Hi - I posted on this earlier - and I have listened again, and again....  the more I hear it - I have to say - to me it sounds like something "bad".   

You changed the strings - and you tell us that with the new strings - it seems the point at which this occurs is slightly different from the previous strings - maybe by a half or full semi-tone - if I follow you correctly?

I just begin to suspect (as I may have suggested previously as one of the possibilities) that there is some unwanted "mechanical" vibration occurring  either on, or (worse) within the body of the violin..  If there is a visible crack ( and you will want to look REAL closely - it won't necessarily be an "open" visible crack - investigate top and bottom plates very very closely, as well as the ribs all the way round ) on the surface, or, more difficult to see, if some "internal part" has come loose / cracked / separated from its glued contact ( like the bass bar as @cdennyb suggests - has come loose etc maybe just at the end or whatever ) - at certain frequencies you might JUST be getting that "imperfection" to vibrate and rattle ----  very hard to tell.   And what makes me think about coming back to this point is that, although you say the point (frequency) at which it occurs with DIFFERENT strings is slightly different - different strings will likely have a different tension - pulling and straining the body under tension in a slightly different way...  that's what made me come back to this and comment again....   but - I am no luthier and I have NEVER taken a violin apart - I'm just interested in the science / physics of this...

I know that did not help.....  just thought I'd drop my 2 cents worth....

EDIT: I don't recall exactly - but the tension of a normally strung violin amounts to something like 16-to-20kg - and MORE actually - try lifting a 20kg bag of cement...  it's a LOT....  and the tension of different string types does of course vary a bit which may just account for the change at which the noise appears - dunno - just guessing...   the violin although appearing solid and robust is indeed a "finely tuned" ( excuse the pun ) mechanical device...  and EVERYTHING needs to be right for it to work and sound OK...