Given this paragraph about the history of Beijing, when did Beijing get the current name?

Over the past 3,000 years, the city of Beijing has had numerous other names. The name Beijing, which means "Northern Capital" (from the Chinese characters 北 for north and 京 for capital), was applied to the city in 1403 during the Ming dynasty to distinguish the city from Nanjing (the "Southern Capital").[35] The English spelling Beijing is based on the government's official romanization (adopted in the 1980s) of the two characters as they are pronounced in Standard Mandarin. An older English spelling, Peking(北平 in Chinese meaning), was used by Jesuit missionary Martino Martini in a popular atlas published in Amsterdam in 1655.[36] Although Peking is no longer the common name for the city, some of the city's older locations and facilities, such as Beijing Capital International Airport, with the IATA Code PEK, and Peking University, still retain the former romanization.

In 1403, during the Ming dynasty, Beijing got its current name.