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When you say the pitch with the R13 is "much better," does that mean "in tune" or "sharp" or just "less flat?"

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Are you sure it's playing that flat or has the needle on your tuner gone full circle making it appear to read 10 cents flat, but is in fact just under a semitone sharp with such a short barrel?

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Yeah, that's pretty short in the panoply of clarinet barrels. But as far as what I do with a 10 cent difference would be to go down at least a couple of millimeters.......round off to 60mm. If it is a little sharp, pull out (but you can't squish smaller if it turns out to still be too long).

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See also this thread http://test.woodwind.org/clarinet/BBoard/read.html?f=1&i=486919&t=486905

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Clark Fobes used to make barrels for the Signature. Perhaps you can contact him for his thoughts.

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If you check the many videos Brad Behn has put up on his Facebook page in the last 8 to 10 months, you will see that in addition to his Buffet clarinets he is also now playing and testing mouthpieces on a Selmer Signature Clarinet. So I would assume he has found or made a barrel that works well on that model and can do the same for you.