I highly recommend Gödel's Proof (Nagel and Newman) as a supplement. I got a lot out of Hofstadter's book, but honestly found much of the discussion of the incompleteness theorem more confusing than it needed to be. If you're mostly interested in the Gödel part, I actually recommend you read it instead of Hofstadter's book.
Gödel's Proof is very concise, and more clearly describes the core argument.
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The mechanisms that give rise to the illusion are not well understood, but it seems to have something to do with hemispheric asymmetry. Individual differences in how the illusion is heard often correlate with handedness (ie whether you're left or right-handed)
Although you characterize it as only being able to "emulate old composers", I may have been referring to the program you are thinking of. Although it is true that it learns and extrapolates from musical input, so do human composers. Its algorithms can learn from anything, and by feeding it a mixture of styles, it can actually generate some fairly compelling and new sounding works. The programmer is also a composer and has trained the algorithm on some of his own works and the output sounds nothing like old composers.
Apprehension about progress in artificial intelligence is entirely natural. In fact, I think apprehension and denial will lead people to continually redefine their notion of intelligent behavior so that current computers are always excluded.
Not so long ago many people said a computer could not beat a grandmaster at chess without being intelligent. Enter Big Blue. Others have stated computers will never compose music that is emotionally meaningful to humans without being intelligent. Enter Experiments in Musical Intelligence and other widely-acclaimed composition programs.
Until the Turing test is passed people will be able to plausibly deny any advances in artificial intelligence. No matter how advanced such "brain in a box" models becomes, they won't pass the Turing test without being embedded in a rich environment with which they can interact.
Gödel's Proof is very concise, and more clearly describes the core argument.