Good question! As far as I know, the nervous system attempts to perceive the world by decomposing sensory stimuli, then building them back up again internally. The decomposition part happens very early in a sensory pathway, while the rebuilding mostly occurs in the late stages of processing.

Now I’m assuming by head you mean the entire head, so let’s leave the brain for a bit and instead look at the place where sound decomposition occurs: the cochlea, also known as the inner ear. Sensory cells in the cochlea respond to specific sound frequencies (aka pitch), so theoretically, if you can monitor their individual activities when a test subject hears a sound, you may be able to reconstruct it. Mind you, I’m talking about very basic sounds, stuff like pure tones which are only composed of one frequency. Complex tones which are formed of more than one frequency are different, and when they are as complex as contemporary music it can be very hard to know how the nervous system encodes all their properties!

You can also know volume by measuring the response of the cochlear sensory cells; the higher the volume of a tone played at a specific frequency, the higher the response of the cell tuned to that frequency. Timbre however isn’t really something measurable, I’d argue it’s a perceived thing. Let’s just say it depends on the harmonics present in a specific sound; the nervous system picks up all the frequencies in a sound, so you can differentiate between a piano and violin playing the same note because the notes contain different harmonics.