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I think you got the basics bang on. An amplifier is primarily an electrical device more than the artsy sound enhancer. And any measure of performance that applies to an electrical device applies here.
This is common knowledge, there are 4 aspects of an amplifier: frequency response, gain, noise, and distortion.
So an ideal amp would be: the same gain for all frequencies, with minimal noise and distortion.
These features can help find out the more neutral and cleaner amps out there. Generally, the better the specs, the better the amp is, but whether these differences are audible is another matter.
Now, the audible properties of these specs:
Frequency response : same as headphones (bass, treble, mids etc).
Gain: Amplification (volume increase).
Noise : Most commonly heard as hissing sound.
Distortion : Can be Harmonic Distortion, or Clipping.
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Measurements such as noise, total harmonic distortion + noise (THD+N), and to a lesser extent crosstalk, should be close to nothing... either hundreds or thousandths of a percent or, in the case of dB ratings, below the noise floor and out of audible range. Clean/neutral SS amps are easy to come by, but specs can often be incomplete, even on expensive audiophile gear. Professional equipment is less stricken by this affliction, but that's a different discussion for a different thread. If you want the most neutral amp you can get, try to find as many hard numbers as humanly possible, and balance that out with subjective impressions of the amp.
Thanks. Below are the specs for the Schiit Magni, an amplifier I'm using for my HE-500s and which, to my ears, has no particular issues driving the 'phones. Based on what you have said and what I think I know about various specifications, this should be a pretty decently neutral/clean amp, right?
I understand the basics of what high and low impedance means to headphones and higher and lower total wattage and how that happens to mean a high impedance set of headphones might need high voltage, but lower overall current. At the same time, orthos often are lower impedance, but need higher overall current to get the air moving. I've seen people make statements like "HE-500s need at least 1w of power". Is there any headphone specifications that might indicate this to be the case or is this kind of statement based on personal experience? When I look at specs for the HE-500s, for example, I see that they're 38ohm and have an 89DB efficiency. I don't see how I could make an overall current need statement based on those two numbers...
Frequency Response: 20Hz-20Khz, -0.1db, 2Hz-200KHz, -3dB
Maximum Power, 32 ohms: 1.2W
Maximum Power, 50 ohms: 1.0W
Maximum Power, 300 ohms: 260mW
Maximum Power, 600 ohms: 130mW
THD: Less than 0.005%, 20Hz-20KHz, at 1V RMS
IMD: Less than 0.007%, CCIF
SNR: Greater than 100db, unweighted, referenced to 1V RMS
Crosstalk: -70dB, 20 Hz-20KHz
Output Impedance: Less than 0.1 ohms
Gain: 5 (14db)