This sounds like a wave of piloerection moving over the skin.

http://en.wikipedia.org/wiki/Goose_bumps

Sometimes there are no frank goosebumps at all, and it is
just a sensory feeling. I've had this at times now that I am
"old" on my left side. It is a wave that moves down my body.
I've always thought it was a "reflex" of some sort. But I cannot control it, and it is not unpleasant. I do have a pinched nerve in my neck on that side, so I always thought that could be a trigger.

The inside of the brain does not have sensation you know.
There are some nerves around the blood vessels which mediate contraction or dilation for blood flow. This is where headache is felt, but not in the tissue of the brain itself.
Open brain surgeries can be done with people awake, who can talk and interact with the surgeon.

Dopamine is the neurotransmitter of "pleasure" and some people feel this when listening to music, or using certain drugs. Stimulants and cocaine will cause a sudden release of dopamine, which is the reinforcement of it, and leads to addictive behaviors over time.

But listening to music and feeling it intensely is just a less intense dopamine mediated experience. (one can see why raves and concerts can have recreational drug use during them to intensify the experience).

People vary in how much dopamine they can make, and all the neurotransmitters vary in amount genetically from person to person.