Vintage Synth Repair

It's pretty simple:

According to the schematic you need +21 volts (unregulated) in front of the
15 volts regulator (after rectification and filtering). This should be
sufficient to calculate the required output of the secondary.
Typical input voltage for the 7815/7915 regulators is 23 volts according to
the datasheet with a minimum voltage of 17.7 volts. So the 21 volts figure
is pretty much in the ballpark.

Now, the question is should a 2x15 volts or 2x18 volts transformer be used?
Anything higher than this is too risky (remember the 21 volts required in
front of the regulator!).
A transformer with 15 volts effective output would give something like
21.21 volts maximum output. Minus the voltage drop across the rectifier we
get about 19.81 volts; this is smoothed by the cap.

HOWEVER: The effective voltage is only correct for full load; the voltage
for no load or partial load is higher! One can usually add about 10% here.
So what we end up is 19.81 volts plus 10%: 21.79 volts.

So I'd stick with a 2x15 volts transformer.