Where to stay in southern Spain?

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I will be traveling to Spain next year wtih my family and will have 3 nights and 2.5 days in southern Spain. The only thing we know for sure we want to see is the Alhambra. We would like to have a "base" to stay at for those 3 nights that is within walking distance of a city, somewhat relaxing/laid back, has a generally friendly feel (and good for kids), is not too touristy, within several hours of Granada, and has beautiful Spanish/Moorish buildings or sights. I speak at least enough Spanish to get by. Does anyone have any recommendations for which city we should stay in? Right now we are leaning toward Granada, but are also considering Cordoba and Sevilla. Does anyone have any suggestions? Are there other close cities we should consider? Is Granada a good choice or should we stay in a different city and just go there to see the Alhambra?

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