Hi Jenny and Phil,
There are two aspects to the tuning of an instrument - the base note of a scale and the intervals within the scale played by the instrument.
On my non-tuneable Ghost whistle I am responsible for both. On my tuneable whistles, the player is responsible for the base note of the scale, and I am responsible for the intervals within the scale because I decide the position and size of the tone holes.
By default, my whistles are tuned in "Equal Temperament" (ET) which is the way modern orchestras, pianos and guitars are tuned. I'm assuming this is what you mean by "concert tuned." (I feel obligated to note that concerts are given on traditional instruments too!)
"Concert Pitch" will tell you what frequency a particular reference note should be tuned to. A=440 is normally used today, but the ensemble is free to agree on another pitch if they so choose, and everyone can accommodate it. Which is hard for things like accordions and pianos to do.
Once a reference pitch has been agreed, the next question is how are the intervals to be spaced? This is not very often thought about, since the maker of the instrument pretty much builds in the intervals. Fiddles and trombones have much more freedom to play what they will in terms of intervals. And harps and other instruments with individually tune-able strings at least have a chance, but it is a chore to re-tune for each key change.
Note that if A=440 is agreed, we can still have our C land in different places if we don't agree on the intervals we will use for our scale. So, all "concert" instruments today are tuned A=440 and Equal Temperament.
Just Intonation is the "natural" tuning of western instruments before the advent of "classical" music, and keyboard instruments. Simply put, you play a note, and then play a note one octave up, and it will be twice the frequency of the first. The other notes you play will fall on "nice" fractional intervals between the two.
The limitation (I didn't say "problem") of this approach is your playing is pretty much limited to one octave. When you play in the next octave, those perfectly places tone holes need to be in different places. So to allow compositions spanning more than one octave that were being penned by the "classical" composers, they had to agree to average out the location of the notes so they sounded equally good/bad regardless of which octave was being played.
I default my instruments to Equal Temperament because the first thing I expect customers to do is go find their piano or electronic guitar tuner and see how I did making the instrument.
I do offer D whistles in "Just Intonation" (JI) tuning, which I expect you are referring to when you say "trad tuning." These are most often purchased by pipers whose bagpipes are JI tuned, but solo players and people who play with baroque groups like them too.
Can you play a Just Intonation whistle along with a piano or guitar? Can you play an Equal Temperament whistle along with a baroque ensemble? The answer to both is a qualified yes, because player technique can push the notes for each hole around a bit. This is more easily done on a flute than on a whistle, but in each case you do have to actually PLAY the instrument vs. just blow and move your fingers.
An experienced musician, and I'm speaking more about their ear than their lips and fingers, will know if they are flat or sharp compared to the others they are playing with and can make some adjustments, within limits, to the tone they are playing.
Comparing one common interpretation of JI (which I use) to an ET C scale, here are the differences in cents for each note:
So depending on the notes played and their duration, you may get by with an ET whistle in a baroque group. You will have to bend the note one way or the other 16 cents at most. That said, it is of course simpler to have the appropriate instrument to start with. But you still have to listen and play the right tone.
Question: What do you call three flutes playing A?
Answer: A chord.
Why do people bother with JI or "traditional tuning" any more? Because if you live within the limitation of one octave, the music sounds better. And with today's computer technology MIDI instruments can me controlled to play JI in any key with the push of a button. And, knowing the root note, I expect they can produce notes in multiple octaves that are nice round fractions of the root tone.
HERE is a link to a graphic demonstration of JI vs ET intervals. Sadly, this kind of on-the-fly re-tuning of the notes being played is not possible when playing a physical instrument. But you will see the reasons for each tuning I think. Please realize that when the narrator says "Clearly out of tune." he means clearly the intervals are not ideal for the selected root note of the scale. They ARE in perfect tune to the Equal Temerament tuning of the scale, which is what all of our keyboards, guitars, pianos, clarinets etc. come with these days so we can all play together in whatever key the composer (or singer) has selected for the performance.
Let me know if you'd like more info or if I've said something that isn't clear to you. Google will find more that you care to read if you search for "Just Intonation."
(Sadly the best site "justintonation.net" has been hacked and it's content replaced with some blather and I would expect the buttons to download a virus, so don't follow that link when it comes up in your search.)