Why/how does pitch change with temperature?

After many gigs where I've played indoors and then gone outdoors or visa versa, it was curious to me that the colder air always makes the drones go sharp relative to the chanter and warmer air makes them flatter than the chanter. Alternatively, I was never sure that it wasn't the chanter going flat in the cold relative to the drones.  As a pragmatic solution for tuning on the fly (i.e., coming out of a church) I've learned to move the drone tops in the opposite direction of the temperature change (i.e., move the drone tops higher when moving to a lower temperature!).

I've investigated a couple ideas: Early on, I expected that the problem might be due to the changing mass of air in the drone as the density of the air increases with decreases in temperature, so you'd need to shorten the drones, but this doesn't make sense in terms of the correction needed. I've also found that the speed of sound increases with increased temperature (about 1% per degree F), so to hold the pitch constant, the drone length (wavelength) must also increase when temperature increases - again the wrong direction.

I also suspected that the governing factor may be the "elasticity" or "compliance" of the air as a function of temperature and the way it affects the condensations and rarefractions. Oddly, it doesn't affect the drones and chanter in the same way.

Having spend a bit more time investigating and noting that it is the chanter that goes flat and gets harder and harder to blow in cold weather, I now suspect that the culprit is that the chanter reed is becoming less flexible at low temperatures.   Basically, at cold temperatures the reed is less flexible and behaves like a larger mass - hence the lower vibrational frequency and with less amplitude - and greater likelihood of "choking".  In fact, I now have a reed that is very weak at room temperature that plays very nicely in cold weather with my "regular" (i.e., room temperature) setting for drone reeds.

I suspect that humidity and the role of water as a plasticizer in the wooden reed also plays at least a minor role.

If you know more about this aspect, please contact me.