ISPARTA, Turkey—In most countries, shouting songs and banging drums through residential neighborhoods at 2 in the morning could get you arrested. In Turkey, it is a way to earn a little money.

Every year, during the holy month of Ramadan, which ended this month, Turks are roused before sunrise daily by an army of drummers noisily celebrating the arrival of the Sahur: the last meal for pious Muslims before they begin the daylight fast. At the end of the holy month, the beat-makers collect cash donations from residents eager...