on frequency (and that includes a lot of headphones from the lowly PortaPro to the HD650).
A chip/op amp is just exactly that, the way it's built/designed, the E09 and others are like that... Tho the E09k puts out far more than 60mW. Most AVR usually just put a series of resistors to step down the speaker voltage for headphone output, thus using the same amp for both. HK used a completely separate amp there, I think Marantz also does this, not sure... Which is the better approach is completely down to actual implementation tho, only measurements and/or your ears can determine that.
SoAmusing, the S&V article I PM'd you a while ago is probably the clearest explanation I've seen of why output impedance matters (less technical, good analogies).
Edited by Impulse - 4/1/13 at 9:51pm