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bloke wrote:a mouthpiece that incorporates characteristics of both a "funnel" and a "bowl".

... and in fact every currently available tuba mouthpiece is somewhere between a funnel and a bowl, so it's all about the details.

Best thing is to try a few mouthpieces, including a Conn Helleberg or close copy, bearing in mind that other factors like throat and back bore make a big difference too.

bloke wrote:There are many things going on with the dimensions of sax mouthpieces

No kidding. One thing we're vaguely aware of that I've never heard mentioned with tuba mouthpieces is the acoustical role the mouthpiece chamber plays in the effective length of the resonating air column. The basic acoustical principle of a saxophone supposes a conical shape, and somehow (apparently "somehow" involves something called Helmholtz resonance) the mouthpiece chamber serves in lieu of the hypothetical continuation of that conical shape where it's truncated at the mouthpiece receiver. So intonation can be influenced by absurdly small adjustments of the mouthpiece, as a small change in chamber size corresponds to a lot of small diameter hypothetical conical bore. Next time I get a chance, maybe I'll try a few notes on the tuba with as much lip as I can stuff into the mouthpiece - could that raise the pitch by making the "chamber" effectively smaller?

But at any rate, sound is a combination of reed, mouthpiece, player and saxophone. Players tend to eventually sound like themselves on whatever mouthpiece; reeds may be good with one mouthpiece and not another; mouthpieces may be good with one sax and not another; and of course what we sound like to ourselves is not what the rest of the world hears. Sounds a lot like the tuba, plus reeds, but range of possibilities is really quite a bit larger - the difference between one mouthpiece and another can be very striking, which I can't say of the tuba.

What receives too little consideration is the difference in sound of players with bowl shaped heads compared to players with funnel shaped heads.

The Laskey Company makes a few mouthpieces with a combination of the funnel and bowl shape. These are the 30B and the 28B.

I have found the 30B a little more flexible/accomodating in the low range on piston CC tubas than the 30H (I am assuming you are looking for this for a CC or BBb tuba...). Contact Mr. Laskey if you're interested, because I don't think he has added this model to his website yet.

Enjoy!

"All else being equal," (which it never is):
1) Depth of cup: the deeper the cup, the more fundamental that is usually reinforced.
2) V cup: the more "V" the cup, the more some even overtones are truncated in favor of lower odd order overtones.
3) "U" or "Bowl" cup: the more "bowl" the cup, the more that the entire range of overtones will be produced.

But as been said above, there are infinite variables, including the player, and all cups have at least a little bit of modification from the shape in order to have appropriate throat and backbore geometry.