Paul,
The ESC "music" is telling you two things.
1) When You plug in the battery, it first tells you how many cells--the 4 blips in your case.
2) If it sees a zero throttle setting, then the ESC is armed--that is the second tune you (sometimes apparently) hear.
If you don't hear (2), it means either that the ESC isn't getting any pulses from the JMP2 (throttle disconnected), or that it is connected but the JMP-2 isn't giving a zero setting, but some other value. Without first seeing a "true" zero throttle setting (somewhere around a 50Hz, 1ms wide pulse (maybe a little longer), the ESC will refuse to send any power to the motor. This is a safety feature. When I run a RC model, I always set my throttle stick to roughly 1/2 throttle, so that when I plug in the power to the model, the ESC isn't armed. That allows me to put the transmitter neck strap on without having to worry about bumping the throttle. Once on, I lower the stick to zero throttle, then the ESC plays its "I armed and ready to go" tune, and then I move the throttle stick and away I go!
Normally the timer does those functions for you. It does sound like there is some issue with your JMP-2. I am not sure what unplugging and replugging in the battery is doing. I know if the JMP-2 thinks it has gone through a timing cycle, that it won't start a new cycle until it sees the power go off and then on again. Why it apparently thinks it has just finished a cycle when you plug in the first time (my conjecture), I don't know.
But when you say you plug it in again and it goes, I hope you don't mean that it automatically starts off and runs the motor without you not pushing the button! That wasn't clear to me from your post.