To my knowledge, the Great Highland Bagpipes are tuned using just intonation rather than equal tempered tuning. Why is this? What is the advantage of just intonation?
The reason doesn't seem to be a technical advantage but the favored traditional sound of the just tuning that bagpipers describe as more colorful and warmer:
If bagpipers used the equal temperament scale the drones would sound out of tune for every note but the As, although B, D, and E might be close enough. C#, F#, and G would be way off; hence why we need to beauty of the just intonation scale.
In the link above he also explains the difference between the TET and the just intonation.
He explains that the bag pipe is in D major and
there are several resources for learning how to (and understanding the theory behind) tuning a bagpipe with an equal temperament tuner. Now, this sounds like an odd thing to want to do since we don’t use the equal temperament scale. The reason you would want to use an equal temperament tuner to tune a just intonation instrument is simple, economics.
He also refers to Kyle Gann who compares the 2 tuning systems and tells us why he favors the traditional tuning:
I've had interesting experiences playing just-intonation music for non-music-major students. Sometimes they will identify an equal-tempered chord as "happy, upbeat," and the same chord in just intonation as "sad, gloomy." [...]
On a more subtle level, after I've been immersed in just intonation for a couple of weeks, equal temperament music begins to sound insipid, bland, colorless. [...]
[...] playing in just intonation for long periods sensitizes me to a myriad colors, and coming back to the equal tempered world is like seeing everything click back into black and white.
The two links above also explain - as does this one by Ewan Macpherson - the system of the different temperatures - and all agree that the A (pitch 460 and higher) sounds sharper than a B♭ in concert pitch.
So we can assume that the just tuning is rather a tradition and sentimental reason than a technical aspect - despite the lower costs of equal tuning.
The drones are tuned by ear so you expect them to be tuned to just intonation.
The holes in the chanters can't be made exactly enough that you could say they're just or equal tempered. The difference between two different chanters is probably greater than the difference between just and equal temperament.
In general for any kind of non-keyed "folk" wind instruments the whole concept of equal temperament is meaningless.
This answer will be about bag-pipes in general, not about the Great-Highland-Bagpipe in particular. As such I won't make a statement about the tuning of the GHB, but instead about the general tuning of Bag-pipes: There is no standard on how to build a bag-pipe. Bag-pipes in general are handmande instruments, so (in principle), with every instrument, the builder is free to choose a a tuning. It would however be unusual for a builder to not stick to a tuning of his choice, because of that instruments of a kind of one builder will (in usual cases) not have different tunings. Examples of different tunings in case of the german shepphards-pipe I have seen featured "Just Intonation", "Kirnberger II"described here or some weird other selfnamed-intonation (I emailed the guy and got a table with how mutch cents the individual notes would differ from the equal-temperament-version),which was a mixture of just and equal temperament.
There are 2 reasons why bag pipes (including the GHB) are usually tuned in a "more just" intonation than the equal-temperament version: Both reasons are connected with how we perceive Intervals, and that "just intervals", with more simple frequency rations, CAN be perceived as "better sounding". You probably know this, but I still want to reconcile it for the sake of the answer: What sounds better to an ear is as well a product of what one is used to, but what one can say objectively about just intervals is that they are more stable (this video shows that), because the frequency ratios are simpler. The intervals are also perceived as "more in tune" because of that (which doesn't necessarily mean they are "better sounding".
Now comes the reasons for a more just tuning to be choosen for bag-pipes: Bag-Pipes, no matter wether they are played together, or solo, are drone-instruments. A Drone (which is an accompanying tone) will be audible most of the time during the piece one performs, even if it might not "fit" harmonically to the actual passage (for example because the drone-note is not present in the harmony that a melodic line is based on). You might want to play a broken D-minor chord (D F A F), while a C-Drone is constantly audible. In general, such situations will be more pleasant sounding when all notes are justly tuned to the drone. Because the frequency ratios the notes have with the drown are more just, the drone can be perceived as a nice accompanyment, no matter what notes are played. Another example would be the melody featuring a B, while the drone being C. The interval is a 7th, which sounds more harmonic in a just tuning, than in the equally tempered one. That is reason 1.
Reason 2 is that the drone automatically restricts the repertoire to keys close to the drone-note. A bag-pipe doesn't need to transpose wildly through all the keys possible, most keys wouldn't work with the fixed drone-note anyways. So there is no need to use equal temperament (which was invented for exactly this reason).