Vintage Synth Repair

>What I mean is, does the tantalum cap get fried because there is a
>short between the -15v and ground caused at one of the boards, or the
>only possible reason is high secondary transformer voltages which
>overheat the 7915 regulator ?

Short answer: Neither.  The cap fried most likely because there was too much 
voltage applied to it (transformer secondary is too high, excessive mains 
voltage, whatever).  If this is in excess of the rated voltage, it will 
fail, it's got nothing to do with the regulator or the load on the supply.

Longer answer: It might be possible that, given excessive current drain and 
leaky filter capacitors, there was a lot of voltage ripple at that point and 
thus the capacitor was placed under undue strain which made it more 
succeptible to failure.  But I really don't buy that, it's much less likely.

>How do I check the secondary voltages and what should I see there ?...

You'll need an AC RMS voltmeter to directly measure the secondary voltage.  
With no load applied, it will be somewhat higher than it would be under 
normal operating conditions (but the regulated output is still regulated and 
shouldn't vary).  But it's easier to just measure the rectified and filtered 
DC voltage: just measure across the tantalum capacitor.  Since the 7915 has 
a 2 volt margin, the input must be at least 17 volts under loaded conditions 
for it to operate properly.  With no load, maybe 18-19.  Voltages above that 
are fine as well, but the excess must be dissipated by the regulator: the 
higher the voltage, the hotter it runs.  The regulator can handle up to 35V, 
something close to that is going to run very hot for even moderate current 
drain (250 mA * (35-15) = 5 W, this is way more than what you would want).

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