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Some people claim that intonation changes when going to a larger or smaller shank mouthpiece. I can understand overall PITCH changing, as the length of the horn is very slightly increased or decreased. However, I am more interested in how it can alter the intonation.
I would love to hear from anyone with a "shank sensitive" tuba, and what the specific results were.

I have seen plenty of cases where a certain tuba will play significantly better overall with a specific shank size - and I make several more sizes than anybody else seems to. It can change how pitches lock in or not. But I don't have a specific answer to your specific question.

Doug Elliott wrote: It can change how pitches lock in or not.

I think this explains it. Different shanks, for me, seems to change how tightly a horn will slot. Some players want/need a tight slotting horn while some like the wider slots. By making the horn more akin to how a player likes someone might think that the mouthpiece fixed a pitch while really, the different mouthpiece just made it easier for them to play in tune. Does that make sense to anyone else? Makes sense in my head.

My horn (a weird, rare 4 piston B&S called the PT-5, but not the same model that got pulled off the market so publicly) is really sensitive to mouthpieces, and definitely shows different intonation tendencies with different mouthpieces in very clearly audible ways. I've always suspected that cup size was the issue for me, but the tuba also really prefers American shank to European. Or at least I do when I play it to a degree that I haven't noticed before.

I'd agree strongly that larger shanks (bigger gaps) make notes "slot more tightly" and that it's probably the most noticable affect, but I have my suspicions that there might be some secondary intonation changes, too. Might be worth it for me to screw around with a mouthpiece adapter and a tuner this weekend and post again if I have time.

Here is a presentation from Stormvi USA about changes in trumpet intonation produced by changes in gap. I don't know how these results were measured, and I don't know how applicable they are to tubas, but it looks as if there may be a real effect on intonation:

http://www.youtube.com/watch?v=YqryPUViyeo

Maybe this is one cause of occasional complaints about instruments whose octaves are not in tune ?

Here is another (also from Stormvi) about how increasing the gap makes for tighter slotting, which may be what a player needs, but may also get in the way of the upper register by causing the instrument to lock in to a lower note. Again, I don't know how much this applies to tubas.

http://www.youtube.com/watch?v=i_ZGqvvPuLM

Are you talking about the OUTER dimension of the shank (and, indirectly, the "gap") - or the INNER dimension?

I have no opinion on the "gap" question - but it's fairly clear from the mathematics that a larger INNER dimension has the effect of raising the envelope (producing better slots up high) and also WIDENING the slots (making the pitch more dependent on the player). It's not at all clear that varying this dimension has anything at all to do with overall PITCH.