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You said it can't be done - here it is, a perfect fake TeeJazz guitar :O I actually had a lot of fun with it, took me some time to work out how and what, but was worth it.
For the more jazzy sounding first two solos from 1:24 and 3:48 I turned off the tube screamer and kept the volume lower for a cleaner "Jazz" sound. Obvuiously anybody can still hear the Strat, but that's what I got... :)
For the ...