Yet another Grounding Question-Separate 'Earth'

ADDITIONAL, not separate, grounds are a fine idea. IOW, adding ground stakes and attaching them to the new panel is fine as long as the new ground wire also runs to the old panel and adds to its grounding. My 1958-model house has an added 4g. ground wire running throughout the attic. It was attached to a waterpipe at the front of the house and the input panel at the side of the house. When I added 3 dedicated lines last year, the electrician added 2 ground stakes below the panel and grounded the already-grounded ground buss to these stakes. Yours should do the same. BTW I now water the bed of earth containing the new groundstakes about weekly, as I'm in the desert.
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I strongly suggest you check with the local building department or electrical inspector before you let this electrician do any work.

The subpanel is not a separate electrical source since it is in the same building as the service. Therefore it must be grounded at the point of the electrical source which is your main panel. Regardless if you put a separate ground rod for the subpanel, you MUST ground that subpanel back at the main panel. You cannot ground anything upstream of the main circuit breaker on its own ground. In other words, add all the ground rods you want, but you still need to ground back to the main panel.

What I suggest, and what I did in my house (which I do not want to burn down), is to install a 2-pole breaker in the main panel. Run wires from this breaker to your subpanel (no main breaker required in the subpanel). Run a separate neutral from the main panel neutral bus to the neutral bus of the subpanel. In the subpanel, install a separate ground bus. This ground bus MUST NOT be bonded to the subpanel or the subpanel neutral. Run a separate ground wire from the subpanel ground bus all the way back to the main panel and join it to the ground at that point. Connect your dedicated circuit grounds to the subpanel ground bus and the dedicated neutrals to the subpanel neutral. This is effective and legal.

You cannot ground seperately, unless your jurisdiction permits it. The reason for the ground is to trip circuit breakers, as the ground becomes an "escape hatch" for short circuit current in the event of a fault. By using separate or multiple grounds, the possibility exists that a short circuit may be routed away from the breaker, causing a dangerous situation.

>>"But, and here's the really critical question- my electrician has proposed a pair of separate ground rods about 10' from the main ground for the rest of the house electrical system, and in his view, the audio system subpanel would be grounded just to these new ground rods, not connected, by ground or anything else, to the rest of the house."<<
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Find another Electrician! You may want to ask what else he is doing that does not meet NEC and your local codes.

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Gs5556's post is right from the text book, NEC. Follow his post and you will meet NEC code.
The earth shall not be considered as an effective ground-fault current path back to the source.

Many thanks to all of you, and to Nsgarch, for the timely responses. Work will be done by the book, according to Code and to Hoyle. The balanced power idea is also intriguing. I will let you know how it goes and will scream if I have questions in the midst of it. Thanks again, everybody.