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I'm looking to get a tach for my shadow, mostly for aesthetic purposes than anything else. It mentions single or dual fire ignitions. What is meant by this?

Many tachs read the electrical pulse of one cylinder lead firing to work out the engine RPM. Some ignitions fire each cylinder when the piston is near TDC only when on the compression stroke (single fire). Some fire when the piston is at TDC on both comp and exhaust strokes.... hence dual fire, sometimes called wasted spark ignitions as well.

Still others will fire only at TDC compression stroke, but fire 2 or more times within a few degrees of rotation, to make sure of a complete combustion event. This is not as common on bikes (yet, it is coming as fuel and ignition systems get more advanced. Cars have been doing this for years), but firing at TDC on the exhaust stroke is still pretty common on bikes.

Do they fire on TDC exhaust stroke for any real reason, like burning unburned fuel, or is it just some design quirk?

Mostly they waste a spark because it easier to trigger the coil to fire every revolution rather than every other revolution. It's cheaper and easier to trigger the ignition off the crankshaft than the camshaft, with more reliable timing accuracy over time.

Makes sense. So I guess you'd have a switch to change in the case of a dual spark that would take the number of sparks and halve that to get the rpm.

Every 'universal' type tach I have seen has some sort of system to allow for this. It might just have settings for the number of cylinders, so for... let's say a dual fire twin, set the tach to read for a 4 cyl. They all have instructions on how to set them for your application.

If you notice the tach reads about twice (or half) what it should, a quick change of settings should get it.