Tube Troubleshooting

No, tube dampers would not help that problem, and it does not sound like a microphonic tube (or tubes). It would be unusual for both tubes to go bad at the same time though. Before disposing of the "bad" tubes, I would first have them checked. What brand of 6SN7 were they?

Yes, at least one of your 6SN7's are bad. Hopefully you live close to someone who has a tube tester, and they can check them for you. I don't have a tech or someone with a tube tester close, so I am looking at getting one for myself.
Thankfully, you found it right away. Cheers -Don

This sounds like a classic case of tube rush. Tube rush is typically found in older spent tubes. The confusing element here is that the OP spoke of more than one tube. It is very unusual for 2 or more tubes to start rushing simultaneously. Thus the suggestion to have them tested is good one.
You asked about microphony, and whether tube dampers would help? The answer is no this is not what microphony refers to and yes tube dampers may help with true microphony.
Microphony is a sensitivity too feedback sounds/ noise or percussion from the environment. Microphony is usually tested by keeping the suspect tube in circuit with the volume at audible but not loud levels, and tapping the suspect tube. If the tapping noise comes through the circuit to the speakers ( plays out the tapping,) you have microphony. It is almost very common in the earliest 6SN7s meaning the 6SN7GT. You don't have dispose of microphonic tube IMO because you shouldn't be tapping or drumming on your tubes while listening. Tubes that are sensitive to sound pressure feedback, are no good and need to be disposed of. (you could try dampers see below, but in general they are too far gone.)
Dampers can help with microphony but have no effect with tube rush which is a symptom of old age and cannot be reversed or helped. In some heavily microphonic tubes using anti vibratory dampers can help, but you don't hear most microphonuc tubes in thre first place.

Thanks, guys.

The tubes in question are Sophia Electric 6SN7s. I bought them new about three years ago. They've sat unused in their box for the last 14 months, though.

A couple more data points: the rushing air was two to three times louder through one speaker than the other.

Galen Carol was kind enough to let me try a used pair of Sophia Electric 6SN7s, and I get some rushing air through the speakers with those tubes in place, but they are listenable. The stock EH tubes have no detectable rushing air sound.

I'll see if Galen has a tube tester and report back. Thanks again!

Coincidence? I recently purchased my preamp a few months ago which had a pair of these sent along that were reported to be in fine condition. When I put them in I ran into the same problem you are having. They were packaged well for the trip, so I can't figure out why they went bad. Maybe they don't travel well as they age.

Old school method of repairing rush-afflicted and iffy tubes:

Suspend the tube upside down, and thoroughly heat up all the pins with a soldering iron (say 10 seconds per pin), if solder runs out of the end of the pin, stop as you've gotten it. (most tube pins are hollow tubes with the actual contact wire running from inside the glass envelope down into the tube, and sometimes the solder joint between pin sleeve and contact wire gets compromised).

Common repair for tubes dating back to my stint in the Navy.

Abucktwoeighty: fascinating. I've enjoyed the Sophias, but now I might start looking for an alternative.

Mcintech: awesome - thank you! As the son of a former Machinest Mate, I am grateful for both your service and your passing along that handy skill!

Everybody can't be a Fire Controlman.

Mcintech, 2 years in the Navy at a NATO base repairing old tube gear that was handed down from various branches. So vintage just moving it was likely to break connections. Lots of fun with that.
One of my best friends was an MM.