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As always, but jazz / classical mouthpieces tend to have different characteristics, i.e. a smaller tip opening and longer lay for classical and larger tip opening and shorter lay for jazz mouthpieces. The Vandoren website suggests Traditional for classical players (although goes on to name Lee Konitz and Stan Getz as examples of players). It also says that the name Java is an amlagamation of Jazz and Variety because the cut of the reedgives more flexibility.
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Interesting topic. What you say is right, I'm sure, as far as the majority of players are concerned, and it must be true that certain reeds suit certain sorts of mpc. However while there's a risk of me being too pernickety here (not for the first time!) I tend to shy away from calling mpcs 'jazz' or 'classical', because it leads people to think that they must have a particular mpc to play a particular type of music.
It all depends what sound you want and what works for you. No doubt many people who play jazz use more open mpcs (and with bigger baffles) than players who mostly play classical music, but there are exceptions. Some people (me included on alto, along with Kenny Garrett, I think...not that I'm comparing myself with him!) use a Selmer S80 no matter what they're playing... I have used a Lawton on Bari in a Wind Band - though I prefer to use the Yani stock mpc because I can play quieter - and the Bari player in one wind band I play in uses his Lawton (as does the tenor player!). And I can't play Links under any circumstances ... (Link with a vandoren reed - my nightmare!)
Of course there are extremes; I guess you wouldn't use a Runyon to play in a sax quartet (not if you wanted to stay friends!), and a very wide lay (say, above Selmer F) is going to have a limited use, but IMHO there is no clear dividing line between the mpcs used for different styles of music.
This chart is interesting;
http://www.du.edu/lamont/Mpce.htmlnick