My opinion which I am not forcing on anyone is this:
If you know the minimum and maximum impedances of your headphone to be Zmin and Zmax and you want the frequency response error to be less than x dB, the output impedance Rout of the amp shouldn't be more than

Rout < (Zmax * Zmin * (beta - 1)) / (Zmax - beta * Zmin)

where beta = 10^(x/20).

**Example:** Maximum and minimum impedances of your headphone are 50 Ω and 30 Ω and you want frequency error be less than 0.5 dB. We get:

beta = 10^(0.5/20) = 1.06.

Rout < (50 * 30 * (1.06 - 1)) / (50 - 1.06 * 30) = 90 / 18.2 = 4.9 Ω.

Letting frequency response error to be 1 dB, we have

beta = 10^(1/20) = 1.12.

Rout < (50 * 30 * (1.12 - 1)) / (50 - 1.12 * 30) = 180 / 16.4 = 11 Ω.
There is the 1/8 -rule for output impedance, but it's not very accurate in every situation. Which impedance value of the above headphone should be used when calculating 1/8 rule? People check the specs and see that these are sold as 32 Ω cans, so they calculate 32/8 Ω = 4 Ω which in this case is very close to the 0.5 frequency response error criteria, little better in fact. Some headphones have so flat impedance curve, that any output impedance keeps the frequency response flat enough.

Excessive output impedance causes

- frequency response error

- lack of damping

- linear distortion

Of these three, frequency response error seems to be the most demanding. If a headphone has very little mechanical damping (to be sensitive), it needs electrical damping meaning low output impedance for the amp, but lack of mechanical damping also means strong resonance for the driver which creates strong maximum impedance on the electrical side. So, if you need electrical damping, you also need to reduce frequency response error. Linear distortion seems the least of the problems. Linear distortion here means that the control of the amp on the movements of the driver is compromised so that the movements of the driver don't follow the amp voltage as accurately as when the output impedance is zero.

I use Sennheiser HD-598. It is pretty "demanding" in respect of output impedance. Zmax = 275 Ω, Zmin = 60 Ω. Just 10 Ω of output impedance gives 1 dB boost around 100 Hz, the resonance frequency of the driver. The output impedancies of my DIY headphone adapters are 1 Ω and 2.2 Ω. Lately I have been testing increasing the output impedance a little to "modify" the sound. If I put a DIY "UPOC" crossfeeder crossfeed off in between my DIY headphone adapter and headphones the output impedance raises to about 10 Ω. The sound becomes a bit more bassy and also "more relaxed" which I don't fully understand yet. Going back to normal set up makes the sound ultra detailed in comparison. So, tinkering with the output impedance can be used to optimaze the sound in respect of accuracy of detail, degree of relaxation, frequency response and damping. Damping and degree of relaxation might be connected, but my understanding at this point is lacking. My headphone is overdamped up to 56 Ω output impedance. I have designed an output impedance selector (5, 10 and 20 Ω), but I haven't constructed it yet. I wish I could understand this better. I hate it when I don't understand something. I have my suspicions, but I am not sure about them at all.

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