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During a trip to try out a new BBb tuba, I evidently lost the mouthpiece I used in my 1913 Keefer 3 valve Eb tuba.
It was a "Lyle" but there were no other identifying marks on it, so I don't know what kind of mouthpiece it was.
The regular tuba mouthpieces which fit my BBb tuba only insert about 1/2" into the receiver on the Eb tuba,
whereas they insert about an inch into the BBb. The Lyle seemed to work well in the Eb tuba, but was small,
maybe about 28mm ID (I'm guessing, but it was definitely smaller than any of my BBb mouthpieces.) and had a
smaller throat than the other tuba mouthpieces.

Any recommendations on what to try in the Eb? I'm guessing that it will take a contrabass trombone mpc, but that's
just a guess. Should I try using a BBb mouthpiece, perhaps a Kelly or Bach 18 or 25? Or does the shank need to fit the
receiver better?

If your BBb takes a mouthpiece with a "standard American" shank, then by comparison your EEb probably needs a "Small European" (sometimes known in America as the E-flat shank). Denis Wick is probably the only manufacturer who still stocks a range of mouthpieces with this size of shank; If it's a generously proportioned EEb then the usual choice is a Denis Wick 3 (not a 3L, which has a standard shank). For a smaller EEb, the 3 may be a bit wide in the throat, so you might go for a 4, which is smaller (30.5 mm) and a tad narrower, or even a 5 (comparable with a bach 32E). Bach used to make special-order mouthpieces with a "British" shank, they may still be available, or you can buy a standard Bach and have the shank turned down.

I think bloke also sells his "Selmansberger Imperial" with a small-Euro shank, and various mouthpiece makers will make a custom-order mouthpiece to fit.

This topic comes up fairly often, so you could try the search function. Here's a start:

viewtopic.php?f=2&t=50050

What Mike W said. Wick 3 (no letter).