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I have a dual primary (0-120V,0-120V) transformer with dual secondaries (0-6,0-6). For a quick test of an amplifier board, I need a 12-0-12 supply.
What would happen if I wired the 2 primaries in parallel (or used just 1) and connect them to the local 240V supply? Logic says the number of primary windings would halve and I would get the 0-12, 0-12 output that I need.
Or would it just smoke?
If the one of the primary coils is rated for 120V and it is connected to 240V, smoke will most likely ensue.
The problem is transformers have a saturation point, this is by design to limit the amount of power that can be transferred for safety reasons. If there is a short on the secondary side, then the power is limited and the full power of mains is limited which reduces fires and wielding that may occur in a product.
The other ratings that shouldn't be exceeded is the coil voltage rating and coil current rating on the primary side.
So check the ratings on the transformer and don't exceed them. Don't put more voltage on the primary coils than rated. Don't put more current through the coils than rated, and don't put more power (the VA, or volt-amp rating) on the transformer.
