The origins of quark-hadron duality: How does the square of the sum become the sum of the squares?

Bloom-Gilman duality demonstrates empirically that the electroproduction of $N^*$'s at low momentum transfers averages smoothly around the scaling curve measured at large momentum transfers. The latter is proportional to the sum of the squares of the constituent charges whereas the former involves the coherent excitation of resonances and is driven by the square of summed constituent charges. We determine the minimal necessary conditions for this equality to be realised so that duality can occur and consider the implications for a range of processes that may be studied soon at CEBAF.